Myrann Baker Rasooli

Laurence Pruvost
email: laurence.pruvost@sorbonne-universite.fr

Orbital Angular Momentum entanglement by Spontaneous Four Wave Mixing realized in a vapor by vortex beams of same handedness

Spontaneous Four Wave Mixing (SFWM) carried out in a vapor by two optical vortex beams is analyzed in the case of same handedness input orbital angular momenta. We show that the generated pair of waves carries orbital angular momenta whose entanglement strongly depends on the chosen four-levels atomic scheme, asymmetric or not. The illustration is done with two schemes of rubidium. In addition based on a theory-experiment comparison we show that the output is a partially coherent superposition of modes preserving the entanglement.

I. INTRODUCTION

The route to quantum computing and simulation requires entanglement and control including a minimization of decoherence processes. Among the available quantum systems, the photon offers many quantum variables that can be entangled through light-matter interaction. One of them is the photonic Orbital Angular Momentum (OAM) which is associated to optical vortex beam.

The first ideas about optical vortex were introduced by Coullet et al. [START_REF] Coullet | Optical vortices[END_REF] before Allen et al. defined its OAM [START_REF] Allen | Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes[END_REF]. The first experimental realizations followed [START_REF] Beijersbergen | Helical-wavefront laser beams produced with a spiral phaseplate[END_REF][START_REF] Padgett | An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes[END_REF]. Then, many applications of vortex beams appeared in many fields. A review can be found in [START_REF] Shen | Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities[END_REF].

An optical vortex carries an helical phase whose number of branches and the handedness are given by an signed integer ℓ. Upon electromagnetic propagation and the Poynting vector analysis, the OAM is equal to ℏℓ per photon. Beside the spin angular momentum, namely the polarization, the OAM takes many values. It is the reason why, quite early, OAM was pointed out as a pertinent variable for coding, multiplexing information and for multi-space entanglement. In 2001, Mair et al. realized the first OAM-entanglement by Spontaneous Parametric Down-Conversion (SPDC) in a non-linear crystal [START_REF] Mair | Entanglement of the orbital angular momentum states of photons[END_REF].

In atomic vapors the atom-vortex interaction was first demonstrated [START_REF] Tabosa | Optical Pumping of Orbital Angular Momentum of Light in Cold Cesium Atoms[END_REF] using degenerated four-wave mixing showing the OAM transfer from input to the idler beam. This experiment has opened a new field to exploit the atomic coherence and OAMs [START_REF] Tabosa | Generation of Light Carrying Orbital Angular Momentum via Induced Coherence Grating in Cold Atoms[END_REF] followed by application to quantum memories [START_REF] De Almeida | Storage of orbital angular momenta of light via coherent population oscillation[END_REF][START_REF] Nicolas | A quantum memory for orbital angular momentum photonic qubits[END_REF].

SPDC and SFWM are well-adapted for entanglement, because they produce pairs of photons correlated by physical constraints. SFWM occurs when two lasers resonantly excite atoms to a state which decays with the emission of two photons and via a short lifetime intermediate level. The excitation and decay selection rules determine how the photons are entangled. Often they are polarization-entangled. If the excitation is realized by vortex beams, they are also OAM-entangled.

In this paper we examine OAM-entanglement as SFWM is realized by two input vortex beams of same handedness applied to a four-levels system. We compare asymmetric and symmetric schemes (Fig. 1).

The analysis allows us to link the model to experimental data previously obtained on Rb atoms for the asymmetric scheme [START_REF] Chopinaud | High helicity vortex conversion in a rubidium vapor[END_REF][START_REF] Walker | Trans-Spectral Orbital Angular Momentum Transfer via Four-Wave Mixing in Rb Vapor[END_REF][START_REF] Akulshin | Arithmetic with optical topo-logical charges in stepwise-excited Rb vapor[END_REF][START_REF] Liu | Manipulation of the orbital angular momentum via fourwave mixing in Rb vapor[END_REF]. The model also predicts OAMentanglement signatures for a symmetric scheme.

The model relies on fields decomposition onto Laguerre-Gaussian (LG) modes. It provides the OAM selection rules for SFWM realized with input LG modes. It gives an analytic formula for the mode decomposition of the generated beams. It explains the experimental observations and the observed partially-coherent interference patterns. In addition it allows us to predict about SFWM with vortex beams operated on a symmetric Rb fourlevels scheme, where SFWM leads to OAM-entanglement signatures different from the asymmetric case.

II. FOUR-LEVELS ATOM AND SPONTANEOUS FOUR WAVE MIXING WITH VORTEX BEAMS

A. The considered problem

We consider four-levels atomic schemes as shown in Fig. 1. the atoms are excited by two lasers at wavelengths λ 1 and λ 2 and decay with two emitted waves at λ 3 and λ 4 , so realizing SFWM.

For Rb atom, as treated in this paper, lasers at λ 1 = 780 nm and λ 2 = 776 nm realize the 5S → 5D twophoton transition. SFWM depends on the chosen 5D fine level. With 5D 5/2 level it produces two waves at λ 3 = 5230 nm and λ 4 = 420 nm (Fig. 1(a)). Because these wavelengths are very different this case is called 'asymmetric'. Experiments reported in [START_REF] Chopinaud | High helicity vortex conversion in a rubidium vapor[END_REF][START_REF] Walker | Trans-Spectral Orbital Angular Momentum Transfer via Four-Wave Mixing in Rb Vapor[END_REF][START_REF] Akulshin | Arithmetic with optical topo-logical charges in stepwise-excited Rb vapor[END_REF][START_REF] Liu | Manipulation of the orbital angular momentum via fourwave mixing in Rb vapor[END_REF] have been done with this scheme. With 5D 3/2 level, SFWM produces output waves at λ 3 = 762 nm and λ 4 = 795 nm (Fig. 1(b)). These wavelengths values being similar, this scheme is called 'symmetric'.

For an atomic vapor excited by two co-propagating vortex beams carrying respectively ℓ 1 and ℓ 2 OAMs, SFWM produces two vortex beams with respectively ℓ 3 and ℓ 4 OAMs. If input beams are colinear and copropagating as shown in Fig. 1(c), due to energy and linear moment conservation, the generated vortex are necessary co-propagating. Experiments of [START_REF] Chopinaud | High helicity vortex conversion in a rubidium vapor[END_REF][START_REF] Walker | Trans-Spectral Orbital Angular Momentum Transfer via Four-Wave Mixing in Rb Vapor[END_REF][START_REF] Akulshin | Arithmetic with optical topo-logical charges in stepwise-excited Rb vapor[END_REF][START_REF] Liu | Manipulation of the orbital angular momentum via fourwave mixing in Rb vapor[END_REF] have been realized with single-ring LG modes of same polarization as input, so the outputs are waves of same polarization.

Each generated wave results of the non-linear interaction of the 3 others. The field of the third wave E 3 is proportional to E 1 E 2 E * 4 . The proportional factor involves the third order susceptibility χ (3) which depends of the vapor density. Respectively, E 4 is proportional to the product E 1 E 2 E * 3 .

B. Laguerre-Gaussian modes

All involved vortex beams of our problem are LG modes or combination of LG modes. Let us remind their expressions and main properties. A LG mode at wavelength λ is defined by its OAM ℓ and its radial number p. We denote it by a compact notation ℓ p . In cylindrical coordinates (r, θ, z), a LG mode of waist w propagating along z axis, has a Gouy number α = |ℓ| + 2p + 1 and a Rayleigh range z R = πw 2 /λ. Its field is expressed by

ℓ p = A ℓ,p e iℓθ e iαAtan( z z R ) e i2πz/λ (1)
where A ℓ,p is the amplitude. We do not give the general expression of A ℓ,p that can be found in [START_REF] Chopinaud | Atomes et vortex optiques : conversion de moments orbitaux de lumière en utilisant la transition à deux photons 5S-5D du rubidium[END_REF] because only ℓ ≥ 0, p = 0 modes are involved in our study. The amplitude of a single-ring LG mode (p = 0) is

A ℓ = A ℓ,0 = 2 πw 2 ℓ! 2r 2 w 2 ℓ/2 e -r 2 w 2 (2)
It is normalized as 2π ∞ 0 |A ℓ | 2 rdr = 1 by using [START_REF]The integral is computed by putting X = 2r 2 w 2 and using ∞[END_REF]. Such a LG mode has one ring of light whose radius R increases with ℓ as R = w ℓ/2 and whose width ∆ is given by ∆ = R + -R -where (R ± ) 2 = w 2 (ℓ ± √ ℓ) are the inflection points of A ℓ .

