3.3 Results
3.3.1 OAM entanglement through turbulence simulated by a single phase
w0/r0
Concurrence
w0/r0
Concurrence Concurrence
w0/r0
Concurrence
w0/r0
Figure 3.12: The concurrence plotted against the scintillation strength (w0/r0) when one of the two photons propagates in turbulence. In (a)|ℓ|= 1, in (b) |ℓ|= 3, in (c) |ℓ| = 5 and in (d) |ℓ| = 7. In the legend, S&R: theoretical curve derived by Smith and Raymer in [84] and NS: Numerical data points. The error bars are calculated as discussed in appendix A.
function (see section 2.4.5). In other words, instead of using Eq. 3.5 as the phase structure function, they used
Dϕquad= ( r
r0
)2
. (3.22)
This approximation simplifies the calculations, but it tends to over-estimate the concurrence as the value of |ℓ| increases.
Concurrence
w0/r0
(a)
Concurrence
w0/r0
(b)
Concurrence
w0/r0
(c) Concurrence
w0/r0
(d)
Figure 3.13: The concurrence plotted against the scintillation strength (w0/r0) when both photons propagate through turbulence. In (a) |ℓ|= 1, in (b) |ℓ|= 3, in (c) |ℓ| = 5 and in (d) |ℓ| = 7. In the legend, S&R: theoretical curve derived by Smith and Raymer in [84] and NS: Numerical data points.
Concurrence
w0/r0
(b)
Figure 3.14: The concurrence (a) and the trace of the density matrix before normalisation (b) plotted against the scintillation strength (w0/r0) for different values of ℓwhen both photons propagate in turbulence.
We observe in Fig. 3.12 and 3.13 that both the S&R theory and the numerical results predict that the concurrence takes longer to decay for higher values of |ℓ|. This is more clearly seen in Fig. 3.14 (a) where we plot the curves corresponding to the different values of |ℓ| on the same graph. This suggests that modes with higher |ℓ|-values are more robust in turbulence and could thus give an advantage in a free-space quantum communication system. On the other hand, the plots of the trace [Fig. 3.14 (b)] show that the trace decays to zero quicker for higher
|ℓ|-values. This suggests that for higher |ℓ|-values the scattering into other modes happens more rapidly. The same behaviour was observed in [84, 90].
Scale at which entanglement decays
The S&R theory predicts that the concurrence lasts longer for higher values of|ℓ|, and that the spacing between adjacent curves decreases as |ℓ| increases. This is also true for the numerical simulation and can be seen in Fig. 3.15 where we plot the S&R theory and the numerical results against the scintillation strength on a logarithmic scale. The fact that the concurrence survives longer for higher |ℓ|- values suggests that the scale of entanglement decay will occur around a different point for larger values of ℓ: the scale at which decoherence occurs depends on the value ofℓ.
To find thatℓ dependence, we use the S&R theory to locate the values of ω0/r0 where the concurrence is equal to 0.5 for the different |ℓ|-values considered. The result obtained is shown in Fig. 3.16 where the ω0/r0 values are plotted against the corresponding values of ℓ on a logarithmic scale.
We find ω0/r0 = 1.35√
ℓ in the single photon case and ω0/r0 = 1.03√
ℓ in the two photon case. Thus in both cases the entanglement decay happens within an order of magnitude around the point where ω0/r0 ≈√
ℓ. By using the expression of the Fried parameter [Eq.(3.21)], we find that the distance scale at which OAM entanglement decays as a function of ℓ is
Ldec(ℓ)≈ 0.06λ2ℓ5/6 ω05/3Cn2
. (3.23)
w0/r0 ConcurrenceConcurrence
w0/r0
(a)
(c)
w0/r0
(b)
Concurrence
w0/r0
(d)
Concurrence
Figure 3.15: The concurrence plotted against the scintillation strengthω0/r0for the S&R theory and the numerical results in the single photon case [(a) and (b)] and in the two-photon case [(c) and (d)]. The horizontal axis is plotted on a logarithmic scale.
Thus for a practical free-space quantum communication system using OAM modes as qubits, the distance between repeaters should be shorter than Ldec(ℓ). For example, if one would send OAM entangled photons in a beam withω0 = 10 cm, a wavelength ofλ= 1550 nm, on a horizontal path in moderate turbulence conditions (Cn2= 10−15 m−2/3), the entanglement between the photons will decay around the distances shown in Table 3.1 for the different values of ℓ.
