Francesco Arzani

Claude Fabre

Nicolas Treps

Versatile engineering of multimode squeezed states by optimizing the pump spectral profile in spontaneous parametric down-conversion

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Versatile engineering of multimode squeezed states by optimizing the pump spectral profile in spontaneous parametric down-conversion

I. INTRODUCTION

Quantum Information Processing using Continuous Variables (CV) of the electromagnetic field requires to efficiently generate various kinds of nonclassical states of light, such as squeezed or EPR entangled states. In addition, in CV quantum optics, it is relatively easy to scale up the dimensionality of the quantum system by increasing the number of modes of the field on which the quantum state spans. One of the most efficient and widely used techniques to produce in a deterministic way such multimode non-classical states of light is to use parametric down conversion in a χ (2) crystal. The quantum correlated, or "twin", photons are then produced in different pairs of EPR entangled modes, the number of which depends on the characteristics of the crystal and of the pump beam. These modes can be either spatial [1] or time/frequency [2] modes. In the case of frequency modes, which will be more specifically studied in this paper, the considered system is nothing else than the extension to the quantum domain of the classical WDM (Wavelength Division Multiplexing) technique, which has been so successful to increase by a large factor the performances of classical communications and information processing.

If one uses a single frequency pump, the different entangled signal and idler pairs modes, having frequencies symmetrically disposed with respect to the half pump frequency, are independent from each other. It is no longer the case when the pump spectrum contains more than one frequency. A bi-frequency pump already yields a quantum state with a complex structure of correlations [3][4][5] ; A mode-locked pump [6][7][8], producing trains of ultrashort pulses, i.e. a frequency comb having millions of teeth, produces an even richer state in terms of quantum correlations and multipartite entanglement [9,10]. Both systems are very promising sources of entangled cluster states, a basic tool in the recently rising do- * francesco.arzani@lkb.upmc.fr main of CV Measurement-Based Quantum Computing (MBQC) [11,12].

The problem of optimizing the shape of the pump beam for a most efficient generation of a specific multimode quantum state of light is then of paramount importance for Quantum Computing applications. In the present paper we tackle the problem by the use of an algorithmic approach, having in view the optimized generation of specific cluster states, and in mind the experimental way to shape the pump, which consists in modifying the pump laser spectrum, both in phase and amplitude, using pulse shapers based on the use of Spatial Light Modulators. We stress that such a setup allows to tailor many different pump shapes with no hardware modification to the experiment. Moreover, exploiting all the degrees of freedom provides great flexibility compared to engineering the phase-matching conditions or simply the width of the pump. The latter route has been explored before, often for Gaussian pulses only with at most quadratic spectral phase, especially in connection to the heralded production of single photons [13] or Fock states [14]. The focus of most earlier works on the subject was on the purity and entanglement of the signal and idler photons, which could be engineered to some extent tuning few parameters. This simplifications allowed to treat the problem analytically. The degree of control on the output state was correspondingly low and would not allow, for example, the optimization of a specific CV cluster state.

In the CV regime, which we are concerned with, the system is characterized by the quantum fluctuations in each mode. In this context, an analytic approach to general pump spectra with no spectral phase was developed for both spatial and temporal modes in [15]. However, the resulting theoretical profiles were not achievable with realistic experimental configurations. We show here that numerical optimizations can be fruitfully used to find the pump profiles producing multimode squeezed states with the properties needed for many different protocols. We also show that the numerical routines can be modified to take into account the physical limitations of a realistic pulse-shaper, ensuring that the optimized profiles are also experimentally realizable.

The article is organized as follows: in the next section we introduce mathematical tools based on Autonne-Takagi and Bloch-Messiah factorizations, that allow us to derive the covariance matrix of the output state in the frequency-mode basis for a pump field with an arbitrary spectral profile. We argue that these techniques are more suited to study type I collinear down-conversion, in which signal and idler photons are indistinguishable, than the singular value decomposition commonly used to treat non-degenerate down-conversion [16]. As an example we apply this formalism to the case of a spectrally chirped pump. In Sec. III we detail how the noise properties of an arbitrary set of temporal modes can be computed from the covariance matrix in the frequency basis. We introduce CV cluster states and explain how to compute their nullifiers for a given pump profile and a given set of modes. In Sec. IV we apply the described methods to the problem of the numerical optimization of the pump spectrum for several tasks: first we concentrate on the gains of the parametric down conversion. We then optimize the pump to produce cluster states whose nodes coincide with specific temporal modes having non-overlapping spectra. In Sec. V we summarize our results and comment on the generality of our approach, outlining how it may be fruitful for tackling problems beyond those detailed in the present work.