C. Qualitative aspects

An optimal SFWM process is obtained if, at first, the input beams overlap J in = A ℓ1 A ℓ2 rdrdθ is maximum. Using Eq.2 one gets

J in = ((ℓ 1 + ℓ 2 )/2)! √ ℓ 1 ! ℓ 2 ! (3)
which is maximum for ℓ 1 = ℓ 2 . J in allows also to define the working diagram of Fig. 2(a). The yellow region indicates the favorite OAM couples (ℓ 1 , ℓ 2 ) to be used.

The working diagram can be understood by qualitative geometric considerations. Indeed we can impose an overlap condition for the input LG modes by comparing their radii and widths as Similarly, the output LG overlap can be evaluated. Assuming that output vortex beams are single-ring LG modes (demonstrated in a next section), J out is defined by J out = A ℓ3 A ℓ4 rdrdθ and its value is

R 2 + ∆ 2 /2 > R 1 -∆ 1 /2 and R 2 -∆ 2 /2 < R 1 + ∆ 1 /2. This gives the following condi- tions ℓ 2 + √ ℓ 2 > ℓ 1 - √ ℓ 1 and ℓ 2 - √ ℓ 2 < ℓ 1 + √ ℓ 1
J out = (ℓ 3 + ℓ 4 )/2)! √ ℓ 3 ! ℓ 4 ! w ℓ3+ℓ4+2 out w ℓ3+1 3 w ℓ4+1 4 ( 4
)
with 2

w 2 out = 1 w 2 3 + 1 w 2 4
. Applying Boyd-Kleinman criterion [START_REF] Boyd | Parametric Interaction of Focused Gaussian Light Beams[END_REF], known to optimize non-linear wave mixing, which states that all involved light beams have the same Rayleigh range, we rewrite Eq.4. Boyd-Kleinman criterion implies w 2 3 /λ 3 = w 2 4 /λ 4 = w 2 /λ; we have put λ = λ 1 = λ 2 because the input wavelengths differ by only 0.4%. In addition because 1/λ 3 + 1/λ 4 = 2/λ, it comes w out = w and then

J out = ((ℓ 3 + ℓ 4 )/2)! √ ℓ 3 ! ℓ 4 ! λ ℓ3+ℓ4+2 λ ℓ3+1 3 λ ℓ4+1 4 (5)
Eq.5 shows that the output overlap J out strongly depends on the output wavelengths.

For the asymmetric scheme, as λ 3 ≫ λ 4 , J out is maximum at small values of ℓ 3 as shown by the yellow region of Fig. 2(b). So ℓ 3 = 0 is the most expected value in the generated pair up to ℓ 4 ∼ 10. For ℓ 4 ≥ 12 pairs with ℓ 3 = 1 appear. The frontier is related to the wavelengths ratio λ 3 /λ 4 ∼ 12.

For the symmetric scheme, because λ 3 /λ 4 ∼ 1, the most expected output pairs satisfy ℓ 3 ∼ ℓ 4 as shown by the yellow region of Fig. 2(c).

Fig. 2(b) and (c) illustrate how OAM-sharing in the output pair strongly depends on the four-levels scheme. We will check at this expectation by computing the output modes probability amplitudes.

III. MODEL FOR SPONTANEOUS FOUR WAVE MIXING WITH VORTEX BEAMS A. Model and assumptions

Our approach consists in decomposing the generated waves E 3 , E 4 , onto a LG modes basis, the coefficients of the decomposition providing the mode probability amplitudes. Then, by propagating each mode, the intensity of the generated waves is determined at any z position. The approach differs from [START_REF] Lanning | Gaussian-beampropagation theory for nonlinear optics involving an analytical treatment of orbital-angular-momentum transfer[END_REF] where the model is based on the Green's function.

In the LG-decomposition model, the output pair is expressed as ℓ3,p3ℓ4,p4 c(ℓ 3 , p 3 , ℓ 4 , p 4 ) ℓ3 p3 ℓ4 p4 where c(ℓ 3 , p 3 , ℓ 4 , p 4 ) is the probability amplitude of the ℓ3 p3 ℓ4 p4

pair. If the mode superposition is coherent the output fields are

E 3 = ℓ3,p3,ℓ4,p4 c(ℓ 3 , p 3 , ℓ 4 , p 4 ) ℓ 3 p 3 (6)
E 4 = ℓ3,p3,ℓ4,p4 c(ℓ 3 , p 3 , ℓ 4 , p 4 ) ℓ 4 p 4 (7)
Fortunately, selection rules associated to SFWM process will reduce the number of involved modes and so the number of c(ℓ 3 , p 3 , ℓ 4 , p 4 ) to be determined (see section III.B). Moreover, the mathematical form of the LG modes will provide an analytical expression (III.C).

In addition, we impose assumptions related to experiments. We assume the atoms to be excited by single-ring LG beams (p = 0) of same waists w and carrying OAMs of the same handedness, ℓ 1 ≥ 0, ℓ 2 ≥ 0. We also assume that the medium length is short compared to the Rayleigh range z R , so we neglect any beam divergence. In respect with the experiments, all the waves involved in SFWM have the same polarization, so the non-linear optics model is scalar.

The last assumption is Boyd-Kleinman criterion as discussed in section II.C.

B. Selection rules and allowed output pairs

SFWM is ruled by phase-matching of the involved waves from the relationship

E 3 ∝ E 1 E 2 E * 4 . The azimutal phase-matching implies OAM conserva- tion ℓ 3 + ℓ 4 = ℓ 1 + ℓ 2 .
For modes with same Rayleigh range (Boyd-Kleinman criterion), the Gouy phase-matching implies the conservation of Gouy number as

α 3 + α 4 = α 1 + α 2 .
These two conditions will determine the allowed pairs of LG modes at the output. For single-ring LG modes with positive OAMs as input it gives

ℓ 3 + ℓ 4 = L (8) |ℓ 3 | + 2p 3 + |ℓ 4 | + 2p 4 = L (9)
where L = ℓ 1 + ℓ 1 ≥ 0 defines the total input OAM.

Let us analyze the solutions of Eq.8,9. If ℓ 3 and ℓ 4 are negative, it is not possible to satisfy Eq.8. If one of them is negative (e.g. ℓ 4 ) Eq.9 gives ℓ 4 -p 3 -p 4 = 0 implying p 3 = p 4 = ℓ 4 = 0 because p i ≥ 0. So, the only solutions are ℓ 3 ≥ 0, ℓ 4 ≥ 0 and p 3 = p 4 = 0. The ouputs waves are thus composed by single-ring LG modes with ℓ 3 ranging [0, L] and ℓ 4 = L -ℓ 3 .

The output is composed by L+1 pairs, so the fields are

E 3 = L ℓ3=0 c(ℓ 3 , L -ℓ 3 ) ℓ 3 0 (10)
E 4 = L ℓ4=0 c(L -ℓ 4 , ℓ 4 ) ℓ 4 0 . (11)

C. Probability amplitudes of output LG modes

The overlap of the four involved LG modes

J(ℓ 3 , ℓ 4 ) = A ℓ1 A ℓ2 A ℓ3 A ℓ4 rdrdθ (12)
is the key-quantity to get the probability amplitude c(ℓ 3 , ℓ 4 ) of the modes in the output, c 2 (ℓ 3 , ℓ 4 ) being the probability to find the pair ℓ3 0 ℓ4 0 . Using the normalization factor J defined by J 2 = L ℓ3=0 J 2 (ℓ 3 , L-ℓ 3 ) we have c(ℓ 3 , ℓ 4 ) = J(ℓ 3 , ℓ 4 )/J. Before computing Eq.12 we qualitatively determine the J(ℓ 3 , ℓ 4 ) maximum. The product of A ℓ1 A ℓ2 is maximum at r = w L/4. To get a good overlap the generated light has to be localized in this region giving an order of magnitude for the output mode radii R 3 and R 4 and of the associated OAMs. The comparison provides w 3 ℓ 3 /2 ∼ w 4 ℓ 4 /2 ∼ w L/4. With Boyd-Kleinman criterion we get ℓ 3 ∼ λ 2λ3 L and ℓ 4 ∼ λ 2λ4 L. For the asymmetric scheme it leads to ℓ 3 ∼ 0.07L and ℓ 4 ∼ 0.93L. So, for L less than ∼ 12 the output would mainly contain the (0, L) pair, the OAM being preferentially transferred to wave 4 (at 420 nm). For larger values of L the pair (1, L-1) would appear, then the (2, L-2) one and so on. A completely different OAM-sharing would occur in the symmetric scheme because ℓ 3 ∼ 0.5L and ℓ 4 ∼ 0.6L.

