Michel Planat

On small proofs of the Bell-Kochen-Specker theorem for two, three and four qubits

Keywords: numbers: 03.65.Ta, 03.65.Fd, 03.65.Aa, 03.67.-a, 02.10.Ox, 02.20.-a Bell-Kochen-Specker theorem, quantum contextuality, multiple qubits

The Bell-Kochen-Specker theorem (BKS) theorem rules out realistic noncontextual theories by resorting to impossible assignments of rays among a selected set of maximal orthogonal bases. We investigate the geometrical structure of small v -l BKS-proofs involving v real rays and l 2n-dimensional bases of n-qubits (1 < n < 5). Specifically, we look at the parity proof 18 -9 with two qubits (A. Cabello, 1996 [3]), the parity proof 36 -11 with three qubits (M. Kernaghan & A. Peres, 1995 [4]) and a newly discovered non-parity proof 80-21 with four qubits (that improves work of P. K Aravind's group in 2008 [5]). The rays in question arise as real eigenstates shared by some maximal commuting sets (bases) of operators in the n-qubit Pauli group. One finds characteristic signatures of the distances between the bases, which carry various symmetries in their graphs.

Introduction

Contextuality is an important hallmark of quantum mechanics. In a contextual world, the measured value of an observable depends on which other mutually compatible measurements might be performed. In this line of thought, the Bell-Kochen-Specker (BKS) theorem is fundamental because it is able to rule out non-contextual hidden variable theories [START_REF] Kochen | The problem of hidden variables in quantum mechanics[END_REF][START_REF] Peres | Quantum theory: concepts and methods[END_REF] by resorting to mathematical statements about coloring of rays located on maximal orthonormal bases in a d-dimensional Hilbert space (d ≥ 3).

A non-coloring BKS proof consists of a finite set of rays/vectors that cannot be assigned truth values (1 for true, 0 for false) in such a way that (i) one member of each complete orthonormal basis is true and (ii) no two orthogonal (that is, mutually compatible) projectors are both true [2, p. 197]- [START_REF] Aravind | Two noncolourable configurations in four dimensions illustrating the Kochen-Specker theorem[END_REF] ‡. The smallest state-independent proofs in three dimensions are of the 31 -17 type (31 rays located on 17 orthogonal triads) and the (closely related) 33 -16 type corresponding to a very symmetric arrangement of rays located on a cube of edge √ 2 [2, fig. 7.2, p. 198], see also [START_REF] Arends | On searching for small Kochen-Specker vector systems[END_REF]. ‡ Throughout the paper, the word proof is not taken in the strict mathematical sense as a list of logical statements, but as a set v -l of v rays and l maximal bases satisfying the BKS postulates/constraints.

The BKS theorem is intimately related to the coloring of a graph whose vertices are the rays and whose edges are the bases [START_REF] Cabello | Simple unified proof of state-independent contextuality[END_REF].

A parity proof of BKS theorem is a set of v rays that form l bases (l odd) such that each ray occurs an even number of times over these bases. A proof of BKS theorem is ray critical (resp. basis critical) if it cannot be further simplified by deleting even a single ray (resp. a single basis), see [12, p. 9] and [START_REF] Pavicić | Kochen-Specker vectors[END_REF] §. The smallest BKS proof in dimension 4 (resp. 8) is a parity proof and corresponds to arrangements of real states arising from the two-qubit (resp. three-qubit) Pauli group, more specifically as eigenstates of operators forming Mermin's square (2) (resp. Mermin's pentagram [START_REF] Cabello | Simple unified proof of state-independent contextuality[END_REF]) [START_REF] Mermin | Hidden variables and the two theorems of John Bell[END_REF]. In what follows, we shall investigate in detail the structure of the 18 -9 two-qubit proofs [START_REF] Cabello | Bell-Kochen-Specker theorem: A proof with 18 vectors[END_REF][START_REF] Waegell | Parity proofs of the Kochen-Specker theorem based on the 24 rays of Peres Found[END_REF], that of the 36 -11 three-qubit proofs [START_REF] Kernaghan | Kochen-Specker theorem for 8-dimensional space[END_REF], and the related small proofs. Moreover, we shall improve the earlier four-qubit 80 -265 proof [START_REF] Harvey | BKS theorem and Bell's theorem in 16 dimensions[END_REF] by simplifying it to a 80 -21 one.

Our overall goal in this paper is to gain a deeper understanding of the algebraic and geometrical structure of the minimal BKS n-qubit proofs. This is not a straigthforward task because there exists a plethora of quantum states appearing as eigenstates shared by the maximal commuting sets of operators in the n-qubit Pauli group. The total number of states is dL, where d = 2 n and L = n i=1 (1 + 2 i ) is the number of maximal commuting sets, see for example [13, eq. (16)]. The number of real states is found to be L R = n i=1 (2 + 2 i ), corresponding to the sequence {4, 24, 240, 4320, • • •} of kissing numbers in the Barnes-Wall lattice B n of dimension 2 n . One can ultimately expect a deep relationship between n-qubit BKS proofs and the B n 's (in the spirit of [START_REF] Planat | Multipartite entanglement arising from dense Euclidean lattices in dimensions 4 -24[END_REF]), but our goal here is more modest. We shall restrict the reservoir of real states to those generated by Mermin's square (24 states for two qubits), Mermin's pentagram (40 states for three qubits) and the magic rectangle [START_REF] Planat | Multipartite entanglement arising from dense Euclidean lattices in dimensions 4 -24[END_REF] found in [START_REF] Harvey | BKS theorem and Bell's theorem in 16 dimensions[END_REF] (80 states for four qubits) .

Apart from the use of standard graph theoretical tools for characterizing the ray/base symmetries, we shall employ a useful signature of the proofs in terms of Bengtsson's distance D ab between two orthonormal bases a and b defined as [15, eq. ( 2)]-[16]

D 2 ab = 1 - 1 d -1 d i,j | a i |b j | 2 - 1 d 2 . ( 1
)
The distance (1) vanishes when the bases are the same and is maximal (equal to unity) when the two bases a and b are mutually unbiased, | a i |b j | 2 = 1/d, and only then. We shall see that the bases of a BKS proof employ a selected set of distances which seems to be a universal feature of the proof.

The next three sections 2, 3, and 4 focus on two-, three-and four-qubit proofs built from the operators in the corresponding Pauli groups. We denote by X, Z and Y the Pauli spin matrices in x, y and z directions, and the tensor product is not explicit, i. § The authors of [START_REF] Pavicić | Kochen-Specker vectors[END_REF] discuss the KS theorem in terms of so-called KS sets and sometimes arrive at different counts for the minimal numbers of vectors required.

The BKS theorem also admits many proofs with complex rays as already shown for the two-qubit case [START_REF] Waegell | Parity proofs of the Kochen-Specker theorem based on 60 complex rays in four dimensions[END_REF]. e. in (2) one denotes Z 1 = Z ⊗ I, Z 2 = I ⊗ Z and ZZ = Z ⊗ Z, in (8) one denotes Z 1 = Z ⊗ I ⊗ I and so on, with I being the identity matrix of the corresponding dimension.

The symmetries underlying the proofs and the distances between the involved bases are revealed ¶. In some sense, quantum contextuality encompasses quantum complementarity by having recourse, not only to the maximal distance corresponding to mutually unbiased bases, but also to another set of distances which is a signature of the proof. Knowing the particular set of distances used in a proof of a given type, one is able to derive all proofs of the same type and their overall structure (at least for two and three qubits).

