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Effect of non-spherical atmospheric charged particles and atmospheric visibility on performance of satellite-ground quantum link and parameters simulation
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The Journal of China Universities of Posts and Telecommunications December 2017, 24(6): 39–48 www.sciencedirect.com/science/journal/10058885
http://jcupt.bupt.edu.cn
Effect of non-spherical atmospheric charged particles and atmospheric visibility on performance of satellite-ground quantum link and parameters simulation Shi Li1 (
), Nie Min1, Yang Guang1,2, Pei Changxing3
1. School of Communication and Information Engineering, Xi’an University of Posts and Telecommunication, Xi’an 710121, China 2. School of Electronics and Information, Northwestern Polytechnical University, Xi’an 710072, China 3. State Key Laboratory of Integrated Service Networks, Xidian University, Xi’an 710071, China
Abstract In order to study the relationship between the non-spherical atmospheric charged particles and satellite-ground quantum links attenuation. The relationship among the particle concentration, equivalent radius, charge density of the charged particle, the attenuation coefficient and entanglement of the satellite-ground quantum link can be established first according to the extinction cross section and spectral distribution function of the non-spherical atmospheric charged particles. The quantitative relationship between atmospheric visibility and communication fidelity of satellite-ground quantum link were analyzed then. Simulation results show that the ellipsoid, Chebyshev atmospheric charged particle influences on attenuation of the satellite-ground quantum link increase progressively. When the equivalent particle radius is 0.2 µm and the particle concentration is 50 µg/m3, the attenuation coefficient and entanglement of the satellite-ground quantum link is 9.21 dB/km, 11.46 dB/km and 0.453, 0.421 respectively; When the atmospheric visibility reduces from 8 km to 2 km, the communication fidelity of satellite-ground quantum link decreases from 0.52 to 0.08. It is shown that the non-spherical atmospheric charged particles and atmospheric visibility influence greatly on the performance of the satellite-ground quantum link communication system. Therefore, it is necessary to adjust the parameters of the quantum-satellite communication system according to the visibility values of the atmosphere and the shapes of the charged particles in the atmosphere to improve reliability of the satellite-ground quantum link. Keywords satellite-ground quantum link, non-spherical atmospheric charged particles, atmospheric visibility, degree of quantum entanglement
1 Introduction Quantum satellite communication has the advantages of wide coverage and so on. It is an important part of constructing global quantum communication network. It has become a hot research topic in this field. At 1:40 on August 16(th), 2016, the successful launching of ‘Mozi’ scientific experimental satellite, marks Chinese quantum satellite communications and space science research has
Received date: 04-08-2017 Corresponding author: Shi Li, E-mail: [email protected] DOI: 10.1016/S1005-8885(17)60240-1
taken an important step. In 2012, Pan Jianwei and his team completed the ‘free space entanglement photon distribution’ experiment [1], the first successful realization of hundreds of miles of free space quantum teleportation and entanglement distribution [2] for the launch of the world’s first ‘quantum communication satellite’ laid the technical foundation. Yearly, the team has successfully realized the lower limit measurement of quantum entanglement associated collapse velocity without localized vulnerability [3], which laid the necessary theoretical basis for large-scale theoretical basis test of quantum science experiment satellite. In 2009, Guo
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et al. built a multi-level ‘quantum government network’ in Wuhu city of Anhui province [4] through the network can be completed between any two points of unconditional security and confidential communications. The shape of the charged particles is very irregular. Under normal circumstances, the particles are regarded as spheres. The scattering properties are calculated by Mie scattering theory [5]. In fact, the atmospheric charged particles are not strictly spherical. The application of Mie scattering theory to the non-spherical atmospheric charged particles will cause a large error. The calculation of non-spherical aerosol scattering properties, generally uses T matrix method [6], discrete dipole approximation (DDA) [7], finite-volume time-domain (FVTD) [8] and other methods, which T matrix method is recognized as a more effective method. In this paper, the T matrix method is used to calculate the scattering properties of two kinds of non-spherical charged particles, named ellipsoid and Chebyshev. When the quantum satellite communicates with the ground station, atmospheric charged particles have the effect of scattering and absorption on the photon quantum signal. As the concentration of the charged particles increases, the extinction effect becomes more and more obvious. Accordingly, it affects the high fidelity transmission of the signal, and then influences the communication performance of the satellite-ground quantum link communication system seriously. The atmospheric visibility directly affects the physical properties of the atmospheric charged particles, and then influences the extinction effect and signal transmission. Thus, it is of great significance to study the effect of non-spherical charged particles and atmospheric visibility on the performance of satellite-ground quantum link communication. In 2008, Pei Changxing team of Xidian University studied the propagation characteristics of stratospheric quantum communication system [9]. Their results demonstrated that the ground air path propagation effect has an important impact on the quantum key distribution. The Ref. [10], studied the effect of PM2.5 air pollution on the free space quantum communication performance, and laid a theoretical foundation for the improvement of free space quantum communication quality under PM2.5 air pollution. The Ref. [11], studied the effect of ice water mixed cloud on the performance of quantum satellite communication, which provides a reference for the performance of the hybrid cloud
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environment. The Ref. [12], studied the influence of mesoscale dust storms on satellite communications and laid the foundation for the study of quantum satellite communication under dust conditions. Based on the T matrix theory, the Ref. [13], studied the effects of non-spherical aerosol particles and atmospheric relative humidity on the performance of free space quantum communication. Nevertheless, previous studies did not take into account the impact of particle charge on the performance of quantum channel communications, is to study the influence of Mie scattering on light quantum for spherical particles, either studied the influence of Mie scattering on light quantum for spherical particles, or to studied the influence of T matrix on the photon for non-spherical particles. In this paper, we studied two kinds of non-spherical atmospheric charged particles, named ellipsoid and Chebyshev. We investigated the relationship between the attenuation and the entanglement of the particles with different charge density and different particle concentration, analyzed the quantitative relation between the atmospheric visibility and the communication fidelity of satellite-ground quantum link. We aimed to provide a reference for the orderly operation of the satellite-ground quantum link communication system in different non-spherical atmospheric charged particles and different atmospheric visibility backgrounds.