Using Eq.2 and note [START_REF]Starting from[END_REF] we establish

J(ℓ 3 , ℓ 4 ) = 2 π L! √ ℓ 1 !ℓ 2 ! w 2L+2 t w L+2 1 √ ℓ 3 !ℓ 4 ! 1 w ℓ3+1 3 w ℓ4+1 4 (13) with 2 w 2 t = 2 w 2 + 1 w 2 3 + 1 w 2 4
. For a given input set, the ℓ 3 , ℓ 4 dependence in Eq.13 comes from the two last terms. We can thus ignore the pre-factor which will disappear in the normalization and so avoid to compute time-consuming factorials.

Furthermore, because ℓ 4 = L-ℓ 3 , the amplitude c(ℓ 3 , L-ℓ 3 ) can be expressed as

c(ℓ 3 , L-ℓ 3 ) = 1 c L! ℓ 3 !(L-ℓ 3 )! 1 w ℓ3 3 w L-ℓ3 4 with c 2 = L ℓ3=0 L! ℓ3!(L-ℓ3)! ( 1 w 2 3 ) ℓ3 ( 1 w 2 4
) L-ℓ3 . We recognize the binomial formula c 2 = ( 1

w 2 3 + 1 w 2 4 ) L ≡ ( 2 w 2 out ) L . Then c(ℓ 3 , L-ℓ 3 ) = L! ℓ 3 !(L-ℓ 3 )! (w out / √ 2) L w ℓ3 3 w L-ℓ3 4 ( 14
)
Eq.14 provides an analytical expression of the amplitude probability, which is valid for any four-levels schemes of atoms even different of rubidium.

It is convenient to express Eq.14 versus wavelengths rather than waists. Applying Boyd-Kleinman criterion and energy conservation (2/λ = 1/λ 3 + 1/λ 4 ) it becomes

c(ℓ 3 , L-ℓ 3 ) = L! ℓ 3 !(L-ℓ 3 )! λ L 2 L λ ℓ3 3 λ L-ℓ3 4 ( 15
)
Eq.15 shows a crucial dependence of the output versus the ratio λ4 λ3 . This is illustrated in the following section.

IV. APPLICATION TO ASYMMETRIC AND SYMMETRIC SCHEMES OF RUBIDIUM

In this section the LG-decomposition method is applied to rubidium schemes of Fig. 1, to compare with available experimental data (asymmetric) or to predict patterns in the case of symmetric scheme.

A. Probability amplitudes

Fig. 3 presents the probability amplitudes c(ℓ 3 , L-ℓ 3 ) versus the total input OAM L for ℓ 3 = 0 to 4. Each plot color represents a pair family, blue for the (0, L) one, orange for (1, L-1) one and so on. The insets show the first L for which probability amplitudes are equal, namely L = 12 for the asymmetric scheme and L = 1 for the symmetric one.

For the asymmetric scheme, Fig. 3(a) shows that the (0, L) pair dominates in the output if L < 12. In this case the input OAM L is mostly transferred to wave 4 (i.e. the blue light). For L > 12 the pair (1, L-1) becomes the most probable in the output. Around L = 12 the output contains mainly the two pairs, [START_REF]) if ℓ is not integer, Γ being the gamma function[END_REF][START_REF] Walker | Trans-Spectral Orbital Angular Momentum Transfer via Four-Wave Mixing in Rb Vapor[END_REF] and [START_REF] Coullet | Optical vortices[END_REF][START_REF] Chopinaud | High helicity vortex conversion in a rubidium vapor[END_REF], the other being of weaker probability. In this L region, ℓ 3 -ℓ 4 being entangled, the question of family entanglement is open.

The symmetric scheme (Fig. 3(b)) has a completely different behavior. Plots probability amplitudes exhibit maximums in succession, each maximum corresponding to the pair (L/2, L/2), which indicates an equitable OAM-sharing in the output waves. Fig. 3(b) shows more crossing of the families, also at low values of L, indicating a multimode character which appears at low L values. For example, at L = 1 the pairs (0, 1) and (1, 0) are 50/50 mixed, which is different in a asymmetric case.

B. Experimental SFWM with vortex beams

Some elements are given here to describe the detected signals in experiments and their signatures.

The principle of experiments reported in [START_REF] Chopinaud | High helicity vortex conversion in a rubidium vapor[END_REF][START_REF] Walker | Trans-Spectral Orbital Angular Momentum Transfer via Four-Wave Mixing in Rb Vapor[END_REF][START_REF] Akulshin | Arithmetic with optical topo-logical charges in stepwise-excited Rb vapor[END_REF][START_REF] Liu | Manipulation of the orbital angular momentum via fourwave mixing in Rb vapor[END_REF] consists to apply to a rubidium vapor at about hundred degrees, two lasers beams previously phase-shaped into a LG mode, in general using a spatial light modulator. The beams are superimposed colinear co-propagating and with the same circular polarization in order to optimize the two-photon transition.

The waves generated by SFWM are extracted by wavelength separation (filter or prism) and analyzed in intensity and in phase. In the mentioned experiments the generated blue beam at 420 nm has been detected. The intensity after propagation (at z = ∞) has been recorded on a CCD camera. The phase of the beam is analyzed by optics interference.

Fig. 4 gives a scheme of the detection used by A. Chopinaud in his PhD work [START_REF] Chopinaud | Atomes et vortex optiques : conversion de moments orbitaux de lumière en utilisant la transition à deux photons 5S-5D du rubidium[END_REF] with intensity detection on CDD1 and phase detection on CCD2 after passing through a modified Mach-Zehnder interferometer. The "Dove-Mach-Zehnder" (DMZ) interferometer includes a Dove prism which returns the field. If the DMZ interferometer is addressed by a pure LG beam carrying the OAM ℓ then it produces an azimuthally-modulated pattern (flower pattern) having a 2ℓ spatial frequency. Fig. 4 shows the intensity and interference pattern of the blue SFWM-generated beam obtained for L = 2.

C. OAM-entanglement signatures

The LG-decomposition model is used to compute intensity and DMZ interference patterns in order to compare with experimental ones. We examine two situations, one where the generated fields are fully coherent, the other with partial coherence.

For both cases the field is considered at z = ∞. The modification to put concerns the Gouy phase which change for each LG mode due to the propagation. For the ℓ 4 component the Gouy phase is then π(ℓ 4 + 1)/2.

For fully coherent fields the beam intensity is

I ∞ = L ℓ4=0 c(L-ℓ 4 , ℓ 4 ) ℓ 4 0 2 (16)
and the DMZ pattern is

I DMZ ∞ = L ℓ4=0 c(L-ℓ 4 , ℓ 4 ) ℓ 4 0 + -ℓ 4 0 2 (17)
Eqs.16-17 computed for L = 2, 10, 20 with experimental parameters given in [START_REF] Chopinaud | Atomes et vortex optiques : conversion de moments orbitaux de lumière en utilisant la transition à deux photons 5S-5D du rubidium[END_REF] provide plots of Fig. 5(a). They have been scaled to be compared to Fig. 5(c) pictures. The wave 4 intensities exhibit shapes as crescent moon, which is not the case in Fig. 5(c). Even if we introduce a relative azimuthal phase between the LG modes component (also random) the intensity is never perfectly annular like in Fig. 5(c). The DMZ interference patterns show large modulations on the bright fringes. For L = 10 and 20 the bright fringes intensity goes down the mean intensity value. Also for L = 20 a part of the pattern is nearly dark. Such modulation is not observed on the experimental data. That suggests a partial coherence.

With the assumption of partially coherent mode composition the intensity is

⟨I ∞ ⟩ = L ℓ4=0 c 2 (L-ℓ 4 , ℓ 4 ) ℓ 4 0 2 (18)
and its DMZ pattern is In addition we observe blurred regions and an alternating contrast along the light ring. Such an observation is viewed in the experimental patterns, making it difficult to count the fringe number. The alternating contrast is well-reproduced if the relative phase between the modes equals zero. This indicates that the generated modes keep the same starting point for their azimuthal phase. It is imposed by the input vortex beams and the third order non-linear effect.