II. DERIVING THE OUTPUT STATE FROM THE PUMP SPECTRAL PROFILE

We study type I spatially degenerate parametric downconversion of an electric field containing a discrete set of N frequencies inside a bulk crystal with χ (2) nonlinearity. These frequencies may be, for example, the teeth of a frequency comb, or they may come from the discretization of a continuous broad band field, but our treatment can be applied to more general distributions of frequencies. Denoting by a j the annihilation operator at the frequency ω j , the quantum description of the process is given by the following hamiltonian in the pump interaction picture [15]

H I = i η 2 N j=1 k=1 L jk a † j a † k + h.c (1)
where h.c. means Hermitian conjugate. The real constant η depends on the single-photon energy, the nonlinear susceptibility, the intensity and the geometry of the pump field. We assume that the spectrum is narrow enough so that photons at all the relevant frequencies have approximately the same energy. We consider only one spatial mode and only one polarization for the pump. Due to the collinear Type I design, the signal will also be in a single spatial mode and its polarization will be orthogonal to that of the pump. Frequency is then the only relevant degree of freedom. The coupling matrix L is the well known joint spectral distribution and is given by

L jk = sinc (φ (ω j , ω k )) α (ω j + ω k ) . (2)
The first factor is the phase matching function, with sinc (x) = sin (x) /x, and φ the phase mismatch angle

φ (ω j , ω k ) = (k p (ω j + ω k ) -k s (ω j ) -k s (ω k )) l 2 (3)
k p (ω j + ω k ) being the wave number of the pump field at frequency ω j + ω k , k s (ω k ), the wave number of the signal field at frequency ω k and l denoting the length of the crystal. The dispersion relations for pump and signal fields can be computed using Sellmeier's equations (see Appendix A). The second factor in Eq. ( 2) is the complex spectral amplitude of the pump field. Earlier theoretical [17] and experimental [5,6] works have proven that hamiltonian (1) accurately describes the physics of SPDC in the squeezing regime we will consider. The joint spectral distribution has been widely studied in the specific case in which α (ω j + ω k ) is real for any j, k [15,16,18], namely when the pump has no spectral phase up to a global phase factor. Since by construction L is symmetric, if the pump has no spectral phase L can be diagonalized with an orthogonal matrix, leading to uncoupled modes which are independently squeezed [15]. A slighly more sophisticated treatment is required to include pump shapes having arbitrary spectral phases. Examples of non trivial spectral phases can be met in fairly common situations, for example in the presence of a quadratic phase (spectral chirp). Two different approaches are possible: either diagonalizing the joint spectral distribution by congruence [19][START_REF] Takagi | Japanese journal of mathematics: transactions and abstracts[END_REF][START_REF] Siegel | [END_REF] or applying the Bloch-Messiah decomposition [22,23] to the symplectic transformation corresponding to a finite-time evolution of the system under the effective hamiltonian of the field inside the crystal. Diagonalization of a complex symmetric matrix by a congruence transformation through a unitary matrix is also known in the literature as Autonne-Takagi factorization or symmetric singular value decomposition. BM decomposition is another special case of singular value decomposition for symplectic matrices and is also known in the literature as Euler decomposition for symplectic matrices. We shall now detail both approaches and show how they allow to find modes of the electric field whose evolution is decoupled inside the crystal.

A. Autonne-Takagi factorization

Every complex symmetric matrix can be diagonalized by a congruence transformation with a unitary matrix [START_REF]Numerical routines for Autonne-Takagi factorization can[END_REF]. This means that for any L in Eq. ( 2) one can find a unitary matrix V such that

V LV T = Λ (4)
with Λ a diagonal matrix with real, non-negative entries. Suppose such matrix V is known for a given L, then one can define the vector of annihilation operators

b ≡ V † a (5)
with a = (a 1 , a 2 , ..., a N ) T , each of which is a linear superposition of the single-frequency annihilation operators. Since V is unitary, the operators b correspond to a set of orthonormal modes whose spectral profile is given by the rows of V . Substituting in Eq. ( 1) and using Eq. ( 4) one finds

H I = i η 2 k Λ kk b † k 2 + h.c. ( 6
)
showing that the modes b k evolve independently, each according to a squeezing hamiltonian. These modes are referred to as supermodes in the literature. The singular values Λ kk (multiplied by the parameter η) correspond to the gains of the downconversion process. Note that having the same matrix V on both sides of L in Eq. ( 4) was crucial to find the same decoupled modes for signal and idler (and thus a single creation operator b † k for each k in Eq. ( 6)). Ordinary (non-symmetric) singular-values decomposition would in general lead to different mode bases for the two. This is not a concern when treating non-degenerate SPDC in either polarization or spatial mode, since the signal and idler photons are distinguishable. However, for the problem at hand, the parametric interaction is more naturally described in terms of independent single-mode squeezers. Using singular-values decomposition, as opposed to Autonne-Takagi factorization, would require additional steps to achieve this.