2 Non-spherical atmospheric charged particle model We Consider a uniform, isotropic, non-magnetic and non-spherical particles [14] in the atmosphere or air environment. There is a certain atmospheric electric field, set the electric field in the vertical direction, showed Ee. Because of the collision between particles, the surface charge density can be represented as σ1. Taking ellipsoid and Chebyshev as two kinds of common non-spherical atmospheric charged particles, two kinds of particles and incident light quantum signal and particle orientation (elliptical particles as an example) as shown in Fig. 1:
(a) Ellipsoidal aerosol particles
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x = sin ϕ cos ϕ r + cos ϕ cos θ φ − sin θ θ
(2)
From the Maxwell equation and boundary conditions, we can see that the scattering coefficient pmn , qmn , and the incident wave coefficient amn , bmn , are linearly dependent [6]: nmax
(b) Chebyshev partieles
pmn = ∑
n′
∑
11 12 Tmnm ′n ′ am′n ′ + Tmnm ′n ′ bm′n′
(3)
n ′ =1 m ′ =− n ′
nmax
qmn = ∑
n′
∑
21 22 Tmnm ′n′ am′n′ + Tmnm′n′bm′n′
(4)
n′=1 m′=− n′
(c) Direction of incident optical signal and aerosol particles Fig. 1 Non-spherical atmospheric charged particle model and aerosol particles
In Fig. 1, a and b denote the length of the ellipsoid particles, c is the radius of the same surface area or volume sphere as the Chebyshev particle, φ is the zenith angle, θ is the azimuth angle, r, θ, φ is the base vector of coordinate system. Due to the difficulty of particle shape geometry and the computational complexity, the scattering properties of rotationally symmetric particles, ellipsoid, cylindrical and Chebyshev particles, mainly adopt the methods of computing T matrix. The T matrix is a transmission matrix between the incident field and the scattering of electromagnetic waves field, which is only related to the inherent refractive index, size and other characteristics of the particle, and the incident field and the scattering field has nothing to do. The vector spherical wave function of the atmospheric charged particles in field Ein has the following expansion formula [15]: 3ε E − σ 1 σR Ein = 0 1 x + 1 eff φ (1) ε1 + 2ε 0 ε0 where ε0 is the dielectric constant of medium background, ε1 is dielectric constant, E1 is incident wave field strength, Reff represents the equivalent spherical radius with the same surface area particles. This article takes 0.1 µm≤Reff ≤1 µm . And:
Written in matrix form is: 11 12 Tmnm pmn am′n′ Tmnm ′n ′ ′n′ am ′n ′ = T = (5) 21 22 qmn bm′n′ Tmnm′n′ Tmnm′n′ bm′n′ The extinction cross section [16] of the non-spherical atmospheric charged particles can be obtained by Eq. (5): nmax n 1 amn ( pmn )* + bmn ( qmn )* = Cex = 2 Re ∑ ∑ 2 −k E n =1 m =− n in
nmax n 2π 11 22 Tmnmn Re + Tmnmn ∑ ∑ −k 2 n =1 m =− n
(6)
wherein k is vacuum wave number, and can be expressed as: 2π k= (7)
λ
where λ is the wavelength of the incident quantum signal. Denote the ground state of the quantum system is 0 A , excited state as 1 A , the initial state of the non-spherical atmospheric charged particles environment as
1
N
,
Because of the collision and entanglement between the two, the qubit will transition from 1 N state to 0 A state with probability P, the environmental quantum state jumps into state 1
N
with probability P. This process can be
used to represent the unitary evolution of quantum and non-spherical atmospheric charged particles environments: 0 0 → 0 0 A N U AN : A N (8) 1 0 → 1 − P 1 0N+ P 0A 1N A N A Then the two Kraus operators of the super-operator Π in the environment base of non-spherical atmospheric charged particles can be calculated [17]: 0 1 M0 = 1− P 0 (9) 0 P M1 = 0 0 where P is the probability that the quantum state is
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erroneous due to the interaction with the non-spherical atmospheric charged particle environment. It is closely related to the extinction cross section Cex of the
The second order dispersion of natural light broadens the waveform of the pulse during transmission. The pulse broadening factor can be expressed as
non-spherical atmospheric charged particles. If the quantum system A initializes the density matrix as: ρ01 ρ ρA = 00 (10) ρ10 ρ11 It evolved into the following equation: ρA → ρA′ ≡ Π ( ρA ) = M 0 ρA M 0 + +M1 ρA M1+ =
τ 2 = 1 + ( L β 2 / T0 2 ) 2
ρ 00 + P ρ11 1 − P ρ01 ( 11 ) 1− Pρ (1 − P ) ρ11 10 In the satellite-ground quantum link, the source of the quantum satellite can be set as { pi , ρi } , where pi is the probability when the quantum character takes ρi , and
∑p
i
=1 .
If
take
ρ0 = 0 0 , ρ1 = 1 1
the input quantum character , after colliding and entanglement
with the non-spherical atmospheric charged particles environment, the original quantum state will evolve into: 0 p0 + Pp1 ρA → ρA′ ≡ Π ( ρA ) = (12) 0 (1 − P ) p1 It can be seen that the quantum state of satellite-ground quantum link will inevitably be affected by the physical properties of the non-spherical atmospheric charged particle.
3 Relationship between the non-spherical atmospheric charged particles and attenuation of the satellite-ground quantum link Natural light dispersion causes crosstalk of light quantum pulse in the transmission, lowers the quantum measurement efficiency of gating and the phase interference contrast. The first order dispersion of natural light affects arrival time of the two polarized components of the pulse. Propagation delay results in expansion of the pulse, i.e., polarization mode dispersion, which can be represented as
τ1 =
( L β1x − β1y ) 2 = DPMD L
(13)
Wherein L is transmission distance of light quantum, β1x , β1y is the first order dispersion of light field along the polarization direction of x and y respectively, DPMD is the average of the polarization dispersion parameters [18].
(14)
where β 2 is second order dispersion coefficient, T0 is initial pulse width. When the photon quantum signal is transmitted in a non-spherical atmospheric charged particle environment, the charged particles absorb the energy of the photons incident on them through the collision, resulting in attenuation of the energy of the photon quantum signal. Atmospheric charged particle spectrum characterizes the quantitative distribution of charged particles in different particle size intervals, and the distribution of the atmospheric charged particles can be fitted by the natural logarithm distribution [6]: − ( ln R − ln R )2 1 eff g n ( Reff ) = exp (15) 1 2 2 ln ∆ g (2π) 2 Reff ln ∆g where Rg is average radius, ∆g is standard deviation. When photons pass through atmospheric charged particles, consider the effect of natural light on light quantum propagation, the extinction coefficient of charged particles can be expressed as [19]: 3.25 N 0VCex (16) A= + τ1 + τ 2 Re where Re is effective radius and can be expressed as:
Re = ∫
1 0.1
2 πReff n( Reff )dReff
(17)
where V is the volume of non-spherical atmospheric charged particles, N 0 is the number of particles per unit volume. Fig. 2 shows that the transmission distance d in the atmospheric environment, the random distribution of the same dielectric properties, particle parameters of different non-spherical atmospheric charged particles. The initial state of the quantum state is I0, the incident plane is a plane of xz, and the boundary plane of the parallel plane boundary media is between z = 0 and z = d . As we can see, when the light quantum state passes through the scattering and absorption of atmospheric charged particles, the quantum state amplitude inevitably produces attenuation, and the degree of attenuation caused by the ellipsoid and Chebyshev atmospheric charged particles is different.
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Simulation results are shown in Fig. 3. Table 1
Simulation parameters
Variable
Values
Ee /(kV ⋅ m −1 )
20
Rg / µm
0.45
∆g / µm
2.41
σ 1 / (µC ⋅ m ) −1
d /km DRMD /(ps ⋅ km -1/2 )
0.2 10 0.5
β2
1
(a) Ellipsoidal atmosphere charged particles
(b) Chebyshev atmosphere charged particles Fig. 2 Transmission model of quantum signaling in non-spherical atmospheric charged particles
When the quantum state is transmitted on satelliteground quantum link, the amplitude decay of the quantum state [20] caused by atmospheric charged particles can be expressed as: I = I 0 e − Ad (18) where I0 is initial amplitude, I is amplitude of the quantum state which has propagated distance d. According to the attenuation factor’s definition, the attenuation coefficients of the quantum state of the atmospheric charged particles on the satellite-ground quantum link:
Aatt = 10 lg
I = 10 Ad lg e I0
(19)
According to the T matrix method can be accurately calculated within the range and combined with the measured data in the literature [21]. Using λ = 1.5 µm wavelength light quantum signal for the communication on the satellite-ground quantum link, the remaining parameters are shown in Table 1. The relationship among the equivalent radius, particle concentration of ellipsoid and Chebyshev atmospheric charged particles, and the attenuation coefficient of satellite-ground quantum link are simulated respectively.