⟨I DMZ ∞ ⟩ = L ℓ4=0 c 2 (L-ℓ 4 , ℓ 4 ) ℓ 4 0 + -ℓ 4 0 2 (19)
Even if the output is a partially coherent superposition of modes, signatures of the mode composition and thus of the output pair families still persist in the DMZ interference pattern. They are the fringes, their number and the contrast. In principle the decomposition of the blue beam over LG modes could be extracted either by a fit procedure or by a decomposition of the DMZ azimuthal profile as a Fourier series. It would be tested in a forthcoming work using systematic data and a wide range of input values. The key to get the information relies on the generation in phase in SFWM, of the two generated waves. To end this study, we aim to predict at OAMentanglement signatures for the symmetric scheme. For that, the DMZ interference patterns have been computed using this LG-decomposition model. The results for L = 2 and 10 are shown in Fig. 6(b,c).

The DMZ interference patterns have completely different forms from those obtained for asymmetric case. They exhibit fringes with a stronger blurring and the number of fringes is far from 2L. It is due to the output decomposition (orange bars in Fig. 6(a)) which is broad and has a maximum at ℓ 4 ∼ L/2. The probabilities for the asymmetric scheme is rather maximum at ℓ 4 ∼ L (blue bars in Fig. 6(a)).

The prediction could be validated with experiments. The deduced decomposition of the blue beam over LG modes using a Fourier series transform of the DMZ interference patterns could be tested.

V. CONCLUSION

The LG-decomposition method to determine generated pairs in the SFWM process has been developed providing an analytical formula (Eq. 14) of mode probability amplitudes. The formula is valid for any four-level scheme of atomic systems.

The method has been applied to analyze experimental data previously obtained for the asymmetric scheme of rubidium. It shows that SFWM produces OAMentangled pairs which are partially-coherent but which have the same azimuthal phase starting point. This confirms the strong constraint in phase imposed by the third order non-linear process and that its 'spontaneous' character is not intuitive.

In addition, the LG-decomposition method gives prediction about the symmetric four-levels scheme, with sig-natures in the DMZ pattern expected to be different. This opens to experiments where multi-dimensional entanglement would appear.

FIG. 1 .

 1 FIG. 1. Rubidium four-levels schemes and SFWM. (a) asymmetric scheme decaying via 6P 3/2 level; (b) the symmetric one decaying via 6P 3/2 level; (c) experiment scheme of SFWM operated with co-propagating vortex beams.

FIG. 2 .

 2 FIG. 2. Vortex beams overlap at the input and the output. (a) overlap at the input giving the working diagram; (b) overlap at the output for the asymmetric scheme; (c) overlap at the output for the symmetric scheme. The dark lines result from qualitative considerations as explained in section II.C.

FIG. 3 .

 3 FIG.3. Mode analysis of the output : probability amplitudes of modes versus the total input OAM. (a) for the asymmetric scheme; (b) for the symmetric scheme. Blue, orange, green, red and purple curves correspond to pair families (ℓ3, ℓ4) for = 0 to 4. The insets show details where the families (0, L) and (1, L-1) are crossing.

FIG. 4 .

 4 FIG. 4. Detection of the intensity of wave 4 (CCD1) and of its phase by a Dove-Mach-Zehnder interferometer (CCD2). Intensity and interference patterns recorded for L = 2.

Fig. 5 (

 5 Fig.5(b) shows the patterns computed by Eqs.18-19 with the same experimental parameter as before. As expected the wave 4 intensities are ring-shaped. The DMZ interference patterns keep azimuthal modulations, but the bright fringes never go down the mean intensity value. It corresponds to observed data of Fig.5(c).In addition we observe blurred regions and an alternating contrast along the light ring. Such an observation is viewed in the experimental patterns, making it difficult to count the fringe number. The alternating contrast is well-reproduced if the relative phase between the modes equals zero. This indicates that the generated modes keep the same starting point for their azimuthal phase. It is imposed by the input vortex beams and the third order non-linear effect.Even if the output is a partially coherent superposition of modes, signatures of the mode composition and thus of the output pair families still persist in the DMZ

FIG. 5 .

 5 FIG. 5. Intensities and DMZ interference patterns of wave 4, for the asymmetric scheme, for L = 2, 10 and 20. (a) fully-coherent emission; (b) partially-coherent case as explained in IV.C. (c) experimental data from [15], intensities on the left and DMZ patterns on the right. Red curves are azimuthal intensity profiles along the above indicated red dotted lines.

FIG. 6 .

 6 FIG. 6. DMZ interference patterns for the symmetric scheme, for L = 2 and 10. (a) Output mode probabilities, symmetric compared to asymmetric (blue). (b) Patterns for a fully-coherent emission. (c) Patterns for a partially-coherent emission. Red curves are azimuthal intensity profiles along the above indicated white dotted lines.

Acknowledgments: The authors thank Région Ile de france, DIM SIRTEQ (Science et Ingénierie en Région Ilede-France pour les Technologies Quantiques) and DIM QuanTiP (Quantum Technologies in Paris Region) for financial supports especially for PhD grant and for scientific equipments.

L.P. thanks Université Paris-Saclay (previous affectation) for supports allocated by the Labex PALM and by IQUPS, Laboratoire A Cotton (LAC, UMR 9025) for past technical supports, and Institut des sciences moléculaires d'Orsay (ISMO, UMR 8214) for the great hospitality during the laboratory renovation.


Selin Sariyer

Prof. Dr Serkan Yeşilot

Dr Nazmiye Kılıç

Dr Arpita Ghosh

Prof. Dr Ozlem Sel
email: ozlem.sel@college-de-france.fr

Dr Rezan Demir-Cakan
email: demir-cakan@gtu.edu.tr

Polyphosphazene Based Inorganic-Organic Hybrid Cathode Containing Pyrene Tetraone Sides for Aqueous Zinc-Ion Batteries

Keywords: Aqueous electrolyte Zn-ion batteries, hybrid electrodes, polyphosphazene, pyrenetetraone, EQCM

Aqueous rechargeable zinc-ion batteries (ARZBs) have been intriguing for electrochemical energy storage applications because of their safety and cost-effectiveness. Regarding cathode materials, rapid development has been observed with the organic-based cathode materials that offer much higher structural integrity upon successive (de-)insertion of charge carries ions. Even though promising results demonstrated with organic electrodes, they still suffer from the short cycle-life due to their discharge products solubility in electrolyte. Herein, electrochemical performance and charge storage mechanism of the synthesized polyphosphazene-based inorganic-organic hybrid electrode containing pyrene-4,5,9,10-tetraone (PTO) redox active lateral group, poly [(bis(2-amino-4,5,9,10-pyrenetetraone)], abbreviated as (PPAPT), were investigated in ARZBs. The charge storage mechanism of PPAPT was examined by various ex-situ (Fourier Transform Infrared Spectroscopy (FTIR), X-ray Diffraction (XRD), Scanning Electron Microscopy and Energy Dispersive X-ray Spectroscopy (SEM-EDS), X-ray Photoelectron Spectroscopy (XPS)) and in-situ (pH change with bromocresol green indicator and Electrochemical Quartz Crystal Microbalance (EQCM)) characterization techniques as well as computational Density functional theory (DFT) revealing that the PPAPT electrode (de-)coordinates both zinc and proton. The electrode and its discharge product are insoluble in the electrolyte demonstrated by UV-vis analysis and exhibited a stable cycling performance with a discharge capacity of 125.4 mAh g -1 after 1000 cycles at a current density of 10 C.