The BKS parity proofs for two qubits

The simplification of arguments in favour of a contextual view of quantum measurements started with Peres' note [START_REF] Peres | Incompatible results of quantum measurements[END_REF] and Mermin's report [START_REF] Mermin | Hidden variables and the two theorems of John Bell[END_REF]. Observe that in (2), the three operators in each row and each column mutually commute and their product is the identity matrix, except for the right hand side column whose product is minus the identity matrix. There is no way of assigning multiplicative properties to the eigenvalues ±1 of the nine operators while still keeping the same multiplicative properties for the operators + . Paraphrasing [START_REF] Peres | Incompatible results of quantum measurements[END_REF], the result of a measurement depends "in a way not understood, on the choice of other quantum measurements, that may possibly be performed".

| | || -Z 1 -Z 2 -ZZ- | | || -X 2 -X 1 -XX- | | || -ZX-XZ-Y Y - | | || (2)
The next step to be able to see behind the scene, and to reveal the simplest paradoxical/contextual set of rays and bases, was achieved by A. Cabello [START_REF] Cabello | Bell-Kochen-Specker theorem: A proof with 18 vectors[END_REF]. It is a 18 -9 BKS parity proof that can be given a remarkable diagrammatic illustration fitting the structure of a 24-cell [START_REF] Waegell | Parity proofs of the Kochen-Specker theorem based on the 24 rays of Peres Found[END_REF]. More generally, it is already known [START_REF] Waegell | Parity proofs of the Kochen-Specker theorem based on the 24 rays of Peres Found[END_REF][START_REF] Pavicić | Kochen-Specker vectors[END_REF] that there exist four main types of parity proofs arising from [START_REF] Liang | Specker's parable of the overprotective seer: A road to contextuality, nonlocality and complementarity[END_REF] Peres rays [START_REF] Peres | Quantum theory: concepts and methods[END_REF], that are of ¶ The notations we use are standard ones: the symbols × and mean the direct and semidirect product of groups, S n is the n-letter symmetric group and D n is the 2n-element dihedral group. + It is intriguing that such a property can be given a ring geometrical illustration by seeing Mermin's square as the projective line over the ring F 2 × F 2 (where F 2 is the field with two elements) and the right hand side column as the locus for pairs of units or pair of zero divisors of R. Ultimately, the geometry of the 15 two-qubit operators in the Pauli group has been found to mimic the generalized quadrangle GQ(2, 2), see [START_REF] Planat | Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function?[END_REF] and references therein.

the type 18 -9, 20 -11, 22 -13 and 24 -15. Types 20 -11 and 22 -13 subdivide into two non-isomorphic ones A and B as shown in Table 1.

For the list of the unnormalized eigenvectors (numbered consecutively) we use the same notation as [START_REF] Waegell | Parity proofs of the Kochen-Specker theorem based on the 24 rays of Peres Found[END_REF] 1 : (3)

The 24 complete orthogonal bases are as follows

Normalizing rays (3), a finite set of distances (1) between the 24 bases is found to be

D = {a 1 , a 2 , a 3 , a 4 , a 5 } = { 1 √ 3 , √ 7 √ 12 , √ 2 √ 3 , √ 5 √ 6
, 1} ≈ {0.577, 0.763, 0.816, 0.912, 1.000}.

Table 2 provides a histogram of distances for various parity proofs vl. Tables 1 and2 give all essential information about the proofs. First, a proof of a given type possesses a seemingly universal pattern in terms of the distances. Observe that the smallest proof does not contain any pair of mutually unbiased bases. Second, a given proof can be seen from its symmetry subsets, each one attached to a selected crossing graph (see the captions of Tables 1 and2). Then, one can create a graph having the bases as vertices and an edge joining two vertices if the two bases are in the proper range of distances. The cliques of the latter graph (not all maximal), of the selected odd size l, are candidates for a proof of the vl type, but not all of them provide proofs. This way, we could explicit all the proofs, 16 proofs of the 18 -9 type (as for the 24 -15 type) and 240 proofs of the 20 -11 type (as for the 22 -13 type), as reported, for example, in [START_REF] Waegell | Parity proofs of the Kochen-Specker theorem based on the 24 rays of Peres Found[END_REF]Table 2].

The 16 proofs of the 18 -9 type can be displayed as the 4 × 4 square (5) in which two adjacent proofs share three bases. Observe that each 2 × 2 square of adjacent proofs has the same shared base, which is taken as an index (e.g. the upper left-hand-side 2 × 2 square has index 7 and the lower right-hand-side square has index 10 G 72 . For the 16 proofs of the 24 -15 type, the symmetry is the same. Basically, still the same group governs the 240 = 96 + 144 proofs of the 20 -11 type (as well as the 240 proofs of the 22 -13 type), although there also exist some extra abelian symmetries. (For a nice geometrical display of the proofs, see [START_REF] Waegell | Parity proofs of the Kochen-Specker theorem based on the 24 rays of Peres Found[END_REF]).

# common elements

0 1 2 distance a 5 , a 4 a 2 a 1 24 -15 type G 72 Z 9 2 (D 4 × S 6 ) G 72 22 -13A type D 4 G 72 × S 4 D 4 22 -13B type D 4 Z 2 3 (Z 4 2 D 6 ) D 4 20 -11A type D 6 Z 2 3 P 1 Z 3 3 P 1 20 -11B type D 4 Z 2 3 P 1 Z 4 2 D 6 18 -9 type G 72 G 72 no overlap
The 16 proofs of the 24 -15 type (not shown) also form a 4 × 4 square in which two proofs share seven elements, comprising a common part of the six reference bases 1 -6 and an isolated base.

Diagrams for the proofs

How can we account for the distance signature of a given proof? A simple diagram does the job.

The diagram for the 18 -9 proof is simply a 3 × 3 square. Below we give an explicit construction of the first proof that corresponds to the upper left-hand-side corner in [START_REF] Harvey | BKS theorem and Bell's theorem in 16 dimensions[END_REF]. The 9 vertices of the graph are the 9 bases of the proof, the one-point crossing graph between the bases is the graph [START_REF] Aravind | Two noncolourable configurations in four dimensions illustrating the Kochen-Specker theorem[END_REF], with aut = G 72 = Z 2 3 D 4 . There are 18 (distinct) edges that encode the 18 rays, a selected vertex/base of the graph is encoded by the union of the four edges/rays that are adjacent to it.

As for the distances between the bases, two bases located in the same row (or the same column) have distance a 2 = 7/12, while two bases not in the same row (or column) have distance a 4 = 5/6 > a 2 , as readily discernible from Table 2 and the histogram in Table 1. Indeed, any proof of the 18 -9 type has the same diagram as [START_REF] Aravind | Two noncolourable configurations in four dimensions illustrating the Kochen-Specker theorem[END_REF].

Similar diagrams can be drawn to reflect the histogram of distances in proofs of a larger size. Below we restrict to the case of a 20 -11A proof (where only the distance between two bases is made explicit, but not the common rays of the bases) The proof consists of 11 bases, 9 of them have the same mutual diagram as in [START_REF] Aravind | Two noncolourable configurations in four dimensions illustrating the Kochen-Specker theorem[END_REF] and their mutual distance is a 2 = 7/12 (as shown) or a 4 = 5/6 (not shown), depending on whether they are located in the same row (or the same column) of the 3 × 3 square, or not. The extra two bases of the right-hand-side column are mutually unbiased (with distance a 5 = 1), their distance to any base of the same row is 1/ √ 3 and their distance to any base of the first row is a 4 (as shown).