Fig. 3 Relationship among the particle density, equivalent radius and satellite-ground link attenuation coefficient
In Fig. 3, Reff represents the equivalent radius of the ellipsoidal or Chebyshev atmospheric charged particles, N0 represents particle concentration of atmospheric charged particles, Aatt1, Aatt2 represents the satellite-ground quantum link attenuation coefficient under the influence of ellipsoid and Chebyshev atmosphere charged particles, respectively. From the figure, we can see that when the particle concentration and the equivalent radius are 0, the attenuation coefficient of the satellite-ground link is 0 approximately, which is consistent with the ideal state. With the increment of particle concentration and equivalent radius, the trend of the attenuation coefficient of the satellite-ground quantum link remains the same. When the concentration of atmospheric charged particles remains constant, the equivalent radius of the ellipsoidal atmospheric charged particles can be expressed as follows:
ab b =a (20) 2 a As can be seen from the equation mentioned above, keeping the long axis a of the ellipsoidal atmospheric charged remain unchanged, with the increasing of orientation ratio b/a, the probability and number of photons passing through atmospheric charged particles increases, extinction effect is more obvious. Decoherence
Reff1 =
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leads to the loss of quantum information among in quantum states, attenuation coefficient of the satellite-ground quantum link also increases. The equivalent radius of the Chebyshev atmospheric charged particles Reff2 is equal to the radius c. With increase of the radius of the same surface area or volume sphere c, attenuation coefficient of the satellite-ground quantum link also increases. But the trend of c is always faster than b/a, thus, the change trend of attenuation coefficient Aatt2 is faster than Aatt1 on satellite-ground quantum link. Correspondingly, the attenuation value Aatt2 of the satellite-ground quantum link is larger under the same conditions. From the simulation results as shown in Fig. 3, we can find that when the equivalent particle radius is 0.2 µm and the particle concentration is 50 µg/m3, under the influence of ellipsoid and Chebyshev atmospheric charged particles, the attenuation coefficient of the satellite-ground quantum link is 9.21 dB/km and, 11.46 dB/km, respectively.
4 Relationship between the non-spherical atmospheric charged particles and the entanglement of the satellite-ground quantum link Winds are dynamic transmission conditions of atmospheric charged particle. In this section, we put non-spherical atmospheric charged dust particles contained in the turbulence of turbid atmosphere as an example, and investigate its relationship with entanglement of quantum channel. By Ref. [22], entanglement of quantum channel can be expressed as follows: E = S ( ρA1 ) = S ( ρB1 ) = − tr( ρA1lb ρA1 ) (21) where A1 represents the non-spherical atmospheric charged dust particles, the B1 represents the satellite-ground quantum link, S(ρA1) and S(ρA2) represent the quantum entropy of A1 and B1, respectively, ρA1 and ρA2 represent the density matrix of A1 and B1. The initial state of the interaction between the non-spherical charged dust particle environment and the quantum state of the satellite-ground quantum link is: ψ = e1 ϕ1 = ∑ Pi n e1 (22) i
where n
is a complete set of Hilbert space, e1
is the
initial state of the non- spherical charged dust particle environment, is quantum state of the ϕ1 satellite-ground quantum link,
Pi
is the nonzero
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eigenvalues common to the reduced density matrices of subsystems A1 and B1. With the time evolution, non-spherical charged dust particle environment and system entanglement eventually evolved into: ψ = ∑ Pi n es (23) i
where es
is the final state of the interaction between the
quantum state of the satellite quantum channel and the non-spherical charged dust particle environment. As can be seen from Eq. (23), with charged dust environment, the entanglement of quantum channel of the satellite-ground quantum link is related to the physical properties of non-spherical charged dust particle. Quantum channel entanglement in charged dust environment can be defined as follows [12]: wt E= (24) (0.3) f σ 1 exp 0.01(1 − Reff ) −1 ( h − h0 ) where w is charged dust starting wind speed, t is the transmission diffusion time of charged dust, σ1 is charge density on the surface of non-spherical charged dust particles, Reff is the equivalent radius of spherical charged dust particles or non-spherical charged dust particles with the same surface area, h is the height of charged dust diffusion, h0 is satellite ground station height, f is coefficient of charged dust disaster. Parameter values are shown in Table 2. Table 2 Entanglement parameters of satellite-ground quantum channel Variable
Values
w / (m ⋅ s −1 ) a/µm σ 1 / (µC ⋅ m −1 ) h/km h0 / km f
4.0 × 10−2 0.1 0.2 10 1 1
Fig. 4 demonstrates relationship among the equivalent radius of the ellipsoid and Chebyshev atmospheric charged particles, the transmission diffusion time, and the entanglement of the quantum channel. In Fig. 4, Reff represents the equivalent radius of the ellipsoidal or Chebyshev atmospheric charged particles, ranging from 0 to 1 µm; t represents the transmission diffusion time, ranging from 0 to 20 h; E1, E2 represents the entanglement of quantum channel under the influence of ellipsoid and Chebyshev atmosphere charged particles, respectively, ranging from 0 to 1. From the figure, it is shown that the same point for the two surface plots are: when
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the equivalent radius of the atmospheric charged particles remains constant with the increasing of diffusion time, quantum entanglement shows an increasing trend. This is because the floating density of charged dust particles is getting smaller and smaller with time. However, the growing trend of the two graphs is clearly different. When transmission diffusion time t=10.2 h, equivalent radius of the atmospheric charged particles is 0.21 µm and 0.84 µm respectively, the entanglement of quantum channel is 0.454 and 0.427 under the influence of ellipsoidal charged particles, and the entanglement of quantum channel is 0.426 and 0.270 under the influence of Chebyshev atmosphere charged particles. It can be seen that the effect of Chebyshev atmospheric charged particles on the entanglement of the satellite-ground quantum link is more significant than that of the ellipsoidal atmospheric charged particles.
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diffusion time, ranging from 0 to 24 h; E1, E2 represents the entanglement of quantum channel under the influence of ellipsoid and Chebyshev atmosphere charged particles, respectively, ranging from 0 to 1. It is easy to see, when the transmission diffusion time remains constant with the increment of charge density of the atmospheric charged particles, i.e., the probability of missing photons increases, quantum entanglement presents an attenuation trend. But the growing trend of the two surface plots is clearly different. When transmission diffusion time t=10.2 h, charge density of the atmospheric charged particles is 0.21 µC/m and 0.42 µC/m, entanglement of quantum channel is 0.413, 0.407 under the influence of ellipsoidal charged particles, and the entanglement of quantum channel is 0.361, 0.356 under the influence of Chebyshev atmosphere charged particles. It is found that the impact of chargeability of Chebyshev atmospheric charged particles is more significant than that of ellipsoidal atmospheric charged particles on the entanglement of the quantum channel.
5 Relationship between the atmospheric visibility and communication fidelity of satellite-ground quantum link
Fig. 4 Relationship among the effective radius, diffusion time and quantum entanglement
Fig. 5 shows the comparison among the charge density of the ellipsoid and Chebyshev atmospheric charged particles, the transmission diffusion time, and the entanglement of the quantum channel.
In order to use the quantum satellite communication platform and ground station to accomplish satellite-ground quantum link. When the photon quantum signal is transmitted to the stratosphere, communicates with the ground can be achieved by means of a communication platform suspended in the stratosphere. Under these circumstances we can consider the complex atmospheric environment on the impact of light quantum signal. Communication mode is shown in Fig. 6.