INTRODUCTION

Rechargeable batteries with aqueous electrolyte have received considerable attention due to their intrinsic safety, cost-effectiveness, environmentally benign nature as well as strong power capability resulting from high ionic conductivity. [1] In addition to the aqueous electrolyte batteries currently at the marketplace, i.e., lead-acid, nickel-iron, nickel-metal hydride or alkaline battery systems, many attempts have been made to development of aqueous rechargeable batteries (ARBs) based on monovalent (Li + , Na + and K + ) and multivalent metal ions (Zn 2+ , Ca 2+ Mg 2+ and Al 3+ ) chemistries. [1][2] Notably, multivalent ion ARBs are promising candidates in large-scale energy storage in which metal ions can transfer more than one electrons in redox reactions, thus attracting great interest due to their higher capacity and energy density. [3] Despite the presence of multivalent metals possessing highly abundant polyvalent cations in earth, ARBs composed of Al, Ca or Mg have limitations in finding suitable electrodes for reversible plating/stripping of these metals and some difficulties for accommodating large hydrated radii in the host structure. [2,4] Fortunately, zinc metal (-0.76 V vs SHE) has been directly employed as anode in ARBs due to its high reversibility in mild acidic aqueous electrolyte (3< 𝑝𝐻 < 7), high over-potential versus hydrogen evolution, high theoretical specific capacity (820 mAh g -1 ) as well as low manufacturing cost. [5][6] Hence, the merits of zinc anode make aqueous rechargeable zinc-ion batteries (ARZBs) an up-and-coming system for grid-scale storage. [7][8] Regarding the development of cathode materials, various transition metal oxide-based (vanadium oxide [9] , manganese dioxide [10] ) or organic electrode (quinone-based [11] , conductive polymer [12] ) materials are proposed that play a significant role for providing improved electrochemical properties and cycling performance. [13] Considering the environmental problems and the depletion of natural resources, it is highly desirable to explore renewable electrode materials for ARZBs, consisting of abundant C, H, O, N and S elements on Earth, which can be obtained artificially or directly from biomass sources. [14] To do so, followed by polyaniline (PANI) [15] conductive polymer used in zinc ion batteries back in 1986, several carbonyl [16] or imine compounds [17][18] , organic radicals such as nitronyl nitroxide [19] , organosulfur polymers [20] , metal organic frameworks (MOF) [21] and covalent organic frameworks (COF) [22] have been extensively studied in ARZBs owing to the diverse molecular structures and exhibit remarkably different electrochemical behaviors. [1,14,23] Among all these electroactive green and sustainable electrode materials, with carbonyl electrodes, i.e., quinones, anhydrides, imides, carboxylates or ketones, electrochemical charge storage occur through the multi-electron enolization reactions in which the good reversibility and high theoretical capacity can be achieved. [1,[24][25] Initially, quinone-based calix(4)quinone (C4Q) cathode composed of eight carbonyls, which reversibly coordinates zinc ions, exhibiting long-cycle stability (87 % capacity retention at 500 mA g -1 after 1000 cycles) and high capacity (335 mAh g -1 at 20 mA g -1 ) was reported. [11] Later, pyrene-4,5,9,10-tetraone (PTO) electrode was reported in ARZBs, particularly providing insoluble discharged compound of PTO in the zinc sulphate electrolyte. [16] In this context, PTO has been utilized as a redox-active cathode in ARZBs because of alluring features not only for its high rate capability, high theoretical capacity but also sustainability and flexibility. [26] In another work, two-dimensional (2D) covalent organic framework (COF) (Tp-PTO-COF) cathode synthesized by condensing 1,3,5-triformylphloroglucinol (Tp) with diamino-PTO via acid-catalyzed solvothermal method which efficiently and reversibly stored zinc ions, leading to a noteworthy capacity of 301.4 mAh g -1 at 0.2 A g -1 by courtesy of large number of nucleophilic active sites and porous structure. [27] A similar recent study has reported that a covalent organic framework using benzenetricarboxaldehyde (BT) and a PTO active monomer, (BT-PTO COF), comprised of carbonyl groups and π-π stacking conjugation, produced by condensation solvothermal synthetic route which exhibited excellent cycling stability (after 10000 cycles 98 % capacity retention) and superior rate capability (108 mAh g -1 at 1785 C), facilitating Zn 2+ and H + insertion. [22] Moreover, Guo et al. reported on the involvement of hydronium ions capture/release in the redox reaction mechanism of PTO anode, and the full cell of PTO//MnO2@graphite felt possessed highly stable long cycle performance. [28] Sun et al. showed that Zn//PTO aqueous electrolyte containing battery system, in which the electrochemical performance at a very low temperature of -70 °C exhibited high rate performance (71 mAh g -1 at 3C) without significant capacity loss after 100 cycles. [29] In this work, we focus on polyphosphazenes, which present an inorganic main chain construction whose side groups can be either organic or inorganic units, consist of phosphorus and nitrogen atoms alternately. [30] They provide unique features such as flame-retardantcy [31] , electro-optical properties [32] , corrosion resistance [33] , thus they have been widely implemented into diverse applications from biomaterials [34] to battery components as anode [35] , cathode [36] , electrolyte [37] and separator [38] . In particular, while inorganic-organic hybrid materials of polyphosphazenes and its derivatives used in electrochemical applications [39] , i.e., supercapacitors, batteries or fuel cells, they have not been subjected into aqueous rechargeable zinc-ion batteries.

Herein, electrochemical performance and charge storage mechanism of poly[(bis(2-amino-4,5,9,10pyrenetetraone)] quoted as (PPAPT) [40] bearing pyrene-4,5,9,10-tetraone (PTO) as electroactive group, were investigated in 2 M ZnSO4 aqueous electrolyte. The electrochemical charge storage performance of PPAPT is evaluated by cyclic voltammetry (CV) and galvanostatic charge-discharge (GCD). Furthermore, assorted ex-situ characterization techniques such as Fourier Transform Infrared Spectroscopy (FTIR), X-ray Diffraction (XRD), Scanning Electron Microscopy and Energy Dispersive X-ray Spectroscopy (SEM-EDS), X-ray Photoelectron Spectroscopy (XPS) as well as several in-situ methods such as time-dependent pH change monitoring and electrochemical quartz crystal microbalance (EQCM) have been employed to elucidate charge storage mechanism of PPAPT in addition to the density functional theory (DFT) calculations.

Materials and Methods

Analysis Methods

PPAPT polymer was synthesized according to our previous literature report and their structural characterization was studied in detail. [40] Dissolution tests were performed in 2 M ZnSO4 solution for the pristine electrode as well as the electrode terminated after the first discharge. All dissolution tests were recorded with a Shimadzu 2101 UV spectrophotometer with quartz cuvettes at room temperature.

The ex-situ FT-IR analysis (Bruker Alpha-P, ATR) was performed to investigate the change in redoxactive groups. The XPS spectra were recorded using a Specs Flex-Mod system with a monochromatic Al Kα X-ray source. Theoretical calculation of the structures were performed by the B3LYP functional (Becke's Three Parameter Hybrid Functional using the LYP Correlation Functional) with the 6-31G(d,p) basis set. The visualization of molecular structure obtained from DFT was performed by Gaussian 5 package. The details were given in the supporting information (SI). Scanning electron microscopy (SEM) (FEI PHILIPS, XL30 SFEG SEM) conjugated with energy dispersive X-ray spectroscopy (EDS) was utilized to investigate the surface morphology and elemental analysis of the PPAPT electrode after the first discharge process. XRD patterns of pristine and fully discharged PPAPT electrode were recorded by using a Bruker-D8 Advance diffractometer (Bruker AXS).

Electrochemical Measurements

The PPAPT/KB electrode, which was prepared by mixing PPAPT and KB (Ketjen black) at a ratio of 30:70 by weight, was used directly as a cathode. Cyclic voltammetry (CV) and galvanostatic chargedischarge (GCD) measurements of the cell were performed in 2 electrode cells where PPAPT/KB was used as working electrode while zinc metal was used both as reference and counter electrode.

Electrochemical tests were performed in 2 M ZnSO4 aqueous electrolyte at a voltage range between 0.1-1.6 V vs Zn/Zn 2+ . In order to monitor the pH changes upon cycling, 4-5 drops of bromocresol green indicator solution, which was prepared by dissolving indicator powder in ethanol, was added into a standard aqueous electrolyte solution of 2 M ZnSO4. Then, the cuvette cell composed of PPAPT cathode (24:56:20 PPAPT, KB, PVdF (Polyvinylidene fluoride), respectively), zinc anode was cycled at C/2 in 2 M ZnSO4 aqueous electrolyte with pH indicator. The color change was observed during the first discharge-charge cycles. Bromocresol green indicator [41] was chosen because of its working pH changes, 3.8 (green color) and 5.5 (blue color). All electrochemical studies were performed with Bio-Logic VMP3 potentiostat/galvanostat. EQCM analyses were carried out using gold patterned AT-cut 9 MHz quartz resonators, acquired from AWS, Valencia, Spain. A slurry containing 24:56:20 wt% of PPAPT:KB:PVDF were prepared in NMP, which was drop-casted on the gold electrode (0.2 cm 2 ) of the quartz resonators. After drop-casting, the loaded resonators were dried in a hot-plate at 80 °C overnight. The mass loadings were estimated by measuring the resonant frequency change (f) of the quartz crystal before and after coating, which was converted to m, using Sauerbrey equation (Δf = -ks × Δm, where ks is the sensitivity coefficient). [42] Typical total loading of the quartz resonators ranges between 6-10 µg. The coated resonators, platinum wire and Ag/AgCl electrode (3M NaCl) were used as the working, counter and reference electrodes, respectively. EQCM experiments were conducted in 1 M ZnSO4 aqueous electrolyte, using a Bio-Logic electrochemical workstation (Biologic SP200) coupled with a SEIKO QCM922A device.