The BKS parity proofs for three qubits

Quantum contextuality of a three-qubit system is also predicted in Mermin's report [START_REF] Mermin | Hidden variables and the two theorems of John Bell[END_REF] in terms of its famous pentagram. Below we display it in a sligthly different form in order to underline its kinship to the four-qubit "magic" rectangle (14). Mermin's rectangle/pentagram (8) features the same (real) operators as in [START_REF] Saniga | Mermin's pentagram as an ovoid of P G[END_REF] * .

| | | | Z 1 Z 1 X 1 X 1 | | | | Z 2 X 2 Z 2 X 2 | | | | Z 3 X 3 X 3 Z 3 | | | | = ZZZ = ZXX = XZX = XXZ = | | | | (8)
Following [START_REF] Mermin | Hidden variables and the two theorems of John Bell[END_REF], ( 8) is a parity proof of the BKS theorem because mutually commuting operators in the four columns multiply to the identity matrix while operators in the single row multiply to minus the identity matrix. Since each operator appears twice in this reasoning, it is impossible to assign truth values ±1 to the eigenvalues while keeping the multiplicative properties of the operators.

The list of (unormalized) eigenvectors coming from the five bases in ( 8) is (in the notation of [START_REF] Kernaghan | Kochen-Specker theorem for 8-dimensional space[END_REF]) These rays form 25 maximal orthogonal bases Table 4. The symmetries involved in various three-qubit parity proofs of the BKS theorem. The first row gives the number of common elements between the bases, this number being related in the second row to the distance between the bases. The five-letter symmetric group S 5 is an important building block symmetry of the proofs.

The finite set of distances involved is

D = { √ 3 √ 7 , √ 9 √ 14 , √ 6 √ 7 } ≈ {0.654, 0.801 , 0.925}.
It contrast to the two-qubit case, there is no set of mutually unbiased bases. Three types of parity proofs may be found, the 36 -11 type discovered in [START_REF] Kernaghan | Kochen-Specker theorem for 8-dimensional space[END_REF] and the two extra types 38 -13 and 40 -15. The same result (and much more) is found in [START_REF] Waegell | Proofs of the Kochen-Specker theorem based on a system of three qubits[END_REF]. Tables 3 and4 gather the main properties. As in the two-qubit case, one uses computer to construct a graph having the bases as vertices and an edge joining two vertices/bases at the proper distances. Then one extracts all sets of cliques, not necessarily maximal, of a given odd cardinality (that is eleven, thirteen and fifteen) and keeps those having the desired property of being parity proofs of the BKS theorem. Doing this, one gets an explicit list of 64 proofs of the 40 -15 type, 640 proofs of the 38 -13 type and 320 proofs of the 36 -11 type, totalling to 2 10 distinct parity proofs.

Below, we provide a short list of 36 -11 proofs: the 16 proofs containing bases 1, 2 and 3

Diagrams for the proofs

To be more explicit, the first parity proof in [START_REF] Peres | Incompatible results of quantum measurements[END_REF] consists of the eleven 8-ray bases [START_REF] Waegell | Parity proofs of the Kochen-Specker theorem based on the 24 rays of Peres Found[END_REF], where the four rays 12, 18, 25 and 38 do not appear and the remaining ones occur 2 or 4 times each As in the previous section, a simple diagram illustrates how distances between the bases are distributed. Let us look at the 36 -11 parity proof [START_REF] Waegell | Parity proofs of the Kochen-Specker theorem based on the 24 rays of Peres Found[END_REF]. The 11 bases are displayed as a pentagram (13) plus the isolated reference base 1.

Two adjacent bases of the pentagram have two rays in common. The reference base has with each of the bases on the horizontal line of the pentagram four rays in common and is disjoint from any other base. One can further observe that each line of the pentagram shares a set of four rays that is disjoint from the set of four rays shared by another line. The automorphism group of this configuration is isomorphic to S 5 .

The maximal distance, a 3 , is that between two disjoint bases, and amounts to 6/7. The intermediate distance, a 2 = 9/14, occurs between two bases located in any line of the pentagram. Finally, the shortest distance, a 1 = 3/7, is that between the reference base and each of the four bases on the horizontal line of the pentagram. Similar diagrams can be produced for any proof.

The BKS proofs for four qubits

The BKS theorem for four qubits was investigated in [START_REF] Harvey | BKS theorem and Bell's theorem in 16 dimensions[END_REF]. The "magic" rectangle [START_REF] Planat | Multipartite entanglement arising from dense Euclidean lattices in dimensions 4 -24[END_REF] (also shown in a pentagram form in [START_REF] Raynal | Mutually unbiased bases in six dimensions: the four most distant bases[END_REF]) is a parity proof similar to (2) and [START_REF] Cabello | Simple unified proof of state-independent contextuality[END_REF] because each operator appears twice, the mutually commuting operators in any column multiply to give the identity operator and the operators in the single row multiply to In [START_REF] Bengtsson | Mutually unbiased bases and Hadamard matrices of order six[END_REF], each ray is paired with a partner ray (possibly itself), which is obtained by inversion of the entries in the original ray. The concept of a partner ray allows us to convert a BKS proof (about contextuality) into a proof of Bell's theorem (about non-locality), as described in [START_REF] Aravind | Impossible colorings and Bell's theorem[END_REF].

In [START_REF] Harvey | BKS theorem and Bell's theorem in 16 dimensions[END_REF], a non-parity BKS proof 80-265 was proposed. Here we find a smaller 80-21 one. Our strategy is as follows. Let us consider the set

D = { 1 √ 5 , √ 3 √ 10 , √ 2 √ 5 , 1 √ 2 , √ 3 √ 5 , √ 7 √ 10 , 2 √ 5 } ≈ {0
.447, 0.547, 0.632, 0.707, 0.774, 0.836, 0.894}, that characterizes the allowed distances between the 625 bases. We randomly select a minimal set B of l bases within the 625's such that (a) there is at least one distance of each type among the selected bases, (b) there is at least one subset of B containing 5 bases partitioning the 5 × 16 = 80 rays (this criterion is adopted to reach the result with only 16 5 = 1048576 checks), (c) the set B satisfies the BKS postulates (i) and (ii) listed in the introduction. We found a mimimal cardinality l = 23 for the set B. It was further simplified to l = 22, a set still satisfying the criterion (b), then to l = 21. The 80 -21 proof, given in [START_REF] Planat | Three-qubit entangled embeddings of CPT and Dirac groups within E8 Weyl group[END_REF], does not satisfy criterion (b), although there exist two sets of four disjoint bases. The main properties of 80 -23, 80 -22 and 80 -21 proofs are summarized in Tables 5 and6 2 and4, where only critical proofs were displayed.

a 80 -23 Z 1 D 6 Z 2 Z 2 2 (A 7 Z 2 ) Z 5 2 Z 6 Z 5 2 × S 18 Z 2 × S 80 -22 Z 1 D 6 Z 2 Z 2 2 S 6 Z 3 2 S 6 Z 2 × S 20 Z 2 × S 80 -21 Z 1 Z 2 2 Z 2 Z 2 2 S 5 Z 3 2 S 5 Z 2 × S 19 Z 2 × S
For the sake of completeness, we mention that the 22-base and 23-base proofs follow by adding to [START_REF] Planat | Three-qubit entangled embeddings of CPT and Dirac groups within E8 Weyl group[END_REF] the following two rays, respectively {2, 10, 12, 14, 35, 37, 43, 45, 65, 69, 71, 74, 76, 78, 79, 80}, {49, 50, 55, 56, 58, 60, 62, 64, 68, 70, 72, 74, 76, 77, 78, 79}. (18) That ( 17) is a BKS proof of the four-qubit system can be easily checked with the help of a computer by checking that for all 16 4 * 80 = 5242880 possibilities of assigning the truth value 1 to a quintuple of rays (i, j, k, l, m) with i, j, k, l and m being the indices in one set of four mutually disjoint bases and in an arbitrary base of index m of ( 17), at least one basis does not satisfy the constraint (ii) of the introduction. The same conclusion holds for the set of 22 bases that contains the set of the 21's, and for the set of 23 bases that contains the set of the 22's. No further simplification of the 21-base set could be obtained while keeping the BKS proof.