Fig. 5 Relationship among the charge density, diffusion time and quantum entanglement Fig. 6
In Fig. 5, σ1 represents the charge density of the ellipsoidal or Chebyshev atmospheric charged particles, ranging from 0 to 0.8 µC/m ; t represents the transmission
A typical optical system of quantum communication system
In Fig. 6, L is the transmission distance between the ground station and the light quantum signal platform, and
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can be expressed as: 2
2
L = R sin α + H + 2 RH − R sin α (25) where R is earth curvature radius, H is the vertical distance between the ground station and the optical quantum signal platform, α is transmission or acceptance dip angle. When the photon quantum state Q changes to Q ′ after arriving at the stratospheric communication platform, it can be calculated as [23]: Q ′ = Q e( − iωη / c − ( χ )/ 2) L + φ (26) where ω is light wave angular frequency, η is photoelectric detection quantum efficiency, χ is system attenuation coefficient, φ
represents a state that has
been scattered outside the Q substrate. The state of the ground station is finally detected: Q ′′ = Q ′ − φ
(27)
From Eq. (25), the communication fidelity between satellite-ground quantum link can be expressed as [24–25]:
F = tr where
Q ρ Q
Q ρ Q
entangled states
= e
2 L / [ ( ln 0 .0 2 ) V
]
(28)
is the inner product of multiphoton
Q
and ρ Q , V is the atmospheric
visibility. Fig. 7 shows the comparison among the atmospheric visibility, transmission distance, and the communication fidelity of satellite-ground quantum link.
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through atmospheric charged particles is decreasing, the extinction effect becomes smaller and smaller, decoherence leads to less quantum information in quantum states, the communication fidelity of satellite-ground quantum link traffic is increasing. And when the atmospheric visibility V is known with the increment of transmission distance, the extinction effect of atmospheric charged particles becomes more and more obviously. Decoherence causes the quantum information in quantum states to loss, the larger the attenuation caused by energy which leads to the decline of communication fidelity of satellite-ground quantum link. Simulation results demonstrate that when transmission distance L= 20 km, atmospheric visibility is 2 km and 8 km, the communication fidelity of satellite-ground quantum link is 0.08, 0.52, respectively. Thus, in order to ensure the quality of communication, fidelity and transmission distance need to be compromised.
6 Data analysis and discussions In this section, we will use the data given in this paper to do comparative analysis with research results work published previously. For comparison, the parameters are in range where the T matrix method can be accurately calculated and combined with the selected data in the literature. We Select the ice water mixed clouds [11] and non-spherical aerosol particles [13], and compare with the data in this paper. The equivalent radius of the different particles under this spectral distribution is 0.3 µm, and the light quantum wavelength is 1.5 µm. The attenuation coefficients of the quantum channel under the influence of different particles are shown in Table 3. Table 3 Comparison of the effects of different particles on quantum channels Particle shape
Fig. 7 Relationship among the atmospheric visibility, transmission distance and quantum fidelity
In Fig. 7, V represents the atmospheric visibility, ranging from 0 to 10 km; L represents the transmission distance, ranging from 0 to 40 km; F represents the communication fidelity of satellite-ground quantum link, ranging from 0 to 1. From the figure, we can see that when the transmission distance L is known with the increment of atmospheric visibility V, the probability and number of photons passing
Spherical particles(a/b=1) ellipsoid particles(a/b=1/2) Chebyshev particles Charged ellipsoidal particles(a/b=1/2) Charged Chebyshev particles
Channel attenuation coefficient 4.61 6.82 7.25
Channel entanglement 0.31 0.26 0.24
9.29
0.18
11.55
0.15
As can be seen in Table 3, the quantum channel attenuation coefficient of the ellipsoidal particles increase by 48%, and the quantum channel attenuation coefficient of the Chebyshev particles increase by 57% compared to the spherical particles. This indicates that the effect of the particle shape on the quantum channel is more significant. Compare with the charged ellipsoid particles and the
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non-charged ellipsoidal particles, the attenuation coefficient and entanglement of the quantum channel are 36% and 30%, respectively. Compare with the charged Chebyshev particles and the non-charged Chebyshev particles, the attenuation coefficient and entanglement of the quantum channel are 59% and 38%, respectively. This indicates that the particles have a greater impact on the quantum channel after charged. In this paper, the factors of particle chargeability are taken into account in analyzing the influence of environment on quantum channel for the first time. We find that the non-spherical charged particles destroy the quantum channel most seriously compared with the previous research results. In order to further study the non-spherical aerosol particles in the charged environment on the satellite –ground quantum link, free space quantum communication provides a theoretical basis. Nevertheless, we do not consider the effect of such factors on the capacity of quantum channels, and also lacks relevant experimental and practical applications. This is the weakness of this paper.
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quantum link also presented approximately logarithmic attenuation. Compared with previous research results, it is proved that the non-spherical atmospheric charged particles have the most serious damage to the communication performance of the satellite-ground quantum link. Thus, the quantitative relationship between the parameters proposed in this paper can provide reference for quantum satellite communication. It makes the quantum satellite communication system adjust the communication parameters adaptively, and reduces the impact of non-spherical atmospheric charged particles and atmospheric visibility on the performance of quantum satellite communication finally. Acknowledgements This work was supported by the National Natural Science Foundation of China (61172071, 61201194), the International Scientific and Technological Cooperation and Exchange Program in Shaanxi Province, China (2015KW-013) and the Scientific Research Program Funded by Shaanxi Provincial Education Department, China (16JK1711).
7 Conclusions References In this paper, we studied the effect of non-spherical atmospheric charged particles and atmospheric visibility on the performance of the satellite-ground quantum link. According to the extinction cross section and spectral distribution function of the non-spherical atmospheric charged particles, the relationship among the particle concentration, equivalent radius, charge density of the charged particle and the attenuation coefficient of the satellite-ground quantum link can be established. This paper gave the relationship among the equivalent radius of ellipsoidal and Chebyshev atmospheric charged particles, charge density, transmission diffusion time, and the entanglement of satellite-ground quantum link. We also analyzed the quantitative relationship between atmospheric visibility and quantum fidelity. Simulation results showed that with the increment of the equivalent radius and charge density of the charged particles, the attenuation coefficient and the entanglement of the satellite-ground quantum link is correspondingly changed in attenuation. However, the effect of the ellipsoid and Chebyshev atmosphere on the performance of the satellite-ground quantum link increases progressively. With the increment of atmospheric visibility, the communication fidelity of satellite-ground
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the performance of quantum. Acta Photonica Sinica, 2016, 45(9): 0927004 (in Chinese) Nie M, Shang P G, Yang G, et al. Influences of mesoscale sandstorm on the quantum satellite communication channel and performance simulation. Acta Physica Sinica, 2014, 63(24): 240303 (in Chinese) Nie M, Ren J M, Yang G, et al. Influences of non-spherical aerosol particles and relative humidity of atmosphere on the performance of free space quantum communication. Acta Physica Sinica, 2016, 65(19):190301 (in Chinese) Hu S, Gao T C, Liu L. Analysis on scattering characeristics and equivalent Mie scattering errors of non-spherical atmospherical aerosols. Journal of the Meteorological Sciences, 2014, 34(6): 612−619 (in Chinese） Min X, Li X C, Li X W, et al. Effects of charged sandstorm on electromagnetic wave propagation. Acta Optica Sinica, 2015, 35(1): 413−420 (in Chinese） Mishchenko M I, Travis L D. Capabilities and limitations of a current Fortran implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers. Journal of Quantitative Spectroscopy and Radiative Transfer, 1998, 60(3): 309−324 Nie M, Tang S R, Yang G, et al. Influence of the ionospheric sporadic E layer on the performance of quantum satellite communication in the mid