Results and Discussion

Electrochemical Performance of PPAPT/KB

Zinc ion storage performance of PPAPT [40] inorganic-organic hybrid electrode, synthesized by the following steps shown in Scheme 1, was performed in a Swagelok type cell using zinc metal as both counter and reference electrodes and glass fiber separator in 2 M ZnSO4 aqueous electrolyte. The combinations of organic active materials with conductive carbon matrix are commonly preferred to avoid unfavorable rate performance caused by poor conductivity of organic cathodes. [43] Additionally, the discharge capacity of PTO-based electrode has been shown to be highly sensitive to the amount of conductive additives. [44] Similarly, the influence of the type of carbon additives (CMK-3 and Super P) on electrochemical performance of pentacene-5,7,12,14-tetraone (PT) carbonyl cathode was investigated in aqueous zinc ion battery and it was concluded that PT incorporation in mesoporous CMK-3 carbon improved the capacity and rate capability. [45] Thus, herein, relatively large amount of conductive Ketjen black additive (70 %) was examined.

Scheme 1. Synthesis steps of PPAPT.

Firstly, electrochemical performance of PPAPT/KB cathode in slightly acidic 2 M ZnSO4 aqueous electrolyte was evaluated by cyclic voltammetry (CV) at a scan rate of 0.2 mV s -1 in the potential range of 0.1-1.6 V vs Zn/Zn 2+ . As depicted in Figure 1a, the two pairs of redox peaks in CV curve of PPAPT/KB were obtained at 0.85/0.76 V and 0.94/1.04 V vs Zn/Zn 2+ in the reduction and oxidation scans, respectively. Therefore, due to its relatively low operating voltage, PPAPT/KB can be also utilized as a suitable anode material in a full battery tests. [46] Furthermore, overlapping consecutive CV cycles at the Figure 1a indicate the stable cycle performance, which coincides well with galvanostatic discharge/charge performance at a current density of 5 C (Figure 1c). Likewise, two stage zinc ion (de-)intercalation redox reaction of PTO [16] and Tp-PTO-COF [27] including β-keto carbonyl groups is compatible with the CV result demonstrated here with PPAPT/KB. Moreover, the kinetics of the coordination reaction of charge carrier ions with PPAPT/KB was investigated by CV measurements at various scan rates (Figure S1a). The relationship between the anodic and cathodic peak current values (i, A) and the sweep rate (v, mV s -1 ) is expressed according to power-law i = a v b equation [47] , where a and b are constants. In Figure S1b, b values obtained from the slope of the straight line of the log(v)log(i) graph at different scan rates were found to be 0.71 for the anodic process and 0.70 for the cathodic process, indicating the fast kinetics of the ions. Similarly, it was reported that the mechanism of insertion of proton and zinc ions into orthoquinone covalent organic frameworks (BT-PTO COF) composed of PTO as a redox active monomer is mainly controlled by pseudocapacitance, leading to fast ion transfer. [22] ZnSO4 electrolyte. As seen in Figure 1c, initial discharge capacity was 148 mAh g -1 at 5C (1 C= 359 mA g -1 ). After 100 cycles, the discharge capacity of 139 mAh g -1 was obtained with 99 % coulombic efficiency during whole cycle. The flat and inclined plateau associated with the two pairs of redox peaks was observed in the discharge/charge voltage profile in Figure 1b. C-rate performance of the PPAPT/KB composite electrode at different current densities was shown in Figure 1d and the cathode provided a discharge capacity of 82 mAh g -1 at a current density of 20 C (~1.8 A g -1 ). Electrochemical performance of KB in 2 M ZnSO4 aqueous electrolyte was also evaluated (Figure S2a). At a current density of 0.09 A g -1 (corresponding to ~1 C for the composite), KB exhibited 29 mAh g -1 discharge capacity and its contribution to the discharge capacity is 44%. Different KB ratios were also tested in order to check the effect of carbon content. It is clearly seen that higher discharge capacity values are acquired with the increase in carbon content (Figure S2b). By decreasing the current density to 1 C, the PPAPT/KB electrode exhibited the highest discharge capacity of 153 mAh g -1 . Discharge capacities of 82, 85, 95, 110, 124 and 153 mAh g -1 were obtained at 20 C, 15 C, 10 C, 5 C, 3 C and 1 C current densities, respectively. Returning to 10 C current density during the C-rate performance test, a discharge capacity of 125.4 mAh g -1 was achieved over 1000 cycles, displaying a very stable cycle performance with a capacity retention of 96%.

The investigation of Charge Storage Mechanism of PPAPT/KB

Revealing pH Change by Bromocresol Green Indicator

The energy storage mechanism of numerous organic electrode materials in aqueous electrolyte zinc ion batteries is generally based on the following redox reactions: i) insertion of zinc ions with water molecules [48] , ii) reversible co-insertion [12,22] of H + and Zn 2+ , iii) bare H + or H3O + hydronium (in)coordination [28,49] . Moreover, the proposed reactions of redox active organic electrodes have been demonstrated by various characterization techniques such as in-situ EQCM [12,50] , Raman [22] and ex-situ SEM-EDS [51] , FTIR [51] , NMR [49,52] , XRD [22] , XPS [52] and DFT (density functional theory) computational methods [18] . To study the charge storage mechanism, first of all, we employed ex-situ XRD to observe the change in the pristine PPAPT/KB electrode and after the first discharge process in 2 M ZnSO4 electrolyte. The appearance of a new peak at 9.5 ᵒ in the XRD pattern of the discharged PPAPT/KB electrode indicates the formation of a Zn4SO4(OH)6•xH2O (ZHS) phase (Figure S3). This precipitate on the PPAPT/KB electrode is attributed to the reaction between OH -and ZnSO4 electrolyte as a result of proton insertion. [51] This phenomenon is also observed when the local electrolyte pH is above 5.5. [53] Obviously, the increment in pH during discharge/charge process can be monitored by the addition of pH indicator in electrolyte solution. [41,54] To perform such analysis, PPAPT/KB cathode as a working electrode and zinc anode as both counter and reference electrode assembled in a cuvette cell was cycled at a current density of C/2, and the photos of the cell were taken every one minute throughout the entire electrochemical performance. First of all, the chemical stability of the Bromocresol green indicator as electrolyte additive and its contribution to the redox reaction were performed by cyclic voltammetry test before galvanostatic discharge/charge cycles. The CV results of electrode in the electrolyte with and without pH indicator as seen in Figure S4 indicate that pH indicator has no significant influence on the anodic and cathodic peak positions. At the beginning of the experiment, the color of the electrolyte (2 M ZnSO4 with Bromocresol green indicator additive) was dark green due to acidic nature and the pH was measured to be 3.87 with a pH meter. Upon discharging, especially when the voltage approaches to 0.8 V vs Zn/Zn 2+ , the color started to turn dark blue as clearly seen in Figure 2. When the PPAPT//Zn full cell was completely discharged to 0.1 V vs Zn/Zn 2+ , the color of electrolyte changes to entirely blue.

The pH of the fully discharged electrolyte was 5.47 when measured with a pH-meter and between 5-6 with litmus paper as demonstrated in Figure S5a. However, while the color of the electrolyte remained unchanged during the first charge, the pH value of the electrolyte presents a reasonable decrease to 4.85 at the end of the first charge. All the changes visualized in the cuvette cell are summarized in Figure 2.

These observations imply that the proton can be irreversibly coordinated to PPAPT/KB electrode. To clarify irreversible proton insertion, Coulombic efficiency can be evaluated at slow current density. At C/2, initial discharge and charge capacity of cuvette cell was 250 mAh g -1 and 151 mAh g -1 , respectively and thus initial Coulombic efficiency is quite low (60 %). Moreover, the association of proton insertion with poor Coulombic efficiency as a result of the competitive hydrogen evolution reaction has been reported in the literature. [55] It is important to note that, the back side of zinc anode close to the cuvette wall turns out to be blue as soon as contacting with zinc sulphate electrolyte in the presence of the indicator (Figure S5b). The same behavior was also observed previously with MnO2//Zn cell when the cell was left at open circuit voltage. In the mentioned cell, the color of the electrolyte containing pH indicator changed spontaneously from green to blue due to the reaction between the zinc metal and proton arising from the electrolyte. [54] Figure 2. Voltage-time curve of PPAPT//Zn cell assembled in a spectrophotometer cuvette, colorful marked points representing the corresponding pictures taken at the discharge/charge state (above), pictures of the prepared PPAPT//Zn cell using pH indicator resulting in a pH change during discharge and charge process (below). Red and black clamp connected to PPAPT electrode and zinc anode, respectively.