B. Bloch-Messiah decomposition

The previous approach solved the problem of finding the supermodes and the relative gains directly from the hamiltonian, which describes the differential evolution of the system. Although leading to the same physical results, it is sometimes more practical to work with the input-output relations corresponding to the evolution of the system for a finite time or its propagation over a finite crystal length. The main advantage is that fom this approach it is straighforward to derive the covariance matrix of the output state, encoding its noise properties. This is described in the following.

Consider the equations of motion for the annihilation operators in the Heisenberg picture

d dt a = i [H I , a] = ηL a † . (7)
Complementing this set of equations with their adjoint one has d dt

a a † = η L a a † (8)
where

L = 0 L L * 0 . (9)
Eq. ( 8) is readily integrated for a finite time t a (t)

a † (t) = exp η Lt a (0) a † (0) . ( 10
)
We define the amplitude and phase quadrature operators respectively as

q j = a + a † √ 2 (11)
p j = a -a † i √ 2 . ( 12
)
Introducing the matrix

C = 1 √ 2
I iI I -iI (13)
the quadrature operators of the relevant frequency modes are written as

q p = C † a a † . (14)
With this convention the shot noise ∆ 2 0 is normalized to 1/2. Combining this with Eq. ( 10) we find the expression for the finite-time evolution of the quadrature operators of frequency modes inside the crystal

S = C † exp η Lt C (15)
so that

q (t) p (t) = S q (0) p (0) (16)
the hamiltonian H I can being applied for a time t = l/c. Actually S is a spatial propagator corresponding to the input-output relation for the fields before and after the crystal. Changing the factor η can easily be achieved adjusting the pump power (as long as it stays in the lowgain or below threshold regime in a cavity setup, which is the domain in which H I can be derived in the form used here). Since H I is quadratic in the annihilation and creation operators, S is a symplectic matrix. The matrix C links it to its complex representation S (c) = exp(η Lt), appearing in Eq. ( 10) [22]. We can apply the Bloch-Messiah decomposition and find a factorization [23]

S = R 1 KR 2 (17)
where R 1 and R 2 are both symplectic and orthogonal matrices and K = diag {e r1 , e r2 , ..., e r N , e -r1 , e -r2 , ..., e -r N } is a squeezing matrix, namely a symplectic diagonal matrix. Its diagonal entries are the singular values of S. In our case, single-frequency modes are the input and output of the overall process, so R 2 = R -1 1 = R T 1 and S is symmetric [START_REF] Bloch | Messiah decomposition has a simple physical intrepretation[END_REF]. The spectral profiles of the supermodes are the rows of the unitary matrix U appearing in the complex representation of R 1 [22]

R (c) 1 ≡ CR 1 C † = diag {U, U * } . ( 18
)
As we will see in the next section, the supermodes found in this way are the same as those obtained through the Autonne-Takagi factorization.

In the hypothesis that the system was initially in the vacuum state, the covariance matrix in the frequency modes basis can also be computed from S as [START_REF] Ferraro | Gaussian states in quantum information[END_REF] Γ

ω = 1 2 SS T = 1 2 R 1 K 2 R T 1 . (19)
Note that it is not necessary to compute the Bloch-Messiah decomposition to get the covariance matrix from S.

C. Relating the two approaches

Given the Autonne-Takagi factorization of L, it is straightforward to compute the Bloch-Messiah decomposition of S. In fact, defining

R 1 = C † diag V † , V T C (20)
R 2 = R T 1 ( 21
)
one finds

R T 1 SR T 2 = R † 1 SR † 2 (22) = C † exp ηt 0 V LV T V * L * V † 0 C (23) = exp ηt Λ 0 0 -Λ = K ( 24
)
where K is the same as in Eq. ( 17) (up to permutations of the diagonal elements). The advantage of using Autonne-Takagi factorization is that it is numerically easier to compute with respect to Bloch-Messiah decomposition. The link between Autonne-Takagi factorization and Bloch-Messiah decomposition was also recently noted in [START_REF] Cariolaro | [END_REF].