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latitude region. Acta Physica Sinica, 2017, 66(7): 259−269 (in Chinese) 18. Keiser G. Optical fiber communications. 4th ed. New York, NY, USA: McGraw-Hill, 2011: 95−98 19. Zhou Y H, He Q S, Zheng X J. Attenuation of electromagnetic wave propagation in sandstorms incorporating charged sand particles. European Physical Journal E, 2005, 17: 181−187 20. Shao C C, Ma J J. Based on T-matrix method studied scattering characteristics of non-spherical particles. Journal of Atomic and Molecular Sciences, 2010, 27(3): 475−479 (in Chinese) 21. Kaegi R. Chemical and morphological analysis of airborne particles at a tunnel construction site. Journal of Aerosol Science, 2004, 35(5): 621−632 22. Zhang Y D. Quantum mechanics. Beijing, China: Science Press, 2010: 343 (in Chinese) 23. Marco L. Quantum radar. San Rafael, CA, USA: Morgan & Claypool Publishers, 2011: 15−17 24. Ren J, Nie M, Yang G, et al. Influences of multiple factors of natural environment on the performance of free space quantum communication. Acta Photonica Sinica, 2015, 44(12): 1227003 (in Chinese) 25. Yang G, Lian B W, Nie M. Decoherence suppression for three-qubit W-like state using weak measurement and iteration method. Chinese Physics B, 2016, 25(8): 080310
(Editor: Liu Yin)
The Journal of China Universities of Posts and Telecommunications December 2017, 24(6): 39–48 www.sciencedirect.com/science/journal/10058885
http://jcupt.bupt.edu.cn
Effect of non-spherical atmospheric charged particles and atmospheric visibility on performance of satellite-ground quantum link and parameters simulation Shi Li1 (
), Nie Min1, Yang Guang1,2, Pei Changxing3
1. School of Communication and Information Engineering, Xi’an University of Posts and Telecommunication, Xi’an 710121, China 2. School of Electronics and Information, Northwestern Polytechnical University, Xi’an 710072, China 3. State Key Laboratory of Integrated Service Networks, Xidian University, Xi’an 710071, China
Abstract In order to study the relationship between the non-spherical atmospheric charged particles and satellite-ground quantum links attenuation. The relationship among the particle concentration, equivalent radius, charge density of the charged particle, the attenuation coefficient and entanglement of the satellite-ground quantum link can be established first according to the extinction cross section and spectral distribution function of the non-spherical atmospheric charged particles. The quantitative relationship between atmospheric visibility and communication fidelity of satellite-ground quantum link were analyzed then. Simulation results show that the ellipsoid, Chebyshev atmospheric charged particle influences on attenuation of the satellite-ground quantum link increase progressively. When the equivalent particle radius is 0.2 µm and the particle concentration is 50 µg/m3, the attenuation coefficient and entanglement of the satellite-ground quantum link is 9.21 dB/km, 11.46 dB/km and 0.453, 0.421 respectively; When the atmospheric visibility reduces from 8 km to 2 km, the communication fidelity of satellite-ground quantum link decreases from 0.52 to 0.08. It is shown that the non-spherical atmospheric charged particles and atmospheric visibility influence greatly on the performance of the satellite-ground quantum link communication system. Therefore, it is necessary to adjust the parameters of the quantum-satellite communication system according to the visibility values of the atmosphere and the shapes of the charged particles in the atmosphere to improve reliability of the satellite-ground quantum link. Keywords satellite-ground quantum link, non-spherical atmospheric charged particles, atmospheric visibility, degree of quantum entanglement
1 Introduction Quantum satellite communication has the advantages of wide coverage and so on. It is an important part of constructing global quantum communication network. It has become a hot research topic in this field. At 1:40 on August 16(th), 2016, the successful launching of ‘Mozi’ scientific experimental satellite, marks Chinese quantum satellite communications and space science research has
Received date: 04-08-2017 Corresponding author: Shi Li, E-mail: [email protected] DOI: 10.1016/S1005-8885(17)60240-1
taken an important step. In 2012, Pan Jianwei and his team completed the ‘free space entanglement photon distribution’ experiment [1], the first successful realization of hundreds of miles of free space quantum teleportation and entanglement distribution [2] for the launch of the world’s first ‘quantum communication satellite’ laid the technical foundation. Yearly, the team has successfully realized the lower limit measurement of quantum entanglement associated collapse velocity without localized vulnerability [3], which laid the necessary theoretical basis for large-scale theoretical basis test of quantum science experiment satellite. In 2009, Guo
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et al. built a multi-level ‘quantum government network’ in Wuhu city of Anhui province [4] through the network can be completed between any two points of unconditional security and confidential communications. The shape of the charged particles is very irregular. Under normal circumstances, the particles are regarded as spheres. The scattering properties are calculated by Mie scattering theory [5]. In fact, the atmospheric charged particles are not strictly spherical. The application of Mie scattering theory to the non-spherical atmospheric charged particles will cause a large error. The calculation of non-spherical aerosol scattering properties, generally uses T matrix method [6], discrete dipole approximation (DDA) [7], finite-volume time-domain (FVTD) [8] and other methods, which T matrix method is recognized as a more effective method. In this paper, the T matrix method is used to calculate the scattering properties of two kinds of non-spherical charged particles, named ellipsoid and Chebyshev. When the quantum satellite communicates with the ground station, atmospheric charged particles have the effect of scattering and absorption on the photon quantum signal. As the concentration of the charged particles increases, the extinction effect becomes more and more obvious. Accordingly, it affects the high fidelity transmission of the signal, and then influences the communication performance of the satellite-ground quantum link communication system seriously. The atmospheric visibility directly affects the physical properties of the atmospheric charged particles, and then influences the extinction effect and signal transmission. Thus, it is of great significance to study the effect of non-spherical charged particles and atmospheric visibility on the performance of satellite-ground quantum link communication. In 2008, Pei Changxing team of Xidian University studied the propagation characteristics of stratospheric quantum communication system [9]. Their results demonstrated that the ground air path propagation effect has an important impact on the quantum key distribution. The Ref. [10], studied the effect of PM2.5 air pollution on the free space quantum communication performance, and laid a theoretical foundation for the improvement of free space quantum communication quality under PM2.5 air pollution. The Ref. [11], studied the effect of ice water mixed cloud on the performance of quantum satellite communication, which provides a reference for the performance of the hybrid cloud
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environment. The Ref. [12], studied the influence of mesoscale dust storms on satellite communications and laid the foundation for the study of quantum satellite communication under dust conditions. Based on the T matrix theory, the Ref. [13], studied the effects of non-spherical aerosol particles and atmospheric relative humidity on the performance of free space quantum communication. Nevertheless, previous studies did not take into account the impact of particle charge on the performance of quantum channel communications, is to study the influence of Mie scattering on light quantum for spherical particles, either studied the influence of Mie scattering on light quantum for spherical particles, or to studied the influence of T matrix on the photon for non-spherical particles. In this paper, we studied two kinds of non-spherical atmospheric charged particles, named ellipsoid and Chebyshev. We investigated the relationship between the attenuation and the entanglement of the particles with different charge density and different particle concentration, analyzed the quantitative relation between the atmospheric visibility and the communication fidelity of satellite-ground quantum link. We aimed to provide a reference for the orderly operation of the satellite-ground quantum link communication system in different non-spherical atmospheric charged particles and different atmospheric visibility backgrounds.