After displaying the pH change at the electrolyte/electrode interface, the typical flake-like formation of ZHS (Zn4SO4(OH)6•xH2O) on discharged PPAPT/KB electrode was further characterized by SEM-EDS. SEM image in Figure S6 clearly illustrated the characteristic structure of ZHS on PPAPT/KB electrode. Moreover, EDS (Figure S6) analysis confirmed the atomic ratio of Zn:S as 4, indicating the production of ZHS. [51] As a result, the participation of proton in charge storage mechanism was revealed by pH increase at the electrode/electrolyte interface as well as the formation of ZHS on electrode by XRD and SEM-EDS. Similarly, in a recent study, it was reported that the hydronium was involved in the storage mechanism of the 2,5-dichloro-1,4-phenylene bis((ethylsulfonyl)amide) electrode providing a very stable cycle performance. [56]

Scrutiny of Redox-Active Group

The active materials dissolution is a major issue for organic electrode materials in many battery systems. [57] Therefore, we studied the solubility of the PPAPT/KB electrode by using UV-Vis spectroscopy. Dissolution test was performed by immersion of PPAPT/KB (pristine and discharge state) in 4 mL of 2 M ZnSO4 electrolyte. It was clear that PPAPT/KB pristine and discharge states were colorless in electrolyte solution over five days, (inset Figure 3a). The PPAPT/KB pristine solutions (black line) have very weak absorption peaks in UV-Vis spectra, because of very minor solubility in the electrolyte (Figure 3a). The solubility of organic molecules in water can affect polarity, molecular weight, and hydrogen bond formation. [58] Thus, PPAPT/KB electrode, despite its polymeric structure, resulted in a very low solubility, probably by virtue of the weak hydrogen bond formation. Moreover, no absorption peaks were observed in the discharge solutions (blue line). Consequently, it is not expected that PPAPT/KB can dissolve due to polymeric structure and steric hindrances in aqueous electrolytes. The electrochemical reaction mechanism of the PPAPT/KB was further investigated by various ex-situ characterizations for the different states in Figure 3c. With the characteristic of C=O bonds, PPAPT is assumed to possess the same active sites as previously reported with quinone materials for batteries. [59- 60] To verify that, the time-dependent discharge-charge curve of PPAPT/KB (Figure 3b) in the first cycle was investigated and the samples were analyzed ex-situ by FT-IR. To do so, the peaks intensity variations of C=O and C-O during discharge and charge states was examined. Figure 3c showed that the vibration peak of the C=O bond at 1673 cm -1 gradually diminishes in the discharge process and returns to its similar pristine state in the charging process. Additionally, the vibration of the C-O bond (located at 1391 cm -1 ) appeared in the discharge state, while almost disappearing at the charged state. The opposite of the change in C=O bonds was observed in C-O bonds. To further reveal both the quantification of elements in PPAPT and the surface chemical compositions of PPAPT (without KB), XPS were recorded and shown in Figure S7. XPS survey spectrum in Figure S7a depicts mainly the presence of C, O, P and N in the obtained material. Additionally, quantitative analysis of PPAPT was provided by XPS, indicating a composition of 59% C, 22% O, 7% N and 2% P.

In the detailed analysis of the P2p spectrum (Figure S7b), the peaks at 134.1 eV and 133.1 eV are attributed to P=N and P-N, respectively. [61][62] Deconvoluted N1s spectrum seen in Figure S7c possesses four peaks with binding energy at 398.4 eV, 399.2 eV, 399.8 eV and 400.8 eV, which are assigned to P=N, C-N, P-N and amino groups (N-H), respectively. [63] The C1s spectrum of PPAPT (Figure S7d) displays the binding energies of 286.5 eV and 284.3 eV, which is ascribed to C=O and C-C, respectively. [60] The ex situ XPS was also utilized to demonstrate the charge storage properties of the resulting PPAPT/KB electrode over the first discharge/charge process (Figure 4a). As seen in Figure 4b, an intensity in the increase of C-O and a decrease in C=O peaks were observed at the discharge state in the C1s spectrum. In addition, the O1s spectrum revealed that the C=O and C-O peaks at 533.5 and 532.2 eV presented in the discharge state due to the coordination of cations (Figure 4c). At the same time, the peaks of Zn 2p1/2 and Zn 2p2/3 appear at the discharged state, in which the peaks at 1046.9 and 1023.5 eV are very strong due to the cation coordination into the PPAPT/KB (Figure 4d). [64] Ultimately, XPS data is consistent with the results of FT-IR spectra. [65] As a whole, these results supported that the C=O groups are a redox active side of PPAPT/KB electrode and the Zn 2+ or H + cations could be reversibly coordinated/de-coordinated from the PPAPT/KB electrodes during the discharge-charge process.

EQCM analysis

Interfacial processes occurring during the electrochemical cycling of the composite electrode is studied by EQCM (Figure 5). A cathodic and an anodic scan of the potential result in frequency decrease and increase, respectively (Figure 5b). This observation indicates that reduction and re-oxidation processes lead to a global mass increase and subsequent decrease of the electrode, probably with a charge compensation mechanism by cationic species. To identify the nature of the species, mass change (m) of the microbalance and simultaneous charge variation (Q) during the anodic/cathodic scan can be used, to estimate the mass per electron (M.P.E. = n × F × (m/Q)) values. Figure 5c shows MPE estimation for a negative scan of the potential taken at 20 mV s -1 , at least three slopes can be distinguished in the m versus Q curve, with significantly higher M.P.E. values (i.e., 204, 739, 423 g mol -1 ) than those corresponding to M.P.E of Zn 2+ (32.7 g mol -1 ) or H3O + (19 g mol -1 ). The same observation is also valid for higher scan rates studied as shown in Figure S8a and S8b. It is noted that the above estimation considers that the QCM acts as a gravimetric probe, (a linear relationship between the Δf and Δm). [42] However, this is valid for a dense, flat and thin layer rigidly attached on the resonator.

In that case, the added layer behaves as an extension of the resonator and follows synchronously its motion without dissipation of energy. [42] This condition can be verified by simultaneously monitoring motional resistance change, Rm along with the frequency changes, f of the resonator. [66] Figure S9 shows a reversible frequency decrease/increase which is accompanied by a non-negligable increase/decrease of the Rm values, indicating a deviation from the linear relationship between the Δf and Δm. This can be attributed to at least two factors: (i) viscoelastic changes of the PPAPT film due to Zn 2+ (de-)coordination during an anodic/cathodic scan of the potential and (ii) a concomitant H + (de-)coordination along with Zn 2+ which results in local pH changes and thus, ZHS formation. [12,[67][68] Since, the f and Rm do not show a significant irreversibility (Figure 5b, 5d and S9), we can consider that the initial state of the electrode is restored within one electrochemical cycle. This observation is in perfect agreement with the FT-IR and XPS analyses above demonstrating that Zn 2+ or H + cations can reversibly (de-)coordination the PPAPT/KB electrodes during the discharge-charge process. Furthermore, the PPAPT/KB electrode shows excellent stability over cycling without any loss of the amplitude in current and frequency variation (Figure 5d).