ℓ 1 3 5 7
Ldec(km) 6.7 16.7 25.6 33.7
Table 3.1: Distance scale at which entanglement decays for OAM entangled photons in a beam withω0 = 10 cm, a wavelength ofλ= 1550 nm, on a horizontal path in moderate turbulence (Cn2 = 10−15 m−2/3).
l
w0 r
|
0Figure 3.16: The scintillation strength plotted againstℓ on a logarithmic scale for both the single photon case (diamond dots) and the two-photon case (circular dots). The equation of the fitted lines are log (ω0/r0) = 0.5 log(ℓ) + 0.1303 in the single photon case and log (ω0/r0) = 0.5 log(ℓ) + 0.01284 in the two photon case.
We notice in Table 3.1 that the distance scale at which entanglement decays is relatively short even in moderate turbulence. This suggests that the OAM state of light might not be suitable for long distance free-space quantum communication.
One can try to increase that distance by using a smaller beam radius, but that would increase beam divergence, which in turn reduces the received power for a given receiver aperture. The entanglement decay distance can also be increased by using adaptive optics.
3.3.2 OAM entanglement through turbulence simulated by multiple phase screens
The single phase screen approximation limits the validity of the predictions in the previous Section to the weak fluctuations regime. In order to simulate the turbulent atmosphere accurately, one needs to use a multitude of phase screens as described in Section 2.4.3. Here, we simulate the turbulent atmosphere with a series of consecutive phase screens. The distance between adjacent phase screens correspond to an increment of 0.2 in the value of (w0/r0)5/3 and both photons
propagate through turbulence. We use increments of (w0/r0)5/3 instead of w0/r0 because this quantity is linear with the total propagation z. This allows us to have a fixed distance ∆z between the phase screens. The numerical results will be compared with the infinitesimal propagation equation (IPE) derived in [90]
(discussed in section 2.4.6).
The IPE is a first order differential equation describing the evolution of OAM entanglement in turbulence. It was derived by treating the distortion that an OAM state experiences due to propagation through a thin sheet of turbulent atmosphere as an infinitesimal transformation. It is thus based on multiple phase screens and predicts the evolution of entanglement even in the strong fluctuation regime.
In the weak fluctuation regime, both the single phase screen and the multiple phase screens should return the same results as shown in Fig. 3.17.
Concurrence
Figure 3.17: The concurrence plotted against the propagation distance for both the single phase screen and multiple phase screens in the weak scintillation regime.
As we increase the fluctuation strength, we expect a difference in the predictions made by the single and multiple phase-screens methods.
w0/r0
Concurrence
Figure 3.18: The concurrence plotted against the scintillation strength (w0/r0) for mul- tiple phase screens in the moderate fluctuation regime.
Figure 3.18 shows the evolution of the concurrence against the scintillation strength in moderate fluctuations (σR2 ≈ 0.1 when the concurrence reaches 0) for the multiple phase screens. Already in this regime, the evolution of the concur- rence is different to what was found with the single phase screen approximation in the weak fluctuation regime. For instance, it was observed in the weak fluctuation regime that the concurrence lasts longer for higher values of|ℓ|, here, we see that the concurrence decays to zero around the same value of w0/r0. This suggests that in the moderate to strong fluctuation regime, the evolution of the concur- rence can no longer be characterised by a single dimensionless parameter (w0/r0) like in the weak-fluctuation regime. This confirms what was reported in Ref. [90]
that the evolution of concurrence requires at least two parameter: the normalized propagation distance
t = z
zR = zλ
πw20, (3.24)
which is independent of the turbulent strength, and another parameter K = Cn2w011/3π3
λ3 , (3.25)
which is independent of the propagation distance. The dimensionless parameters w0/r0 and K are just two possible ways of combining the dimension-carrying
parameters. The parametersw0/r0, K and t are related by w0
r0 = 1.37K3/5t3/5. (3.26)
log(K) w0 [m] Cn2 [m−2/3] λ[nm]
1.5 0.2 9.7·10−16 1400 2 0.05 3.2·10−13 1190 2.5 0.1 5.0·10−15 481
3 0.05 1.0·10−12 807.2 3.5 0.1 9.7·10−14 600
4 0.1 5.0·10−12 1494.8 4.5 0.1 9.7·10−13 600
5 0.5 9.7·10−15 618.7
Table 3.2: Parameters used for the plots in Fig. 3.19, 3.20, 3.21 and 3.22
Concu rr ence
(a)
Concu rr ence
t x 10
-3(b) w
0/r
0Figure 3.19: The concurrence plotted againstw0/r0 (a) and against t(b) for ℓ= 1 and for different values ofS= log10(K) in the multiple phase screen method.