One observes from Table 6 (column 2) that the proofs are quite random given the overall symmetry group Z 1 . But many remnant symmetries are present as one can see by looking at the other crossing graphs (in columns 3 to 8 ).

Conclusion

We have performed a systematic investigation of small state proofs of the BKS theorem involving real rays of several qubits. The proofs correspond to some sets of maximal orthogonal bases constructed from Mermin's 3 × 3 square (for two qubits) and from Mermin's pentagram (for three and four qubits). These BKS states belong to a larger set of real states on an (Euclidean) Barnes-Wall lattice B n . It would be desirable to discover the precise status of the KS sets on B n . This is left for a future work.

Another ongoing work of ours concerns BKS proofs with complex rays in the spirit of [START_REF] Waegell | Parity proofs of the Kochen-Specker theorem based on 60 complex rays in four dimensions[END_REF][START_REF] Megill | Probabilistic generation of quantum contextual sets[END_REF] and BKS proofs for more qubits (a particular case of five qubits is investigated in [START_REF] Planat | Five-qubit contextuality, noise-like distribution of distances between maximal bases and finite geometry[END_REF]). A deeper understanding of KS sets may be useful for conceptual questions concerning the EPR local elements of reality, quantum complementarity, counterfactual compatibility and non-contextual inequalities [START_REF] Liang | Specker's parable of the overprotective seer: A road to contextuality, nonlocality and complementarity[END_REF][START_REF] Abramsky | Logical Bell inequalities Preprint[END_REF].

  [1000], 2 : [0100], 3 : [0010], 4 : [0001], 5 : [1111], 6 : [11 11 ] 7 : [1 11 1], 8 : [1 11 1], 9 : [1 111 ], 10 : [1 111], 11 : [11 11], 12 : [111 1] 13 : [1100], 14 : [1 100], 15 : [0011], 16 : [001 1], 17 : [0101], 18 : [010 1] 19; [1010], 20 : [10 10], 21 : [100 1], 22 : [1001], 23 : [01 10], 24 : [0110]

1 :

 1 [10000000], 2 : [01000000], 3 : [00100000], 4 : [00010000], 5 : [00001000], 6 : [00000100], 7 : [00000010], 8 : [00000001], 9 : [11110000], 10 : [11 11 0000], 11 : [1 11 10000], 12 : [1 11 10000], 13 : [00001111], 14 : [000011 11 ], 15 : [00001 11 1], 16 : [00001 11 1], 17 : [11001100], 18 : [1100 11 00], 19 : [1 1001 100], 20 : [1 100 1100], 21 : [00110011], 22 : [001100 11 ], 23 : [001 1001 1], 24 : [001 100 11], 25 : [10101010], 26 : [1010 10 10], 27 : [10 1010 10], 28 : [10 10 1010], 29 : [01010101], 30 : [01010 10 1], 31 : [010 1010 1], 32 : [010 10 101], 33 : [100101 10], 34 : [100 10110], 35 : [10010 110], 36 : [100 10 11 0], 37 : [0110 1001], 38 : [01 101001], 39 : [01 10 100 1], 40 : [0110100 1]. (9)

( 11 ) 3 D 4

 1134 These 16 selected proofs have 4, 5, 6, 7 or 8 bases in common. The 8-base crossing graph is regular, of valency 5, with automorphism group aut = Z5 2 G 72 , where G 72 = Z 2 was already found as an important symmetry group of the two-qubit 36 -11 proofs.

  11 ], 72 : [11 1111 111 1111 111], 73 : [1 1111 111111 1111 1] 74 : [1 1111 11111 1111 11], 75 : [1 111 11 1111 1 111 11], 76 : [111 1111 11 111 1111] 77 : [1 111 1 111111 1 111 1], 78 : [111 1111 11 111 1 111 ], 79 : [1 1111 111 11 1 111 1 1], 80 : [111 1111 1 11111 111 ].

1 :

 1 {1, 2, 3, 4}, 2 : {5, 6, 7, 8}, 3 : {9, 10, 11, 12}, 4 : {13, 14, 15, 16}, 5 : {17, 18, 19, 20}, 6 : {21, 22, 23, 24}, 7 : {1, 2, 15, 16}, 8 : {1, 3, 17, 18}, 9 : {1, 4, 23, 24}, 10 : {2, 3, 21, 22}, 11 : {2, 4, 19, 20}, 12 : {3, 4, 13, 14},

13 : {5, 6, 14, 16}, 14 : {5, 7, 18, 20}, 15 : {5, 8, 21, 23}, 16 : {6, 7, 22, 24}, 17 : {6, 8, 17, 19}, 18 : {7, 8, 13, 15}, 19 : {9, 10, 13, 16}, 20 : {9, 11, 18, 19}, 21 : {9, 12, 22, 23}, 22 : {10, 11, 21, 24}, 23 : {10, 12, 17, 20}, 24 : {11, 12, 14, 15}

Table 1 .

 1 ). All four indices proof v -l # proofs a 1 a 2 a 3 a 4 a 5 The histogram of distances for various parity proofs v -l obtained from Mermin's square. One can check the expected equality 2 a i = l(l -1) in each proof. Let us first observe that the symmetry group of Mermin's graph[START_REF] Peres | Quantum theory: concepts and methods[END_REF] isG 72 = Z 2 3 D 4 .The 16 proofs of the 18 -9 type overlap in 3 or 5 elements. The way the proofs overlap each other (the crossing graph) is that of the square (5) with aut ∼ = Z 4 2

	24 -15	16 18 18	9 54	6
	22 -13A	96 12 18	3 42	3
	22 -13B	144 12 18	4 42	2
	20 -11A	96	6 18	0 30	1
	20 -11B	144	6 18	1 30	0
	18 -9	16	0 18	0 18	0

Table 2 .

 2 The symmetries involved in various two-qubit parity proofs of the BKS theorem. The first row gives the number of common elements between the bases, the second row relates these numbers to distinguished distances. Among the building block symmetries are the group G 72 = Z2 3 D 4 and the single qubit Pauli group P 1 ∼ = D 4 Z 2 , a group underlying the CPT symmetries of the Dirac equation[START_REF] Planat | Three-qubit entangled embeddings of CPT and Dirac groups within E8 Weyl group[END_REF].in each row and in each column correspond to four disjoint bases that together partition the 24 rays.

	⎛	7 8 10	⎞	⎛	7 9 11	⎞	⎛	8 9 12	⎞	⎛	10 11	⎞
	⎝	13 14 16	⎠ -	⎝	14 15 18	⎠ -	⎝	16 17 18	⎠ -	⎝	13 15	⎠ -
		22 23 24			19 20 22			20 21 24			19 21
	⎛	| 7 7 9 11	⎞	⎛	| 20 7 8 10	⎞	⎛	| 12 10 11 12	⎞	⎛	| 23 8 9	⎞
	⎝	16 17 18	⎠ -	⎝	13 15 17	⎠ -	⎝	13 14 16	⎠ -	⎝	14 15	⎠ -
		19 21 23			20 21 24			19 20 22			22 23
	⎛	| 17 8 9 12	⎞	⎛	| 10 10 11 12	⎞	⎛	| 14 7 8 10	⎞	⎛	| 9 7 9	⎞
	⎝	13 15 17	⎠ -	⎝	16 17 18	⎠ -	⎝	14 15 18	⎠ -	⎝	13 14	⎠ -
		19 20 22			22 23 24			19 21 23			20 21
	⎛	| 12 10 11 12	⎞	⎛	| 23 8 9 12	⎞	⎛	| 7 7 9 11	⎞	⎛	| 20 7 8	⎞
	⎝	14 15 18	⎠ -	⎝	13 14 16	⎠ -	⎝	13 15 17	⎠ -	⎝	16 17	⎠ -
		20 21 24			19 21 23			22 23 24			19 20
		| 14			| 9			| 17			| 10

Table 3 .