D. Numerical simulations

Most of our results are obtained through numerical simulations, the details of which can be found in Appendix B. For our examples, we take the unshaped pump α (g) (ω) to be a Gaussian pulse of spectral width about ∆λ ≈ 3.54 nm full-width-half-maximum (FWHM) centered around λ 0 = 397.5 nm, which can be obtained by upconversion of a 10 nm pulse FWHM, corresponding to a duration of about 100 fs, centered around 795 nm. We consider free-space setups and assume the nonlinearity is provided by bulk BIBO crystals of length between 0.5 mm and 2 mm, whose refractive indexes are computed using Sellmeyer's equations (see Appendix A for details). We denote the unshaped spectral profile by

α (g) (ω) = 1 σ ω √ 2π e - (ω-ω 0 ) 2 4σ 2 ω ( 25
)
with ω 0 = 2πc/λ 0 and σ ω = ω 2 0 ∆λ/4πc √ 2 ln 2, c being the speed of light in vacuum.

In previous works considering a real pump with a Gaussian spectrum [15], it was noted that the diagonalization of L leads to alternating signs in the gains, meaning that the supermodes are squeezed in alternating quadratures. This actually comes from imposing that the spectral profile of the supermodes is real, which is possible because the supermodes have a trivial spectral phase. An equivalent choice would be to define the supermodes to be all squeezed in the same quadrature, which amounts to multiplying the spectral amplitudes of half of the supermodes by i. In fact, multiplying a row of V by i in Eq. ( 4) flips the sign of the corresponding diagonal element in Λ and rotates the squeezing direction by π/2 in phase space. Defining the modes such that the phase quadrature is always the squeezed one is more suited to handle the case in which the pump has a non-trivial spectral phase. The reason is that in this case supermodes may have nontrivial specral phases as well, as we shall see, so there is no simple criterion to choose which quadrature should be squeezed based on supermodes.

To keep close to an experimental scenario, we assume that the spectral profile of the pump is modified by a pulse shaper, which can be built with a spatial light modulator in a 4-f configuration [28]. We model this as a device with a finite number of degrees of freedom, parametrized by the real vector u, corresponding to spectral amplitude and phase at given frequencies. The resulting pump profile α ( u) (ω) is found multiplying the unshaped amplitude by a transfer function I ( u) (ω) interpolating these parameters

α ( u) (ω) = α (g) (ω) I ( u) (ω) . ( 26
)
More details can be found in Appendix C. It is worth clarifying how we derive physical values for the squeezing of the supermodes. These are proportional to the factor η in the Hamiltonian of Eq. ( 1), which is generally difficult to compute accurately from first principles. For our purposes, it will be more convenient to adjust it so that the squeezing of the first supermode approximately matches the experimentally measured value. Once the highest squeezing is fixed, the ratio between the squeezing parameters of the supermodes is the same for any pump power below threshold [15].

E. An example: Chirped pump As a first example of a pump with a non-trivial spectral phase we consider a gaussian pump with a quadratic spectral phase, namely a spectrally chirped pump of amplitude

α (ch) (ω) = α (g) (ω) e i φ 2 2 (ω-ω0) 2 (27)
where φ 2 is the quadratic phase. Spectral chirp is fairly common in experimental situations, often as an unwanted effect, so it is interesting to study its impact on the downconversion process. The quadratic spectral phase implies that the pulse is no longer Fourier limited: the duration of the pulse increases while the spectrum remains constant. This makes the duration of the pulse a useful parameter to characterize the amount of chirp. If ∆t = 1/(2σ ω ) is the duration of the un-chirped pulse [START_REF]With the convention ∆t = 1/(2σω), ∆t[END_REF] (φ 2 = 0), the duration after chirp is [30]

∆t = ∆t 1 + φ 2 2∆t 2 2 . ( 28
)
Studying the dependence of the output state it is then natural to ask how much modification is really due to the spectral phase and how much is just a consequence of the increased duration. It turns out that the two situations are very different, as can be seen from the plots in Fig. 1. We compare, for the two cases, the largest parametric gain (Fig. 1a) as well as the first one hundred parametric gains (Fig. 1b) as functions of ∆t /∆t. The plots were obtained for a fixed energy in each pump pulse. We assume the downconversion of a pulse with ∆t ≈ 30 fs takes place in a 0.5 mm BIBO crystal. All the gains are normalized to the highest gain for φ 2 = 0 and ∆t /∆t = 1. In both cases, the gain of the first supermode Λ 11 increases with ∆t at first but then starts decreasing. However, the descent is steeper in the chirped case. Moreover, numerically we find that for increasing quadratic phase

Λ tot = j Λ 2 jj = const. ( 29
)
within machine precision [31], whereas Λ tot monotonically increases for un-chirped pulses of longer duration.