2 Non-spherical atmospheric charged particle model We Consider a uniform, isotropic, non-magnetic and non-spherical particles [14] in the atmosphere or air environment. There is a certain atmospheric electric field, set the electric field in the vertical direction, showed Ee. Because of the collision between particles, the surface charge density can be represented as σ1. Taking ellipsoid and Chebyshev as two kinds of common non-spherical atmospheric charged particles, two kinds of particles and incident light quantum signal and particle orientation (elliptical particles as an example) as shown in Fig. 1:
(a) Ellipsoidal aerosol particles
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x = sin ϕ cos ϕ r + cos ϕ cos θ φ − sin θ θ
(2)
From the Maxwell equation and boundary conditions, we can see that the scattering coefficient pmn , qmn , and the incident wave coefficient amn , bmn , are linearly dependent [6]: nmax
(b) Chebyshev partieles
pmn = ∑
n′
∑
11 12 Tmnm ′n ′ am′n ′ + Tmnm ′n ′ bm′n′
(3)
n ′ =1 m ′ =− n ′
nmax
qmn = ∑
n′
∑
21 22 Tmnm ′n′ am′n′ + Tmnm′n′bm′n′
(4)
n′=1 m′=− n′
(c) Direction of incident optical signal and aerosol particles Fig. 1 Non-spherical atmospheric charged particle model and aerosol particles
In Fig. 1, a and b denote the length of the ellipsoid particles, c is the radius of the same surface area or volume sphere as the Chebyshev particle, φ is the zenith angle, θ is the azimuth angle, r, θ, φ is the base vector of coordinate system. Due to the difficulty of particle shape geometry and the computational complexity, the scattering properties of rotationally symmetric particles, ellipsoid, cylindrical and Chebyshev particles, mainly adopt the methods of computing T matrix. The T matrix is a transmission matrix between the incident field and the scattering of electromagnetic waves field, which is only related to the inherent refractive index, size and other characteristics of the particle, and the incident field and the scattering field has nothing to do. The vector spherical wave function of the atmospheric charged particles in field Ein has the following expansion formula [15]: 3ε E − σ 1 σR Ein = 0 1 x + 1 eff φ (1) ε1 + 2ε 0 ε0 where ε0 is the dielectric constant of medium background, ε1 is dielectric constant, E1 is incident wave field strength, Reff represents the equivalent spherical radius with the same surface area particles. This article takes 0.1 µm≤Reff ≤1 µm . And:
Written in matrix form is: 11 12 Tmnm pmn am′n′ Tmnm ′n ′ ′n′ am ′n ′ = T = (5) 21 22 qmn bm′n′ Tmnm′n′ Tmnm′n′ bm′n′ The extinction cross section [16] of the non-spherical atmospheric charged particles can be obtained by Eq. (5): nmax n 1 amn ( pmn )* + bmn ( qmn )* = Cex = 2 Re ∑ ∑ 2 −k E n =1 m =− n in
nmax n 2π 11 22 Tmnmn Re + Tmnmn ∑ ∑ −k 2 n =1 m =− n
(6)
wherein k is vacuum wave number, and can be expressed as: 2π k= (7)
λ
where λ is the wavelength of the incident quantum signal. Denote the ground state of the quantum system is 0 A , excited state as 1 A , the initial state of the non-spherical atmospheric charged particles environment as
1
N
,
Because of the collision and entanglement between the two, the qubit will transition from 1 N state to 0 A state with probability P, the environmental quantum state jumps into state 1
N
with probability P. This process can be
used to represent the unitary evolution of quantum and non-spherical atmospheric charged particles environments: 0 0 → 0 0 A N U AN : A N (8) 1 0 → 1 − P 1 0N+ P 0A 1N A N A Then the two Kraus operators of the super-operator Π in the environment base of non-spherical atmospheric charged particles can be calculated [17]: 0 1 M0 = 1− P 0 (9) 0 P M1 = 0 0 where P is the probability that the quantum state is
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erroneous due to the interaction with the non-spherical atmospheric charged particle environment. It is closely related to the extinction cross section Cex of the
The second order dispersion of natural light broadens the waveform of the pulse during transmission. The pulse broadening factor can be expressed as
non-spherical atmospheric charged particles. If the quantum system A initializes the density matrix as: ρ01 ρ ρA = 00 (10) ρ10 ρ11 It evolved into the following equation: ρA → ρA′ ≡ Π ( ρA ) = M 0 ρA M 0 + +M1 ρA M1+ =
τ 2 = 1 + ( L β 2 / T0 2 ) 2
ρ 00 + P ρ11 1 − P ρ01 ( 11 ) 1− Pρ (1 − P ) ρ11 10 In the satellite-ground quantum link, the source of the quantum satellite can be set as { pi , ρi } , where pi is the probability when the quantum character takes ρi , and
∑p
i
=1 .
If
take
ρ0 = 0 0 , ρ1 = 1 1
the input quantum character , after colliding and entanglement
with the non-spherical atmospheric charged particles environment, the original quantum state will evolve into: 0 p0 + Pp1 ρA → ρA′ ≡ Π ( ρA ) = (12) 0 (1 − P ) p1 It can be seen that the quantum state of satellite-ground quantum link will inevitably be affected by the physical properties of the non-spherical atmospheric charged particle.
3 Relationship between the non-spherical atmospheric charged particles and attenuation of the satellite-ground quantum link Natural light dispersion causes crosstalk of light quantum pulse in the transmission, lowers the quantum measurement efficiency of gating and the phase interference contrast. The first order dispersion of natural light affects arrival time of the two polarized components of the pulse. Propagation delay results in expansion of the pulse, i.e., polarization mode dispersion, which can be represented as
τ1 =
( L β1x − β1y ) 2 = DPMD L
(13)
Wherein L is transmission distance of light quantum, β1x , β1y is the first order dispersion of light field along the polarization direction of x and y respectively, DPMD is the average of the polarization dispersion parameters [18].
(14)
where β 2 is second order dispersion coefficient, T0 is initial pulse width. When the photon quantum signal is transmitted in a non-spherical atmospheric charged particle environment, the charged particles absorb the energy of the photons incident on them through the collision, resulting in attenuation of the energy of the photon quantum signal. Atmospheric charged particle spectrum characterizes the quantitative distribution of charged particles in different particle size intervals, and the distribution of the atmospheric charged particles can be fitted by the natural logarithm distribution [6]: − ( ln R − ln R )2 1 eff g n ( Reff ) = exp (15) 1 2 2 ln ∆ g (2π) 2 Reff ln ∆g where Rg is average radius, ∆g is standard deviation. When photons pass through atmospheric charged particles, consider the effect of natural light on light quantum propagation, the extinction coefficient of charged particles can be expressed as [19]: 3.25 N 0VCex (16) A= + τ1 + τ 2 Re where Re is effective radius and can be expressed as:
Re = ∫
1 0.1
2 πReff n( Reff )dReff
(17)
where V is the volume of non-spherical atmospheric charged particles, N 0 is the number of particles per unit volume. Fig. 2 shows that the transmission distance d in the atmospheric environment, the random distribution of the same dielectric properties, particle parameters of different non-spherical atmospheric charged particles. The initial state of the quantum state is I0, the incident plane is a plane of xz, and the boundary plane of the parallel plane boundary media is between z = 0 and z = d . As we can see, when the light quantum state passes through the scattering and absorption of atmospheric charged particles, the quantum state amplitude inevitably produces attenuation, and the degree of attenuation caused by the ellipsoid and Chebyshev atmospheric charged particles is different.