 3 The histogram of distances for various parity proofs v -l obtained from

	Mermin's pentagram. Observe that the symmetry group of Mermin's pentagram
	is S 5 . Two proofs of the 36 -11 type share 3, 4, 7, 8 or 9 elements, with crossing graph whose aut ∼ = Z 6 2 S 5 , or 5 or 6 elements with crossing graph having aut ∼ = Z 14 2 (Z 2 × S 5 ). Two proofs of the 40 -15 type have 9, 10, 11 or 12 elements
	in common. The graphs corresponding to 9 or 11 shared elements are complementary,
	with aut ∼ = Z 10 2 aut ∼ = Z 32 2 (Z 5 2 aut ∼ = Z 6 2 S 5 .	(A 2 6 S 6 ) and the graph corresponding to 12 common elements has D 4 ), the graph corresponding to 10 shared elements has

. 1 :

 1 {35, 37, 43, 45, 51, 53, 54, 57, 65, 69, 71, 72, 74, 76, 77, 80}, 2 : {17, 18, 25, 26, 35, 37, 39, 40, 43, 45, 47, 48, 51, 52, 57, 59}, 3 : {35, 40, 43, 48, 50, 52, 58, 59, 61, 62, 63, 64, 68, 70, 78, 79}, proof v -l a 1 a 2 a 3 a 4 a 5 a 6 a 7

	80 -23	1	3 17 19 76 69 68
	80 -22	1	1 17 19 65 64 64
	80 -21	1	1 14 19 60 64 51

Table 5 .

 5 The histogram of distances for various proofs obtained from the square of operators[START_REF] Planat | Multipartite entanglement arising from dense Euclidean lattices in dimensions 4 -24[END_REF].

	# common elements	0	2	4	6	8	10
	distance a 7 a 6 a 5	a 4	a3	a 2

Table 6 .

 6 The symmetries involved in the selected four-qubit proofs of the BKS theorem. The first row gives the number of common elements between the bases, the second row providing the corresponding distances. The proof 80 -23 contains the 80 -22 one, and the latter contains the 80 -21 one. Thus Table6has a slightly different status than Tables

  {5, 11, 13, 15, 21, 22, 23, 27, 28, 29, 30, 32, 53, 54, 61, 63}, 18 : {5, 11, 13, 15, 17, 18, 23, 25, 26, 27, 28, 32, 51, 52, 57, 59}, 19 : {33, 34, 36, 38, 41, 42, 44, 46, 65, 66, 67, 69, 71, 73, 75, 80}, 20 : {17, 18, 20, 21, 22, 24, 25, 26, 28, 29, 30, 32, 67, 69, 75, 80}, 21 : {1, 6, 10, 12, 33, 34, 36, 39, 40, 46, 47, 48, 66, 67, 73, 75}.

	4 : {1, 2, 4, 5, 7, 11, 12, 16, 21, 22, 25, 26, 35, 37, 39, 48},
	5 : {3, 7, 8, 16, 18, 19, 20, 21, 24, 25, 30, 31, 33, 42, 44, 46},
	6 : {1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 16, 19, 20, 24, 31},
	7 : {1, 6, 10, 12, 23, 24, 31, 32, 33, 34, 36, 46, 49, 60, 62, 64},
	8 : {17, 18, 20, 21, 22, 25, 26, 27, 29, 30, 31, 32, 37, 43, 47, 48},
	9 : {20, 27, 31, 32, 33, 34, 36, 37, 38, 41, 42, 43, 44, 46, 47, 48},
	10 : {1, 2, 9, 10, 20, 27, 31, 32, 37, 43, 47, 48, 52, 53, 57, 63},
	11 : {3, 7, 11, 15, 33, 34, 41, 42, 54, 55, 57, 58, 59, 60, 63, 64},
	12 : {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16},
	13 : {2, 3, 10, 11, 12, 13, 14, 16, 65, 66, 74, 75, 76, 78, 79, 80},
	14 : {33, 36, 41, 44, 56, 58, 60, 62, 65, 69, 70, 73, 74, 75, 77, 79},
	15 : {4, 6, 12, 14, 51, 54, 56, 58, 59, 60, 61, 62, 65, 69, 73, 75},
	16 : {18, 21, 26, 29, 49, 50, 55, 64, 66, 67, 68, 71, 76, 77, 79, 80},
	17 :

Acknowledgements

The author thanks the anonymous referees for their constructive remarks. He also acknowledges Metod Saniga for his help in improving the second version of the manuscript.

give minus the identity operator. There is no way of assigning the eigenvalues ±1 while still preserving the multiplicative properties of the operators .

To investigate a state proof of the BKS theorem, we have at our disposal the following set of 5 × 16 = 80 rays [START_REF] Bengtsson | Mutually unbiased bases and Hadamard matrices of order six[END_REF]

• • • partners

For a finite geometrical account of the "magic" rectangle [START_REF] Planat | Multipartite entanglement arising from dense Euclidean lattices in dimensions 4 -24[END_REF], see [START_REF] Saniga | Finite geometry behind the Harvey-Chryssanthacopoulos four-qubit magic rectangle[END_REF].


Yanne K Chembo

Maxime Jacquot

John M Dudley

Laurent Larger
email: laurent.larger@femto-st.fr

P Roux

E Dordor

L Gauthier-Manuel

L Furfaro

R Giust

J.-M Merolla

L Illing

B Beausire

F Berther

J Nguyen

V Arm- Bruster

J Nunes

R Bourquin

J.-C Savoie

P Roblot

L Tournier

Ikeda-like chaos on a dynamically filtered supercontinuum light source

Keywords: numbers: 05.45.Jn, 42.65.Sf, 01.50.My, 02.30.Ks

We demonstrate temporal chaos in a color selection mechanism from the visible spectrum of a supercontinuum light source. The color selection mechanism is governed by an acousto-optoelectronic nonlinear delayed feedback scheme modeled by an Ikeda-like equation. Initially motivated by the design of a broad audience live demonstrator in the framework of the International Year of Light 2015, the setup provides also a unique experimental tool to investigate the dynamical complexity of delayed feedback dynamics. Deterministic hyperchaos is here analyzed from the experimental time-series. A projection method identifies the delay parameter, for which the chaotic strange attractor originally evolving in an infinite-dimensional phase space, can be revealed in a 2D sub-space.

Since the pioneering work of Edward Lorenz [1] who demonstrated the phenomenon of sensitivity to initial conditions as predicted by Henri Poincaré [START_REF] Poincaré | Les méthodes nouvelles de mécanique céleste[END_REF], chaos theory has been the focus of numerous experimental investigations in order to demonstrate its ubiquity in real world. It was indeed rapidly identified in many areas, such as chemistry (Belousov-Zabotinski), electronics (Chua's circuit), fluid mechanics (turbulence), solid mechanics (driven pendulum), life sciences, and also in optics.

In this latter field, a prototype nonlinear optical system of interest was proposed by Kensuke Ikeda more than 30 years ago [START_REF] Ikeda | [END_REF]4]. His system was consisting in a ring optical cavity with a Kerr medium, which was seeded by a continuous-wave (CW) single-mode laser beam. Depending on the input laser intensity level, the phase shift determined by the interference condition at the Kerr medium input can potentially destabilize the optical phase of the intra-cavity light beam, thus leading to a non-trivial phase dynamics. A simplified model ruling the dynamics of the Ikeda ring-cavity results in a scalar first order delay differential equation of the form

τ dx dt (t) = -x(t) + β sin 2 [x(t -τ D ) + Φ 0 ] , (1)
where x is the Kerr phase shift, τ stands for the the ultrafast light-matter interaction of the Kerr effect, τ D is the time delay imposed by the cavity roundtrip time, β is the gain of the feedback loop, and Φ 0 is a static offset phase accumulated along the ring cavity. The solutions of this equation have been shown to lead potentially to chaotic fluctuations. From a dynamical systems viewpoint, this setup is characterized by a nonlinear multiple timescale interaction with significantly different scales, since τ is typically several orders of magnitude smaller τ D . Ikeda's simulations of the scalar delay differential equation successfully led to the observation of a period-doubling route to chaos. Later on, a convenient optoelectronic implementation of the Ikeda dynamics successfully showed the full bifurcation scenario from a stable steady state to chaos through period doubling [5], and soon after, an integrated electro-optic experimental system was proposed as well to explore the nonlinear behavior of Ikeda-like dynamics [6].

