Making entanglement between photonic Orbital Angular Momenta by Spontaneous Four Wave Mixing in an atomic vapor

. Spontaneous Four Wave Mixing (SFWM) which generates photonic pairs is studied if it is addressed by optical vortices carrying an orbital angular momentum (OAM). We show that the output beams are OAM-correlated and that the entanglement depends on the 4-level scheme used to realize SFWM.

Since early publications about optical vortices and the photonic OAM [1] the researches and the applications related to this variable and its properties have expanded to many fields as reported in recent review papers [2].
The OAM has been early be pointed out for its quantum properties and its possible use in quantum entanglement [3].Our work deals with this question but beside using non-linear crystals we explore interaction between vortex beams and atomic vapours which produce correlated OAMs.

Vortex beams
The OAM comes from the helical wavefront carried by an optical vortex.The integer ℓ defines the number of branches of the helix, its sign designs its handedness.Such waves propagate by keeping this property, that means that they obey to the Helmholtz equation.
In addition, Allen and al. showed that a such wave carries an OAM which equals ℏℓ per photon.This quantum variable is questioned about its superposition states and entanglement.Compared to the wave polarisation which takes only two values, OAMs ranges ℤ and open new insight for encoding and entanglement.For that exploring the interplay of OAM with matter is a key.

Laguerre-Gaussian modes
Vortex beams can be classified in families.The Laguerre-Gaussian (LG) modes is one of them very often used especially because they can be currently generated by phase-shaping techniques.
In the optics domain, as used for atomic excitation, LG modes or superposition of LG modes are prepared by using vortex plates or spatial light modulators.Many detection tools, interferometric ones, allow us to measure and determine the OAM of a LG mode or a combination.
LG modes are not only indexed by ℓ but also by the radial number  (radial nodes number).The  !ℓ mode amplitude denoted  !ℓ is Gaussian-shaped structured by a Laguerre polynomial.Its phase is characterized by an azimuthal phase ℓ, and the Gouy phase  (/ # ) with the Gouy number  = 1 + |ℓ| + 2 and  # the Rayleigh range of the beam.These properties mainly determine the nonlinear interaction with atoms.

Spontaneous Four Wave Mixing
Four Wave Mixing (FWM) realised in an atomic is known to transfer OAMs between the three input vortex beams and the output generated one If the third beam is delayed, FWM allows to store the OAM carried by the input beams, so realizing a quantum memory [4].
SFWM is slightly different : the atoms are addressed by only two input waves ( $ ,  % ) and SFWM produces a photonic pair ( & ,  ' ) (Fig. 1).The main advantages of SFWM are the different colours of emitted waves so detectable without ambiguity et the photonic entanglement.

SFWM with vortex beams
As SFWM is addressed by two vortex beams (ℓ $ , ℓ % ), the phase-matching associated to this nonlinear process imposes the generated voter pair (ℓ & , ℓ ' ) to satisfy the OAM and the Gouy number conservation : (2) These two conditions limit the number of output pairs and implies correlation between the output OAMs.

Probability amplitudes of generated pairs
In addition the LG composition of the output pair is determined by the mode overlap  $,% &,' resulting of the nonlinear interaction term and expressed as ) where  is the normalized factor to get a probability sum equal to 1.

SFWM with same handedness OAM
Our first study concerns SFWM realized with single ring LG mode as inputs and of same handedness (ℓ $ .ℓ % ≥ 0 and  $ =  % = 0) as done experimentally [5].The experiments showed that if  = ℓ $ + ℓ % is less than ~11 then SFWM mainly transfer the total OAM to wave 4. For larger  values, as shown in Fig. 2, wave 4 contain many OAMs and ℓ ' takes many values, ,  − 1 and so on.

Probability amplitudes of generated pairs
To extend the study we have applied our model to another Rb scheme where the decay occurs via the 5 $/% level so  & = 762  and  ' = 795 .This scheme being symmetric with  & / ' ∼ 1, so the probability amplitudes are different (Fig. 3) showing that as soon as  = 1 the input OAM is shared between the  & ,  ' outputs with almost the same probability.Such a configuration opens the way to control OAMcorrelation and could bring new possibilities for strong OAM-entanglement and their use in quantum computing.
show  $,% &,' versus .Because blue curve crosses that orange one at  ∼ 12 we can explain why LG mode with ℓ & = 1 appears for this value.This value corresponds to the output wavelengths ratio  & / ' ∼ 12. Indeed the radius of ℓ & = 1 LG mode becomes of same order of magnitude of the ℓ ' = 11 one.