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Simulation results are shown in Fig. 3. Table 1
Simulation parameters
Variable
Values
Ee /(kV ⋅ m −1 )
20
Rg / µm
0.45
∆g / µm
2.41
σ 1 / (µC ⋅ m ) −1
d /km DRMD /(ps ⋅ km -1/2 )
0.2 10 0.5
β2
1
(a) Ellipsoidal atmosphere charged particles
(b) Chebyshev atmosphere charged particles Fig. 2 Transmission model of quantum signaling in non-spherical atmospheric charged particles
When the quantum state is transmitted on satelliteground quantum link, the amplitude decay of the quantum state [20] caused by atmospheric charged particles can be expressed as: I = I 0 e − Ad (18) where I0 is initial amplitude, I is amplitude of the quantum state which has propagated distance d. According to the attenuation factor’s definition, the attenuation coefficients of the quantum state of the atmospheric charged particles on the satellite-ground quantum link:
Aatt = 10 lg
I = 10 Ad lg e I0
(19)
According to the T matrix method can be accurately calculated within the range and combined with the measured data in the literature [21]. Using λ = 1.5 µm wavelength light quantum signal for the communication on the satellite-ground quantum link, the remaining parameters are shown in Table 1. The relationship among the equivalent radius, particle concentration of ellipsoid and Chebyshev atmospheric charged particles, and the attenuation coefficient of satellite-ground quantum link are simulated respectively.
Fig. 3 Relationship among the particle density, equivalent radius and satellite-ground link attenuation coefficient
In Fig. 3, Reff represents the equivalent radius of the ellipsoidal or Chebyshev atmospheric charged particles, N0 represents particle concentration of atmospheric charged particles, Aatt1, Aatt2 represents the satellite-ground quantum link attenuation coefficient under the influence of ellipsoid and Chebyshev atmosphere charged particles, respectively. From the figure, we can see that when the particle concentration and the equivalent radius are 0, the attenuation coefficient of the satellite-ground link is 0 approximately, which is consistent with the ideal state. With the increment of particle concentration and equivalent radius, the trend of the attenuation coefficient of the satellite-ground quantum link remains the same. When the concentration of atmospheric charged particles remains constant, the equivalent radius of the ellipsoidal atmospheric charged particles can be expressed as follows:
ab b =a (20) 2 a As can be seen from the equation mentioned above, keeping the long axis a of the ellipsoidal atmospheric charged remain unchanged, with the increasing of orientation ratio b/a, the probability and number of photons passing through atmospheric charged particles increases, extinction effect is more obvious. Decoherence
Reff1 =
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leads to the loss of quantum information among in quantum states, attenuation coefficient of the satellite-ground quantum link also increases. The equivalent radius of the Chebyshev atmospheric charged particles Reff2 is equal to the radius c. With increase of the radius of the same surface area or volume sphere c, attenuation coefficient of the satellite-ground quantum link also increases. But the trend of c is always faster than b/a, thus, the change trend of attenuation coefficient Aatt2 is faster than Aatt1 on satellite-ground quantum link. Correspondingly, the attenuation value Aatt2 of the satellite-ground quantum link is larger under the same conditions. From the simulation results as shown in Fig. 3, we can find that when the equivalent particle radius is 0.2 µm and the particle concentration is 50 µg/m3, under the influence of ellipsoid and Chebyshev atmospheric charged particles, the attenuation coefficient of the satellite-ground quantum link is 9.21 dB/km and, 11.46 dB/km, respectively.
4 Relationship between the non-spherical atmospheric charged particles and the entanglement of the satellite-ground quantum link Winds are dynamic transmission conditions of atmospheric charged particle. In this section, we put non-spherical atmospheric charged dust particles contained in the turbulence of turbid atmosphere as an example, and investigate its relationship with entanglement of quantum channel. By Ref. [22], entanglement of quantum channel can be expressed as follows: E = S ( ρA1 ) = S ( ρB1 ) = − tr( ρA1lb ρA1 ) (21) where A1 represents the non-spherical atmospheric charged dust particles, the B1 represents the satellite-ground quantum link, S(ρA1) and S(ρA2) represent the quantum entropy of A1 and B1, respectively, ρA1 and ρA2 represent the density matrix of A1 and B1. The initial state of the interaction between the non-spherical charged dust particle environment and the quantum state of the satellite-ground quantum link is: ψ = e1 ϕ1 = ∑ Pi n e1 (22) i
where n
is a complete set of Hilbert space, e1
is the
initial state of the non- spherical charged dust particle environment, is quantum state of the ϕ1 satellite-ground quantum link,
Pi
is the nonzero
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eigenvalues common to the reduced density matrices of subsystems A1 and B1. With the time evolution, non-spherical charged dust particle environment and system entanglement eventually evolved into: ψ = ∑ Pi n es (23) i
where es
is the final state of the interaction between the
quantum state of the satellite quantum channel and the non-spherical charged dust particle environment. As can be seen from Eq. (23), with charged dust environment, the entanglement of quantum channel of the satellite-ground quantum link is related to the physical properties of non-spherical charged dust particle. Quantum channel entanglement in charged dust environment can be defined as follows [12]: wt E= (24) (0.3) f σ 1 exp 0.01(1 − Reff ) −1 ( h − h0 ) where w is charged dust starting wind speed, t is the transmission diffusion time of charged dust, σ1 is charge density on the surface of non-spherical charged dust particles, Reff is the equivalent radius of spherical charged dust particles or non-spherical charged dust particles with the same surface area, h is the height of charged dust diffusion, h0 is satellite ground station height, f is coefficient of charged dust disaster. Parameter values are shown in Table 2. Table 2 Entanglement parameters of satellite-ground quantum channel Variable
Values
w / (m ⋅ s −1 ) a/µm σ 1 / (µC ⋅ m −1 ) h/km h0 / km f
4.0 × 10−2 0.1 0.2 10 1 1
Fig. 4 demonstrates relationship among the equivalent radius of the ellipsoid and Chebyshev atmospheric charged particles, the transmission diffusion time, and the entanglement of the quantum channel. In Fig. 4, Reff represents the equivalent radius of the ellipsoidal or Chebyshev atmospheric charged particles, ranging from 0 to 1 µm; t represents the transmission diffusion time, ranging from 0 to 20 h; E1, E2 represents the entanglement of quantum channel under the influence of ellipsoid and Chebyshev atmosphere charged particles, respectively, ranging from 0 to 1. From the figure, it is shown that the same point for the two surface plots are: when
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the equivalent radius of the atmospheric charged particles remains constant with the increasing of diffusion time, quantum entanglement shows an increasing trend. This is because the floating density of charged dust particles is getting smaller and smaller with time. However, the growing trend of the two graphs is clearly different. When transmission diffusion time t=10.2 h, equivalent radius of the atmospheric charged particles is 0.21 µm and 0.84 µm respectively, the entanglement of quantum channel is 0.454 and 0.427 under the influence of ellipsoidal charged particles, and the entanglement of quantum channel is 0.426 and 0.270 under the influence of Chebyshev atmosphere charged particles. It can be seen that the effect of Chebyshev atmospheric charged particles on the entanglement of the satellite-ground quantum link is more significant than that of the ellipsoidal atmospheric charged particles.
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diffusion time, ranging from 0 to 24 h; E1, E2 represents the entanglement of quantum channel under the influence of ellipsoid and Chebyshev atmosphere charged particles, respectively, ranging from 0 to 1. It is easy to see, when the transmission diffusion time remains constant with the increment of charge density of the atmospheric charged particles, i.e., the probability of missing photons increases, quantum entanglement presents an attenuation trend. But the growing trend of the two surface plots is clearly different. When transmission diffusion time t=10.2 h, charge density of the atmospheric charged particles is 0.21 µC/m and 0.42 µC/m, entanglement of quantum channel is 0.413, 0.407 under the influence of ellipsoidal charged particles, and the entanglement of quantum channel is 0.361, 0.356 under the influence of Chebyshev atmosphere charged particles. It is found that the impact of chargeability of Chebyshev atmospheric charged particles is more significant than that of ellipsoidal atmospheric charged particles on the entanglement of the quantum channel.