In addition to their relevance, Ikeda-like oscillators have found central applications in optical chaos communications [7][8][9] ultra-stable microwave generation [10,11], and more recently, photonic brain-inspired processors based on the concept of Echo State Network (also known as Liquid State Machine or Reservoir Computing) [12][13][14][15].

In the present article, we report a setup which implements physically an Ikeda-like dynamics according to the conceptual idea of wavelength chaos initially reported in [7]. The latter publication was however involving an electrically tunable semiconductor laser diode act- ing as a variable wavelength light source in the infrared range, with a few Angströms tunability around 1.5 µm. In our reported setup, such a wavelength dynamics is transposed into the full 300 nm visible range of a bright white light source, a supercontinuum (SC), because of the targeted live demonstration for broad audience dissemination of photonic and nonlinear dynamics phenomena. The electrically driven wavelength selection mechanism is thus provided by an acousto-optic tunable filter, thereby allowing for a chaotic behavior of a visible color. The experimental system is schematically presented in Fig. 1, which shows the autonomous oscillator architecture with a nonlinear delayed acoustoopto-electronic feedback loop. It consists of:

(i) A supercontinuum (SC) source providing an intense white light (25 mW in the visible range). The infra-red part of the light source, which represents most of the optical power of the supercontinuum, is filtered out since it is not relevant to our primary objectives and it could represent an eye danger during the experiment. We used a commercial SC providing an already collimated beam with ca. 1 mm diameter. Principally white light emitting diode could be used as well, however requiring more electronic feedback gain, and needing a dark environment for the observation of the color dynamics.

(ii) An on-axis acousto-optic tunable filter (AOTF), allowing for the fast (a few µs response time) and nearly linear selection of a color (spectral slice of ca. 4 nm width in the blue range until 16 nm in the red range) within the supercontinuum visible spectrum. The input signal ruling the selection is a 0.5 Watt radio-frequency waveform, spanning from 74 MHz (selecting the red color) to 158 MHz (blue color). Since the AOTF is polarization sensitive, and since the supercontinuum does not provide a polarized light beam, one obtains three angularly separated output beams: The (+1) on-axis order, with the proper AOTF polarization state, and corresponding to the selected color; The (-1) off-axis order, contains the same selected color, but with an orthogonal polarization state, and with a wavelength dependent angular deviation; And the zero off-axis order, gathering all the non-selected colors, i.e. an unpolarized white light beam from which the selected color is removed (the AOTF has more than 90% efficiency for the selected color).

(iii) A commercial X-cut quartz birefringent plate, which thickness was reduced down to 250 µm, such that the corresponding optical path difference over the visible range results to a maximum of 2π phase shift. The wave plate is oriented, as usual for a birefringent filter with maximized contrast, at 45 • from the polarization state of the selected color.

(iv) A polarization beam splitter oriented parallel or perpendicular to the AOTF output polarization state, and providing two complementary interference states at its two outputs. This device generates at its output the two-wave interference modulation transfer function (cos 2 or sin 2 ), fulfilling the role of the nonlinear transformation in the dynamical delayed feedback.

(v) A Silicon photodiode converts the intensity interference into an electrical amplitude, which can then be processed electronically: The signal is delayed in time by an electronic delay line (analogue FIFO memory providing a time delay τ D ≃ 0.5 s, which can be easily followed by the eye response time, again for a broad audience live demonstrator target). The electronic signal is also lowpas filtered to force the resulting wavelength change to also occur within the human eye bandwidth, typically with 3 Hz cut-off frequency (τ ≃ 50 ms response time). Gain and offset can be adjusted through two variable resistors which are defining the constant voltages used in a standard multiplier circuit using an AD633, thus allowing for the tuning of parameters β and Φ 0 in Eq.(1), respectively.

(vi) The output signal of the previous photodetection, delaying, filtering offsetting and amplifying functions, serves then as the input of a voltage controlled oscillator (VCO). This device delivers a constant amplitude RF output which RF frequency (in the 74 MHz-158 MHz range) is proportional to the input voltage amplitude (between 0 and 10 Volts).

(vii) An RF amplifier is finally required before feeding electronically the AOTF, in order to have the 0.5 W RF power needed to operate the AOTF at its maximum diffraction efficiency, above 90%.

Additional technical details need to be considered in order to optimize the matching of the model of this setup with the Ikeda one in Eq. ( 1). First, one needs to filter the detected optical intensity to average the actual pulsed SC light source. Since the global dynamics is limited by a 3 Hz low pass cut-off, and since the SC repetition rate can be set to 20 kHz, a passive first order RC filter right after the photodiode, with a 200 Hz cut-off, easily extracts the average intensity of the interfering SC light pulses: Most of the pulse harmonics are thus removed, and the electronic image of the intensity is properly smoothed to correspond to a continuous time dynamics. Another important issue concerns the combined effect of the nonflatness of the SC spectral density, and the-non constant spectral response of the Silicon photodiode. Without any dedicated compensation, the red side of the spectrum appears to have a much higher gain than the blue one. The adopted solution is to passively compensate this through a properly adjusted linear color dependent gain with a simple multiplier circuit after photodetection. The gain input of the multiplier is thus designed from a proper amplification and offsetting of the voltage driving the VCO (and thus selecting the color, low voltage for red and high voltage for blue).

For appropriate gain (β) and offset (Φ 0 ) to be adjusted on the electronic board, a visual demonstration of the Ikeda dynamics complexity can be obtained. One can of course monitor the input voltage of the VCO, but more interestingly in this setup, one directly sees from the different AOTF light beam outputs the nicely color encoded motion corresponding to the obtained dynamics. There are actually three such possible visualizations: The (-1) AOTF diffraction order, which is unused by the delayed feedback loop, but shows the selected color during the dynamics; The second interferometer output from the polarization beam splitter, which shows the selected color with a complementary interference intensity with respect to the one of concern in the dynamics; And the zero-order at the AOTF output which appears as an intense white light beam -but once spatially spanned by a diffraction grating, a bright rainbow is observed in which a moving dark line is clearly seen, this dark line corresponding to the absence of the color actually selected by the AOTF.

The latter visualization (through the zero order AOTF output) was used in practice in our setup. We have also used white screen projectors to display and monitor the supercontinuum rainbows with the chaotically a moving dark line. In order to generate such rainbows, we have exploited the high brightness of the SC light source, and we used 20% diffraction efficiency gratings, from which the complementary 80% zero order was recycled with mirrors, thus allowing to re-use the remaining non-diffracted light through further gratings with different projection directions and different diffraction orientations. Please see Supplemental Material [16] for a short video of the physical setup in the lab, while operating in a chaotic regime. The movie shows a moving dark line within a grating-diffracted zero order, the arrangement of the different devices forming the full acousto-electrooptoelectronic setup, as well as a short live time trace of an electronic signal proportional to the normalized dynamical variable x(t).