To get a physical picture of Λ tot , consider the perturbative expansion of the evolution for small times/pump power/nonlinearity. The Λ jj are then seen to be proportional to the probability amplitude for a pump photon to be converted into two photons in the supermode j. In fact, applying the evolution operator for a small time δt to the vacuum one gets

U (δt) |0 = ∞ l=0 (-iδtH I ) l l! l |0 (30) = I + δt η 2 k Λ kk b † k 2 + O δt 2 |0 . (31)
The sum of Λ 2 jj is then proportional to the probability of converting a photon of the pump into two photons in any supermode within time δt. This can be interpreted as the conservation of the overall efficiency of the downconversion process for increasing quadratic phase. On the other hand, it is clear that the process is not insensitive to the quadratic spectral phase: more signal modes are excited as the quadratic phase increases, while the highest gain for a single mode decreases. The overall efficiency increases for un-chirped pulses of longer duration, but the magnitude of the gains drops faster. As a consequence, for large ∆t the number of modes with approximately the same squeezing is higher for a chirped pump, as can be seen from Fig. 1b. Chirp can be added easily in experiments at constant pump power, whereas changing the pulse duration generally involves losses. The former may then be more convenient.

Fig. 1c shows spectral amplitude and phase of the first supermode obtained for φ 2 ≈ 2700 fs 2 , the quadratic phase doubling the duration of the pulse. For the plot, we subtracted a linear term from the spectral phase, which only amounts to a temporal delay. Interestingly, the remaining spectral phase is not quadratic, as in the pump. Instead, it is well fitted by a cubic term

φ fit (ω) = e iφ3(ω-ω0/2) 3 . (32)
The same cubic phase fits well the spectral phase of all the supermodes and is thus an important effect to take into account in experiments. The coefficient φ 3 seems to have a non-trivial dependence on φ 2 . A systematic study of the effect of chirp is however beyond the scope of the present work and is left to future investigations. We only wish to highlight here that Autonne-Takagi factorization can be used to study pump fields with arbitrary spectral shapes and this can lead to the discovery of new and interesting features already in quite simple situations.

III. NOISE PROPERTIES OF THE OUTPUT STATE

Here we introduce the formalism we will use to compute the relevant measurable quantities of the output state from the covariance matrix in the frequency basis.

A. Noise of a set of modes

The noise properties of any mode can be computed from the covariance matrix in the frequency basis Γ ω . Take the mode corresponding to the annihilation operator

d = l v l a l ( 33
)
where the v l are complex numbers satisfying l |v l | 2 = 1. v l is the complex amplitude of the electric field mode at frequency ω l . The quadratures of d [START_REF]We will identify modes with their annihilation operator[END_REF] are given by

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1.
q (d) = l (Re (v l ) q l -Im (v l ) p l ) (34)
p (d) = l (Im (v l ) q l + Re (v l ) p l ) . ( 35
)
Consider now a set of M ≤ N orthogonal modes related to the frequency modes by

d = D a ( 36
)
where the matrix D has M × N complex entries. The orthonormalization condition of the modes takes the form

DD † = I M . ( 37
)
The quadratures of modes d are then given by

q (d) p (d) = Re (D) -Im (D) Im (D) Re (D) q p ≡ R D q p .
(38) The covariance matrix of the modes d is then obtained from that of frequency modes as

Γ d = R D Γ ω R T D . ( 39
)
When M < N , the transformation in Eq. ( 39) can be understood as changing the modes to a basis of which d constitute the first M elements and then discarding the remaining modes (which amounts to removing the corresponding rows and columns from the covariance matrix).

B. Cluster states and nullifiers

One of the main goals of our work is to exploit the methods outlined above in optimization routines to find the shape of the pump which is best suited to produce CV cluster states on a given set of modes. In order to do this, we recall that a CV cluster state is a multimode state which, in its ideal version, can be defined as the simultaneous eigenstate of a set of operators called nullifiers. If G is the graph associated with the cluster state, which we will identify with its adjacency matrix, nullifiers can be written as

δ = p (d) -G q (d) . ( 40
)
We assumed here that the nodes of the cluster correspond to the generic modes d of Eq. (36). Although more general situations can be considered [START_REF] Menicucci | [END_REF], we will restrict to unit-weight cluster states. In this case G jk = 1 if and only if modes j and k are nearest neighbours in the graph and all the other entries of G are zero. Different conditions may be required to certify the experimental production of cluster states, but a basic one is that the noise of the nullifier operators lay below the vacuum noise. Standard homodyne detection techniques are sufficient to measure the quantum fluctuations of these operators, as explained in Appendix E.