5 Relationship between the atmospheric visibility and communication fidelity of satellite-ground quantum link
Fig. 4 Relationship among the effective radius, diffusion time and quantum entanglement
Fig. 5 shows the comparison among the charge density of the ellipsoid and Chebyshev atmospheric charged particles, the transmission diffusion time, and the entanglement of the quantum channel.
In order to use the quantum satellite communication platform and ground station to accomplish satellite-ground quantum link. When the photon quantum signal is transmitted to the stratosphere, communicates with the ground can be achieved by means of a communication platform suspended in the stratosphere. Under these circumstances we can consider the complex atmospheric environment on the impact of light quantum signal. Communication mode is shown in Fig. 6.
Fig. 5 Relationship among the charge density, diffusion time and quantum entanglement Fig. 6
In Fig. 5, σ1 represents the charge density of the ellipsoidal or Chebyshev atmospheric charged particles, ranging from 0 to 0.8 µC/m ; t represents the transmission
A typical optical system of quantum communication system
In Fig. 6, L is the transmission distance between the ground station and the light quantum signal platform, and
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can be expressed as: 2
2
L = R sin α + H + 2 RH − R sin α (25) where R is earth curvature radius, H is the vertical distance between the ground station and the optical quantum signal platform, α is transmission or acceptance dip angle. When the photon quantum state Q changes to Q ′ after arriving at the stratospheric communication platform, it can be calculated as [23]: Q ′ = Q e( − iωη / c − ( χ )/ 2) L + φ (26) where ω is light wave angular frequency, η is photoelectric detection quantum efficiency, χ is system attenuation coefficient, φ
represents a state that has
been scattered outside the Q substrate. The state of the ground station is finally detected: Q ′′ = Q ′ − φ
(27)
From Eq. (25), the communication fidelity between satellite-ground quantum link can be expressed as [24–25]:
F = tr where
Q ρ Q
Q ρ Q
entangled states
= e
2 L / [ ( ln 0 .0 2 ) V
]
(28)
is the inner product of multiphoton
Q
and ρ Q , V is the atmospheric
visibility. Fig. 7 shows the comparison among the atmospheric visibility, transmission distance, and the communication fidelity of satellite-ground quantum link.
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through atmospheric charged particles is decreasing, the extinction effect becomes smaller and smaller, decoherence leads to less quantum information in quantum states, the communication fidelity of satellite-ground quantum link traffic is increasing. And when the atmospheric visibility V is known with the increment of transmission distance, the extinction effect of atmospheric charged particles becomes more and more obviously. Decoherence causes the quantum information in quantum states to loss, the larger the attenuation caused by energy which leads to the decline of communication fidelity of satellite-ground quantum link. Simulation results demonstrate that when transmission distance L= 20 km, atmospheric visibility is 2 km and 8 km, the communication fidelity of satellite-ground quantum link is 0.08, 0.52, respectively. Thus, in order to ensure the quality of communication, fidelity and transmission distance need to be compromised.
6 Data analysis and discussions In this section, we will use the data given in this paper to do comparative analysis with research results work published previously. For comparison, the parameters are in range where the T matrix method can be accurately calculated and combined with the selected data in the literature. We Select the ice water mixed clouds [11] and non-spherical aerosol particles [13], and compare with the data in this paper. The equivalent radius of the different particles under this spectral distribution is 0.3 µm, and the light quantum wavelength is 1.5 µm. The attenuation coefficients of the quantum channel under the influence of different particles are shown in Table 3. Table 3 Comparison of the effects of different particles on quantum channels Particle shape
Fig. 7 Relationship among the atmospheric visibility, transmission distance and quantum fidelity
In Fig. 7, V represents the atmospheric visibility, ranging from 0 to 10 km; L represents the transmission distance, ranging from 0 to 40 km; F represents the communication fidelity of satellite-ground quantum link, ranging from 0 to 1. From the figure, we can see that when the transmission distance L is known with the increment of atmospheric visibility V, the probability and number of photons passing
Spherical particles(a/b=1) ellipsoid particles(a/b=1/2) Chebyshev particles Charged ellipsoidal particles(a/b=1/2) Charged Chebyshev particles
Channel attenuation coefficient 4.61 6.82 7.25
Channel entanglement 0.31 0.26 0.24
9.29
0.18
11.55
0.15
As can be seen in Table 3, the quantum channel attenuation coefficient of the ellipsoidal particles increase by 48%, and the quantum channel attenuation coefficient of the Chebyshev particles increase by 57% compared to the spherical particles. This indicates that the effect of the particle shape on the quantum channel is more significant. Compare with the charged ellipsoid particles and the
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non-charged ellipsoidal particles, the attenuation coefficient and entanglement of the quantum channel are 36% and 30%, respectively. Compare with the charged Chebyshev particles and the non-charged Chebyshev particles, the attenuation coefficient and entanglement of the quantum channel are 59% and 38%, respectively. This indicates that the particles have a greater impact on the quantum channel after charged. In this paper, the factors of particle chargeability are taken into account in analyzing the influence of environment on quantum channel for the first time. We find that the non-spherical charged particles destroy the quantum channel most seriously compared with the previous research results. In order to further study the non-spherical aerosol particles in the charged environment on the satellite –ground quantum link, free space quantum communication provides a theoretical basis. Nevertheless, we do not consider the effect of such factors on the capacity of quantum channels, and also lacks relevant experimental and practical applications. This is the weakness of this paper.
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quantum link also presented approximately logarithmic attenuation. Compared with previous research results, it is proved that the non-spherical atmospheric charged particles have the most serious damage to the communication performance of the satellite-ground quantum link. Thus, the quantitative relationship between the parameters proposed in this paper can provide reference for quantum satellite communication. It makes the quantum satellite communication system adjust the communication parameters adaptively, and reduces the impact of non-spherical atmospheric charged particles and atmospheric visibility on the performance of quantum satellite communication finally. Acknowledgements This work was supported by the National Natural Science Foundation of China (61172071, 61201194), the International Scientific and Technological Cooperation and Exchange Program in Shaanxi Province, China (2015KW-013) and the Scientific Research Program Funded by Shaanxi Provincial Education Department, China (16JK1711).
7 Conclusions References In this paper, we studied the effect of non-spherical atmospheric charged particles and atmospheric visibility on the performance of the satellite-ground quantum link. According to the extinction cross section and spectral distribution function of the non-spherical atmospheric charged particles, the relationship among the particle concentration, equivalent radius, charge density of the charged particle and the attenuation coefficient of the satellite-ground quantum link can be established. This paper gave the relationship among the equivalent radius of ellipsoidal and Chebyshev atmospheric charged particles, charge density, transmission diffusion time, and the entanglement of satellite-ground quantum link. We also analyzed the quantitative relationship between atmospheric visibility and quantum fidelity. Simulation results showed that with the increment of the equivalent radius and charge density of the charged particles, the attenuation coefficient and the entanglement of the satellite-ground quantum link is correspondingly changed in attenuation. However, the effect of the ellipsoid and Chebyshev atmosphere on the performance of the satellite-ground quantum link increases progressively. With the increment of atmospheric visibility, the communication fidelity of satellite-ground
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