screen, using a spatial light modulator (SLM). Photon pairs entangled in the OAM mode are generated and one photon from each pair propagates through the tur- bulence while the other is left undisturbed. The entanglement is quantified by the tangle [128, 129] and the results obtained are compared with those presented in the previous chapters.
This chapter is organized as follows: The experimental procedure is presented in section 5.2 followed by the numerical procedure in section 5.3. The results and discussions are presented in section. 5.4. We introduce an experiment where down- converted photons are simulated with back-projected light in section 5.5 and some conclusions are provided in section 5.6.
The results presented in this chapter were obtained by the author with input and guidance from Dr. Filippus S. Roux, Prof. Thomas Konrad and Prof. Andrew Forbes. Melanie McLaren assisted with the experimental setup.
Eq. (2.106) in section 2.4.3. This random phase was added to the phase function of one of the SLMs.
The Kolmogorov spectrum [86, 114]
ΦKn (k) = 0.033 Cn2k−11/3 (5.1) was used to allow for a comparison with existing studies, and subgrid sample points were added, as described in section 3.2.2 [125] to ensure that the random phase functions can reproduce the Kolmogorov structure function. The scintilla- tion strength considered ranged from w0/r0 = 0 to 4, in 0.4 increments. Thirty realisations corresponding to different phase fluctuations were performed for each scintillation strength and a full quantum state tomography [130] was performed for each realization to reconstruct the density matrix describing the state of the two qutrits. These matrices were then averaged to obtain the density matrix cor- responding to each scintillation strength.
The concurrence [122] is the preferred entanglement measure for two-dimensional bipartite systems. Unfortunately the generalisation of the concurrence to multi- dimensional systems is not a trivial problem. The lower bound for the concur- rence can be obtained for multidimensional systems by computing the convex roof [131], however, this is computationally demanding. There are some gener- alisations of the concurrence to multidimensional systems, these include the G- concurrence [132, 133] and the I-concurrence [134]. Here, the tangle is used to quantify the amount of entanglement between the two qutrits [128, 129]. It is defined as
τ(ρ) = 2tr(ρ2)−tr(ρ2A)−tr(ρ2B), (5.2) whereρA andρB are the reduced density matrices of subsystems AandB. Ifρmax
is ad×ddimensional density matrix representing a maximally entangle state, then
τ(ρmax) = 2(d−1)/d. (5.3)
For bipartite two-dimensional states (qubits), the tangle is the lower bound for the square of the concurrence. This is illustrated in Fig. 5.1 where the tangle and
w0/ r0
C2
Figure 5.1: The tangle and the concurrence squared plotted against the scintillation strength (w0/r0). These curve are the S&R theory calculation for the evolu- tion of the OAM entanglement between two qubits (|ℓ|= 1) as they evolve in atmospheric turbulence (section. 2.4.5).
the concurrence squared are plotted against the scintillation strength. These curves are the S&R theory calculation for the evolution of the OAM entanglement between two qubits (|ℓ|= 1) as they evolve in atmospheric turbulence (section. 2.4.5).
So far, only LG modes were considered in this work. In this chapter, we consider Bessel-Gauss (BG) modes instead [21]. The electric field of these modes is given by Eq. (2.31) in chapter 2, it is repeated here for convenience
MℓBG(r, ϕ, z;kr) = izR q(z)Jℓ
(izRkrr q(z)
) exp
[
−kr2zRz 2kq(z)−ikz
]
exp(iℓϕ) (5.4) where q(z) =z+izR, J(·) is the Bessel function and zR =πw20/λ is the Rayleigh range. The radial profile of the beam can be scaled by choosing different values of the radial kr. Just like LG beams, the BG beams also carry an OAM of ℓ~ per photon. The BG modes can thus be used as a basis to represent the quantum state of the two photons after SPDC. In the BG basis, one can write the state of the two photons produced by SPDC as [72]
|Ψ⟩=∑
ℓ
∫ ∫
aℓ(kr1, kr2)|ℓ, kr1⟩s| −ℓ, kr2⟩idkr1dkr2, (5.5) with|aℓ(kr1, kr2)|2 being the probability of measuring the signal and idler photons in the states |ℓ, kr1⟩s and | −ℓ, kr2⟩i respectively. In our experiment, we selected
Coin. counts x103 [s-1 ]
l
Figure 5.2: the OAM spectrum for LG modes (blue); for BG modes with kr = 21 rad/mm (green) and for BG modes with kr = 35 rad/mm (red).
a particular kr value for both the signal and idler photons. We can thus write the state of the two photons as
|Ψ(kr)⟩=∑
ℓ
aℓ|ℓ⟩s| −ℓ⟩i (5.6) The motivation behind using the BG modes instead of LG modes is that BG modes have a broader and flatter OAM spectrum compared to LG modes [72].
This is illustrated in Fig. 5.2 where we plot the OAM spectrum for LG and BG modes on the same graph. The OAM spectrum is flatter for BG modes in the sense that the difference in the coincidence counts between ℓ = 0 and the higher ℓ values is smaller for the BG modes compared to the LG modes. For instance, the difference in the number of counts between ℓ = 0 and ℓ = 3 is 66 counts for LG modes whereas it is 23 counts for BG with kr = 21 rad/mm and 7 counts for BG with kr = 35 rad/mm. Also, the difference in number of counts between ℓ = 0 and ℓ = 5 is 108 counts for LG modes whereas it is 26 counts for BG with kr = 21rad/mm and 6 counts for BG with kr = 35rad/mm. The flatness of the OAM spectrum plays an important role in the reconstruction of the density matrix representing the state of the two photons produced by SPDC. If we consider a state
|Ψ⟩in = 1
√3(|ℓ⟩A|−ℓ⟩B +|0⟩A|0⟩B+|−ℓ⟩A|ℓ⟩B), (5.7) a big difference in the coincidence counts between the OAM values 0 and ℓ will lead to an inaccurate reconstruction of the density matrix. This is illustrated in Fig. 5.3, where we plot the density matrices describing the state of two qutrits represented by photons generated through SPDC for ℓ = 1,3 and 5 and for both LG and BG modes. One can see that the reconstructed density matrix becomes increasingly less accurate for LG modes as the value of ℓ is increased.
LG BG
LG BG
LG l = 1, 0, -1 BG
l = 3, 0, -3
l = 5, 0, -5
Figure 5.3: Real part of the density matrices describing the state of two qutrits repre- sented by photons generated through SPDC for different ℓvalues and both LG and BG modes. The x and y axis represent the basis vectors.