From the nonlinear dynamics viewpoint, the setup also allows for the understanding and monitoring of the temporal chaotic behavior of the visible wavelength dynamics. In particular, the fact that all the timescales are slowed down such that everything can be directly captured by human eyes (timescale of the order of seconds), allows to reveal very easily the numerous transients during convergence towards an attractor (a usu-FIG. 3: (Color online) Plot of the filling factor ρ as a function of the time lag τs, following the definition of Eq. 2. Note that the filling factor sharply drops to a value very close to zero when the lag time fulfills the condition τs = τD = 0.513 s. In that case, the dynamics of the sustem is projected onto the nonlinear transfert function in the plane xτ s -x. On the other hand, when τs is significantly different from the timedelay τD, the dynamics of the system in the plane xτ s -x is chaotically distributed. This plot provides evidence about the fact that the dynamics displayed in Fig. 2 is a chaotic behavior resulting from the nonlinear Ikeda feedback loop.

ally neglected phenomenon, however interesting in many aspects, e.g. the local dynamical properties of an attractor, the real/complex nature of the related eigenvalues, etc.). Numerous complex slow-fast dynamical phenomena could also be explored with this setup, as already reported in experiments belonging to the same class of delay dynamical systems [17][START_REF] Erneux | Laser dynamics[END_REF][START_REF] Erneux | Applied Delay Differential Equations[END_REF][START_REF] Callan | [END_REF][21][22]. We have particularly focused on the exploration of the chaotic regimes. The corresponding timetrace in presented in Fig. 2, and the envelope dynamics displays the characteristic pseudo-random oscillations of chaotic Ikeda systems, with a timescale of the order of seconds. The autocorrelation function shown in Fig. 2 features typical peaks at lag times approximately equal to ±(τ + τ D ), as it is known from chaotic Ikeda systems, which feature flay spectra below the lowpass cut-off frequency as well in the chaotic regime (see Fig. 2).

In order to provide unambiguous evidence of the deterministic chaotic nature of the pseudo-random behavior displayed in Fig. 2, we use the filling factor method for dynamical systems, which is a well known accurate method that can be used to extract the precise delay value from a time series ruled by a delay differential equation [23,24]. The method was used for example in chaos communications cryptanalysis [25], for which the most important parameter to recover first is precisely the time delay. The method is used in this work to evidence unambiguously the deterministic feature in the recorded experimental time traces, and its basic principle is as follows. Despite the fact the dynamics is infinite dimensional, there exists a low dimensional subspace, in which a strong constrain is imposed by the dynamical equation of motion. In the case of a conventional scalar nonlinear delay dynamics (Ikeda, Mackey-Glass, etc.), one simply has to construct a 3D vector trajectory from a time series, which vector coordinates are [ ẋ(t), x(t), x(t-τ s )] where τ s is a time shift. When τ s precisely matches the time delay τ D of the original equation of motion, the trajectory lies on a surface. Ploting the volume filled by the trajectory in this 3D space as a function of τ s will reveal a nearly constant value corresponding roughly to a cube of a volume [max(x) -min(x)] 3 , except when τ s = τ D for which time shift the volume shrinks sharply to zero because one has to have a surface. This technique can be further simplified in a 2D space, plotting the points [x(t k ), x(t k -τ s )], where t k are the dates of the original time trace exhibiting extrema, i.e. ẋ(t k ) = 0. The calculation of the surface covered in the 2D plane [x(t k ), x(t kτ s )] can be approximated, when using a finite number of dates t k for the extrema of the time trace, as:

ρ = 1 L -1 L l=1 max[x l (t k )] -min[x l (t k )] max(x) -min(x) (2)
where the interval in x (that is, [min(x), max(x)]) visited by the trajectory is divided into (L -1) sub-intervals of the kind ∆ l = [x l , x l+1 ] for l = 1 to L. For each value x(t kτ s ) belonging to one such interval, we collect the values

{x l (t k ) = x(t k )|x(t k -τ s ) ∈ ∆ l }.
The definition of ρ involves the amplitude span [max(x l (t k ))-min(x l (t k ))] in each of these (L -1) sets {x l (t k )}. The filling factor ρ is plot as a function of the time lag τ s in Fig. 3. It appears that ρ sharply drops to zero when the lag time matches the delay time, while the dynamics is projected onto the quasi-sinusoidal transfert function in the plane x τsx, thereby providing a proof that the slowly-varying envelope of the supercontinuum spectrum displayed in Fig. 2 follows an Ikeda-like chaotic behavior. The Supplemental Material [16] provides a video file of the moving dots in the plane x τsx, while τ s is finely incremented from 0.3 s to 0.7 s with with 0.25 ms increment every movie frame. The first frame, as well as the last one and the one corresponding to the matching of τ s with the correct delay value τ D , are represented in the insets of Fig. 3.

In conclusion, we have reported the Ikeda-like temporal chaotic behavior of a color dynamically sliced in the visible spectrum of a supercontinuum light source. This complex behavior has been analyzed using the various tools of time series analysis such as the autocorrelation plot, the power Fourier spectrum, and the filling factor curves. Perspectives for future research are numerous for this chaotic rainbows. This system could include as well a high-pass filtering in the electronic feedback [26], resulting in an integral slow time scale responsible for the occurrence and stabilization of many additional dynamical motions, such as chaotic breathers [27] or single delay periodic motions [28]. Further scientific issues could also concern the observation of chimera states [29,30] in this visible wavelength delay dynamics. Such a particular complex and self-organized motion should be also observable in this visible wavelength dynamics, if some of the system features can be adequately modified. One would need first to implement a slow integral time scale through a high-pass filtering in the feeedback loop, and second to replace the birefringent filter by a Fabry-Pérot one which exhibits a strong asymmetry between the minima and maxima of the nonlinear transformation. Other investigations could concern a possible smooth transition scenario between the continuous time delay differential equation model, and its discrete time singular limit map approximation [31], once the time scales of the dynamics can be controlled such that SC light pulses can be time resolved in the delayed feedback loop. The setup can be also used within a teaching perspective in physics, optics, and nonlinear dynamics. Various phenomena can be easily illustrated by direct or indirect visual observation. Finally, since the experiment was initially motivated by dissemination and broad audience scientific demonstration issues in the framework of the international year of light 2015, one might also conclude with a fortuitous and surprising comment linking the history of arts and sciences through the chaotic rainbow setup. Indeed, the famous French painter Paul Cézanne once said "We live in a rainbow of chaos". It is then surprising to realize that Cézanne was a contemporary of Henri Poincaré, one of the precursor of the chaos theory through his seminal work on nonlinear dynamical systems. "Chaos" was however proposed much later only in the 70s, by Li and Yorke [32] as a naming for the kind of dynamical system solutions exhibiting sensitivity to initial conditions.

FIG. 1 :

 1 FIG. 1: (Color online) Schematic representation of the visible wavelength nonlinear delayed feedback loop oscillator. SC: collimated supercontinuum light source; IR: infra-red blocking filter; AOTF: acousto-optic tunable filter; BWP: birefringent quarz wave plate; PBS: polarization beam splitter; MP: mirror prism extracting the zero order AOTF output, for further display of rainbows through mirrors and diffraction gratings; OEC: optoelectronic circuit performing photodetection, filtering, delaying, amplification and offset setting; VCO: voltage controlled oscillator; RF: radio-frequency driver.

FIG. 2 :

 2 FIG. 2: (Color online) Part of a chaotic signal. The two right plots are calculated from a long chaotic time series (up: Autocorrelation function revealing the echo in the chaotic time series at ca. (τ + τD); Down: Amplitude of the Fourier spectrum, showing the nearly flat filling of the Foureir components within the oscillator bandwidth).

This work was supported by the Region Franche-Comté (CORPS project) the Labex ACTION (contract ANR-11-LABX-01-01), and the IDEFI CMI-FIGURE French higher education program (contract ANR-11-IDFI-0010).