Computational Method

Theoretical calculations were performed to get more insights into the electrochemical properties of the PPAPT electrode, in which the repeating monomer units called mPPAPT (mP) in the polymer chain, were assumed as the simplified molecule (Figure 6a). The binding energy with the optimized structure of mPPAPT (abbreviated as mP) units during the multistep binding processes was individually evaluated through DFT simulation (Figure 6b). The binding energy of all molecules bonded with 1-8 electrons are negative, indicating that PPAPT electrodes are able to accommodate a maximum of 4 Zn ions or 8 H ions in theory or in among of thesemixed ion coordination. For this reason, the binding energy of the zinc and hydrogen ions with these monomer molecules was investigated with the different numbers of cations, and all results were given in Figure 6b and Table S1. We calculated a total of 24 possible geometrical configurations of mP and their Gibbs free energies were obtained. The most stable configuration of each chemical formula was selected by comparing their single point energy. The mP structure has three main coordination types (only H + , Zn 2+, or both) and eight main coordination steps as shown in Figure 6b. The tendency to decrease the binding energies formed by the addition of each ion indicates that cation coordination can occur in the structure through a multistep coordination pathway (Table S1). As shown in Figure 6b, the eight-step H + coordination process is seen as the most ideal route for the mP structure, while only Zn 2+ coordination is seen as the most difficult route in cation coordination (Figure S10). However, both H + and Zn 2+ coordination have increased possibility more than that of bare Zn 2+ coordination. After all, the DFT calculation results demonstrate that H + can be efficiently coordinated into the mP structure. The theoretical calculations are based on thermodynamics but do not take into consideration some of the experimental parameters (concentration of Zn 2+ /H + in electrolyte, ionic size, desolvation energy etc). [69] Besides these reasons, while the formation of mPH8 structure is the most possible than the only Zn 2+ containing configurations, both Zn 2+ and H + consisted structure is more probable to obtain under the experimental conditions since the concentration of Zn 2+ is much higher than that of H + in the 2 M ZnSO4 electrolyte. [57,69] For this reason, when the DFT results and experimental conditions are taken into account, different discharge products containing H + and Zn 2+ ions can be obtained. This situation and also the polymer structure of PPAPT will cause many different mechanisms to be formed during the charge-discharge process. PPAPT electrodes are able to accommodate number of ions to charge compensate of 8e -in theory. Considering the maximum number of electrons, possible electrochemical mechanism and the simplified electrochemical process of PPAPT are given respectively in Figure S11a and Figure S11b. Therefore, when other studies with organic electrodes containing pyrene-4,5,9,10tetraone (PTO) group are compared with this current study (Table S2), it can be concluded that the electrochemical performance of PPAPT/KB electrode exhibited comparable results with the literature studies.

Conclusion

The electrochemical performance of poly[(bis(2-amino-4,5,9,10-pyrenetetraone)]-phosphazene (PPAPT) was investigated in 2 M ZnSO4 aqueous electrolyte, exhibiting a stable cycling performance with a discharge capacity of 125.4 mAh g -1 after 1000 cycles at a current density of 10 C. Further, charge storage mechanism of PPAPT was elucidated by various ex-situ and in-situ characterization methods.

The formation of the Zn4SO4(OH)6•xH2O (ZHS) structure, which indicates the coordination of the proton, is evidenced by the formation of a sharp diffraction peak in the XRD spectrum of the discharge electrode and the typical flake-like structure in the SEM image, as well as in situ EQCM analyses. The pH increase caused by the coordination of the proton with PPAPT was also shown with bromocresol indicator. Furthermore, it was confirmed by XPS and FTIR analyses that the redox active site of the PPAPT bearing 4,5,9,10-pyrenetetraone cathode was the C=O bond. DFT computational method further suggested both Zn 2+ and H + cations coordination into PPAPT electrode. Eventually, phosphazene -based cathode materials, which form a novel inorganic-organic hybrid class of materials in aqueous rechargeable zinc-ion batteries, are potential candidate with their good electrochemical performance.

Table of Contents

The

Figure 1 .

 1 Figure 1. Electrochemical performance of PPAPT/KB electrode, (a) Cycling voltammetry at a scan rate of 0.2 mV s -1 , (b) Discharge-charge profile at a current density of 5C, (c) Cycling performance at 5C, (d) C-rate performance (inlet figure shows the first 80 cycles of the C-rate test)

Figure 3 .

 3 Figure 3. (a) UV-Vis spectra of PPAPT/KB in 2M ZnSO4 (inset photo: ZnSO4 electrolyte solution, PPAPT/KB pristine and its discharge state immersed into electrolyte solutions, (b) Time dependent discharge-charge curve of PPAPT/KB at a current density of C/5 the first cycle, (c) Ex-situ FT-IR spectra of PPAPT/KB in different states (pristine, discharged to and charged state).

Figure 4 .

 4 Figure 4. XPS spectra of (a) PPAPT/KB cathode material after the first discharge cycled at C/5 current density, (b) C1s spectra, (c) O1s spectra of pristine and discharge state, (d) Zn 2p spectra of the pristine and discharge state.

Figure 5 .

 5 Figure 5. EQCM analysis of PPAPT/KB cathode material in 1M ZnSO4 electrolyte. (a) CV curves and (b) corresponding frequency changes, measured at different scan rates (20, 50, 100 mV s -1 ); (c) Mass per electron (M.P.E.) calculations, shown for the reduction branch of the EQCM data at 20 mV s -1 and (d) current and frequency response as a function of time, measured at 100 mV s -1 .

Figure 6 .

 6 Figure 6. (a) The structure of mPPAPT (mP), (b) The binding energies with different numbers of Zn 2+ /H + ions in mP structure.

  electrochemical performance of poly[(bis(2-amino-4,5,9,10-pyrenetetraone)] electrode in zinc sulfate electrolyte demonstrated very stable discharge/charge cycling process. In-situ EQCM and visualization of pH change with indicator additive to electrolyte revealed the zinc and proton participation to charge storage mechanism in addition to the DFT calculation.

Figure S3 .

 S3 Figure S3. (a) XRD pattern of PPAPT/KB electrode and fully discharged PPAPT/KB electrode.

Figure S4 .

 S4 Figure S4. CV curves of PPAPT/KB electrode in the electrolyte with Bromocresol green indicator and without electrolyte additive at a scan rate of 0.2 mV s -1 .

Figure S5 .

 S5 Figure S5. (a) The cell completely discharged and its pH is between 5-6 as depicted by a litmus paper, (b) Back side of zinc anode showing blue colour formation direct contact with 2 M ZnSO4 electrolyte containing bromocresol green indicator.

Figure S6 .

 S6 Figure S6. SEM images of PPAPT/KB electrode after the first discharge at different magnification (left side), EDS spectrum and corresponding elemental analysis of discharged PPAPT/KB electrode (right side).

Figure S7 .

 S7 Figure S7. XPS analysis of PPAPT, (a) survey spectrum, high-resolution spectrum of (b) P2p, (c) N1s, (d) C1s.

Figure S8 .Figure S9 .Figure S10 .

 S8S9S10 Figure S8. EQCM analysis of PPAPT/KB cathode material in 1M ZnSO4 electrolyte. Δm vs. ΔQ

Figure S11 .

 S11 Figure S11. a) The possible electrochemical mechanism of PPAPT during the discharge and charge process. b) The simplified electrochemical process of PPAPT.

Acknowledgement

The authors appreciate the financial support from TUBITAK (119N054). TUBITAK ULAKBIM, High Performance and Grid Computing Center (TR-Grid e-Infrastructure) is thanked for the calculations data in the computational chemistry details of this paper.

DFT Calculation

Density functional theory (DFT) was employed to optimize the stabilized structures of mPPAPT with Gaussian 16 software package. [1] All calculation were examined the B3LYP functional with 6-31G(d,p) basis sets and the results were compared with the monomer molecule (mPPAPT). All frequency and single point energy calculations were carried out at the same level in order to verify the optimized structures. In addition, the solvation effect was considered by employing the integral equation formalism variant of the Polarizable Continuum Model (IEFPCM), and water was used as the solvent. [2] The binding energy ( E b ) of mPPAPT-HaZnb was computed based on the following equations [3] .

𝐸b(Binding Energy) = 𝐺 mPHaZnb -𝐺mP -a𝐺 H + -b𝐺Zn2+ 2 M ZnSO4 145 mAh g -1 after 1000 cycles at 3 A g -1 capacity retention 70% [4] PTO-4NH2Ph 3 M ZnSO4 125.4 mAh g -1 at 100C (25 A g -1 ) 83% capacity retention at 5 A g -1 after 5000 cycles [5] Tp-PTO-COF 2 M ZnSO4 218.5 mA h g -1 at 2 A g -1 over 1000 cycles with 95% retention [6] BT-PTO COF 3 M Zn(CF3SO3)2 210 mAh g -1 at 5 A g -1 after 10 000 cycles 98 % capacity retention [7] PPAPT 2 M ZnSO4 139 mAh g -1 at 5 C (0.45 A g -1 ) after 100 cycles with 94% retention, and 125.4 mAh g -1 at 10 C after 1000 cycles with 96% retention

This study