(a)
w
0/r
0Concu rr ence Concu rr ence
(b)
Figure 3.20: The concurrence against w0/r0 (a) and against t (b) for ℓ = 3 and for different values of S= log10(K) in the multiple phase screen method.
Concu rr ence
(b) w
0/r
0Concu rr ence
(a)
Figure 3.21: The concurrence plotted againstw0/r0 (a) and against t(b) for ℓ= 5 and for different values ofS= log10(K) in the multiple phase screen method.
Concu rr ence
w
0/r
0(a)
Concu rr ence
(b)
Figure 3.22: The concurrence plotted againstw0/r0 (a) and against t(b) for ℓ= 7 and for different values ofS= log10(K) in the multiple phase screen method.
Figures 3.19, 3.20, 3.21 and 3.22 show the plots of the concurrence againstw0/r0
and against t for the different values of S = log10(K) and the azimuthal index ℓ considered. The different sets of dimension parameters that were used to produce the different values of K are given in Table 3.2. It can be seen from those figures that the plot of the concurrence againstw0/r0 coincide with one another for larger values of S, that is they lie on a limiting curve.
As a function of w0/r0, the curves of the concurrence lie on the limiting curves for large values of K, but they tend to fall below this limiting curve when K is small. This suggests that there is a value of K beyond which the evolution of the concurrence depends only on w0/r0. This corresponds to the situation that is considered in the Paterson model [41], where the behaviour is completely determined by w0/r0.
On the other hand, for small values of K, the plots of the concurrence devi- ate from the limiting curve, in that they decay faster than the limiting curve as a function w0/r0. This suggests that the Paterson model can not be used un- der these conditions. Two dimensionless parameters are required to describe the behaviour of the concurrence and the trace during propagation under these condi- tions, namely K and t.
Our results are qualitatively similar to those obtained with the IPE [90] (dis- cussed in section 2.4.6), but the detailed behaviour is quantitatively different. The IPE predicts that for a value of the normalised propagation distance t >1/3, the evolution of the OAM entanglement can no longer be described by the single di- mensionless parameterw0/r0. One needs the two dimensionless parametersK and t. Our results [Fig. 3.19 (b), 3.20 (b), 3.21 (b) and 3.22 (b)] on the other hand, show that the value oftbeyond which the Paterson model doesn’t hold depends on the value ofℓ. For instance, we see from Fig. 3.19 (b), 3.20 (b), 3.21 (b) and 3.22 (b) that the value of t beyond which one needs the two dimensionless parameters K andt to describe the evolution of the concurrence (the value oft beyond which the curves of the concurrence againstw0/r0 do not overlap any more) is 0.01 when ℓ= 1, 0.007 when ℓ= 3, 0.003 when ℓ = 5 and 0.001 when ℓ= 7.
Although one can see from the plots in Figs. 3.19, 3.20, 3.21 and 3.22 that one
dimensionless parameter is not enough to describe the evolution of the entangle- ment, they do not reveal whether more than two dimensionless parameters are not perhaps required. For this purpose we consider different sets of dimension parameters that give the same value forK and plot them as a function of t.
w0 [m] Cn2 [m−23] λ[nm]
Set 1 0.04 10−13 633.0 Set 2 0.05 10−14 385.8 Set 3 0.10 10−15 417.8 Set 4 0.20 10−16 452.4 Set 5 0.40 10−17 489.9
Table 3.3: Parameters used for the plots in Fig. 3.23 (K = 91.6).
0 0.02 0.04 0.06 0.08 0.1 t
0 0.2 0.4 0.6 0.8 1
Concurrence
set 1 set 2 set 3 set 4 set 5
Figure 3.23: Plots of the concurrence plotted against t for K = 91.6 when|ℓ|= 1. The values of the parameters used for each plot is given in table 3.3
Figure 3.23 shows the plots of the concurrence as a function of t for K = 91.6 when |ℓ|= 1. Five different sets of parameters (shown in Table 3.3) that produce the same value of K are considered. We see from the figure that regardless of the values of the individual parameters, all the points that correspond to the same value ofK lie on the same curve.