V Plessky

M Lamothe

Hyperbolically frequency modulated transducer in SAW sensors and tags

Introduction:

The idea to use the chirp interdigital transducers (IDT) and corresponding chirp (usually LFM) signals in SAW tags and sensors was proposed some years ago [START_REF] Ianelli | Surface-wave transducer device and identification system with such device[END_REF]. It is especially attractive for the Ultra-Wide-Band (UWB) devices [START_REF] Harma | Feasibility of ultra-wideband SAW RFID tags meeting FCC rules[END_REF], [START_REF] Lamothe | Temperature measurements with Ultra-Wideband SAW sensors[END_REF], Fig. 1, since the significant processing gain B*T can be obtained compared to environmental parasitic reflections and thermal noise, (Bbeing the used frequency band of the chirp signal and T its duration).

Fig. 1 SAW reflective delay line sensor with chirp transducer.

In the "orthogonal frequency coding devices" [START_REF] Malocha | Orthogonal frequency coding for SAW device applications[END_REF] the signals from different sensors are separated due to spread-spectrum technique, including IDTs or reflectors with hopping frequency. The lithium niobate (LiNbO3) substrates often used for such devices have large temperature coefficient of delay and the increase of the delays with temperature is accompanied with decrease of frequencies, generating the change in MHz/μs slope of dispersion of the LFM transducer. That, finally, results in additional losses, broadening of compressed signals and/or complicated reader algorithms, especially because the sensor temperature is usually unknown in advance. The use of hyperbolically frequency modulated (HFM) signals was proposed many years ago [START_REF] Thornton | Temperature-invariant pulse compression using S.A.W. devices[END_REF] for time-invariant compression in SAW compression and expansion filters. However, the paper [START_REF] Thornton | Temperature-invariant pulse compression using S.A.W. devices[END_REF] describes only the signal properties, while design of such devices has not been disclosed. Also, for applications in radars, instead of Reflective Array Compressors, (RACs), such devices might have a visible drawbackthe compressed pulse would shift with temperature. Here we demonstrate that the HFM transducer is ideal for SAW-sensor and tags applications. In such a transducer the local period of electrodes is linearly dependent on the electrode coordinate. It will be shown that it means exponential dependence of the period on the number of the electrode. The thermal variation changes all period proportionally, which for the exponential array means just a shift. Therefore such a transducer remains a kind of invariant to the temperature.

Purely geometric problem: If the period of an array increases linearly with coordinate, how can the coordinate xn of n-th element of this array be calculated? For the geometric structure of electrodes (strips, grooves, etc.) with period linearly changing with coordinate x one can write the following relation:

(1) Wherein xn is the coordinate of the centre of n-th electrode, p0-initial period, and ε is the linear grows or decay (if ε<0) coefficient. Supposing that the initial electrode (n=0) is situated at the origin of coordinate 0X, x0 =0, x1=p0, the formula (1) can be treated and an equation in integer numbers, which has unique solution:

(2) Or, calculating the n-th period:

(3) Two parameters p0 and ε completely determine the array. If we fix the first period p0 and the last period xN+1-xN =pend we can find that (4) re-write the formulas (2) and (3) in the following form: ,

These formulas show that while the period of the structure changes linearly with coordinate it changes exponentially with the number of element.

If the total length L=xN of the structure is known the following equation determines the number of periods N: If in the beginning we have high frequencies, that is low period, the same formula is valid with B<0. The same formulas can be obtained considering phase of HFM signal [START_REF] Thornton | Temperature-invariant pulse compression using S.A.W. devices[END_REF].

HFM SAW transducer:

The above formula (11) is suitable for calculation of position of 90°-reflectors in RAC devices, operation with HFM signals and having 1 reflector per period. For SAW transducer, we must put the series of electrodes of alternating +/polarity electrodes that is 2 per period. We can re-write formula (11) in the following equivalent form (12)

If B>0 the period decreases along the structure, and if B<0, it is linearly increasing along the structure . The same formula could be used for 180° HFM reflector.

Numeric simulations: As an example we simulated numerically using precise FEM/BEM software belonging to GVR Trade SA [6]. We use (Fig. 2 We simulate the temperature effect on the delay line by increasing all dimensions in the propagation direction by 2% and decreasing the characteristic frequencies (F0 and B) by 2% also. These changes roughly correspond to about 250°C increase of the device temperature.

Fig. 3 Simulated impulse responses

Fig. 3 shows the impulse response of the devices. The expanded by 2% delay line has longer response. However, the main part of this response geometrically is not only similar, but identical to the initial response, when being shifted appropriately, as illustrates Fig. 3. Namely this feature of HFM signal allows us using the same all "matched-tosignal" filter [7] for all expanded copies of the initial signal. We have simulated compression procedure numerically using only one, initial, signal for creation the "matched-to-signal" filter. The results are shown in Fig. 4. Both signals (initial, ideally matched to our filter, and extended by 2%) are compressed with the same amplitude and width of the both compressed peaks is identical (close to 1/B =5ns at -3dB ). In practice it means that the result is temperature-invariant: whatever is the temperature of the device its response can be compressed by the same filter.

Fig. 4 Compressed HFM pulses

In SAW sensors, Fig. 1, (or in SAW-tags) such common for all reflectors shift of compressed reflected pulses is not so important, since the information is extracted from distance variation between the pulses. We have repeated the same numeric experiment with similar device but with LFM transducer: having the same frequency band, dispersive delay, etc. If we use inversed in time impulse response for "matched to signal" filter, the compression visibly degraded for 2% expanded LFM device. For larger B*T product the degradation would be even more pronounced.

Conclusion:

The Hyperbolically Frequency Modulated signals and transducers (eventually reflectors) are ideally suitable for SAW-sensors and SAW-tags, since compression of such signals, being temperatureinvariant, can be achieved with always the same matched-to-signal filter, simplifying significantly the interrogation algorithm.

  the coordinate xn can also be presented in the following form:(9) If our structure corresponds to a SAW propagating with velocity V, and the periods of the structure are related to frequency as (10) (here F0 is the centre frequency, |B|-the frequency band) , introducing L= xNtotal length of the structure we can represent (9) in the following form for the case when there is 1 element per period:(11)

  ) 128°-LiNbO3 substrate 1 with HFM IDT 2 operating in frequency range 200MHz-400MHz. The wide band transducer 3 is placed in the acoustic channel 4 of the HFM transducer allowing direct analyzing of signals generated by HFM transducer. The dispersive delay time T is equal to 0.5 μs, B*T product thus being B*T=100. The period of the HFM transducer at high frequency end is 9.6 μm, linearly increasing with distance to 2 time larger value at the low frequency end. The transducer includes 279 fingers. The wide band standard IDT including 5 electrodes with period λ=12.7 μm is situated in 1000μm (Marc: please check!) from HFM transducer on high frequency side of it. Both transducers have aperture W= 600 μm.

Fig. 2

 2 Fig. 2 Geometry used in the numeric simulation.We simulate the temperature effect on the delay line by increasing all dimensions in the propagation direction by 2% and decreasing the

Fig. 5

 5 Fig. 5 Compressed pulses, LFM case. Right curve compressed pulse at initial temperature, left curve -2% expanded chirp.We have repeated the same numeric experiment with similar device but with LFM transducer: having the same frequency band, dispersive

Plessky, V. P.: 'FEMSAW -an extremely accurate and powerful 2D FEM/BEM simulation tool for SAW devices', http://www.gvrtrade.com, accessed 16 nd August 2013

Morgan, D.P.: 'Surface-wave devices for signal processing' (Elsevier,Amsterdam, 1985)

Acknowledgments:

The authors thank S. Ballandras and J.-M. Friedt for constant support and numerous discussions V.Plessky and M. Lamothe (GVR Trade SA, Ch.de la Rose,10, 2025, Gorgier, Switzerland), E-mail: victor.plessky@gvrtrade.com M. Lamothe is also with CNRS, FEMTO ST, Besançon, France.