IV. PUMP OPTIMIZATION

From the previous sections it should be clear that the relation between the spectral profile of the pump and the properties of the output state is far from trivial. As a consequence, it is generally very difficult to find an analytical form for the pump optimizing a given property of the output, such as the entanglement pattern of a given set of modes. Instead, one could run a numerical optimization algorithm to try and improve the desired quantities. The results obtained with this approach make the object of the following sections. For the optimization we used an evolutionary algorithm developed in [34], of which some details are given in Appendix D.

A. Squeezing spectrum

We have already seen in Sec. II E that changing the spectral profile of the pump can impact the squeezing spectrum. We investigate here to which extent this can be used to enhance a given property. Specifically, we look for the spectral profiles that flatten the squeezing spectrum, equalizing the first k gains, or separate the highest gain from the others, effectively concentrating more squeezing in the first supermode. For the first task, we run the optimization for the fitness function

f 1 ( u) = 1 Λ 11 ( u) k j=1 Λ jj ( u) (41)
where u are the shaper's parameters. At this point we are not concerned with the absolute value of the gains, which can in principle be adjusted changing the power of the pump, so we divide all the gains by the largest one.

For the second task, we run the optimization with the fitness function

f 2 ( u) = Λ 11 ( u) Λ 22 ( u) . (42)
Note that since we are only interested in the gains, which are the singular values of the joint spectral distribution, and not the shape of the supermodes, we can use commonly available numerical routines for the SVD.

For the optimization to be meaningful some constraints have to be imposed. Indeed, if no constraint is imposed, the algorithm may converge to solutions which have a very small overlap with the Gaussian pulse that would be obtained without the shaper. This is a problem because, since the shaper is a passive optical component, it means that much of the power in the pulse is thrown away in the process and a very high power would be needed to realize such profiles. Optimization is however interesting because it makes clear that the "amount" of squeezing and its distribution among different modes are very different resources, as will be especially evident in the following sections about cluster states. More realistic profiles can be obtained with a modification to the fitness function which adds a weight hindering convergence towards profiles having a small overlap with the original Gaussian. To this end one can add a function of the power of the shaped pump, renormalized by the maximum of the shaper's transfer function to impose that the shaper is only attenuating [35]. The power of the shaped pump is given by

w ( u) = 1 m ( u) 2 dω α ( u) (ω) 2 (43)
where

m ( u) = max ω I ( u) (ω) . (44)
The fitness functions f 1 and f 2 are then replaced by

f1 ( u) = 1 Λ 11 ( u) k j=1 Λ jj ( u) + a • x (w ( u)) (45) f2 ( u) = Λ 11 ( u) Λ 22 ( u) + b • y (w ( u)) (46)
with a and b positive real numbers. x and y may be arbitrary functions. A possible criterion to choose such a function may be that it should be negligible if the power is above some fraction of the original Gaussian and very rapidly becomes negative and large if the power is below this threshold. Solutions with a power lower than the threshold are then disfavoured but the weight does not influence the optimization as long as the power stays "acceptable". The magnitude of a and b can then be used for fine tuning. Fig. 2 shows the results of two optimizations starting from a reference Gaussian spectrum with a shaper working in a window of about 9 nm, corresponding to ±3 standard deviations (in amplitude) around the central frequency, and a 1.5 mm BIBO crystal for downconversion.

The supermodes resulting from the optimized pumps are shown in Fig. 2b. At first sight both amplitude and phase seem very complicated, except for the first supermode arising from the optimization f2 . This is a good sign, because the first supermode is the most interesting one in this case, being by far the most squeezed. As for the others, the apparent complexity may be explained and overcome by the quasi-degeneracy of the gains. See Appendix F for details. We stress that the optimization algorithm is stochastic and there is no guarantee that the optima are also global optima. Our aim here is to show that optimizing the shaper's cofiguration we could find pump profiles giving a significant improvement on the initial Gaussian. When f1 is optimized, the squeezing spectrum is made flatter, with Λ jj > 0.9Λ 11 for j up to ∼ 80, to be compared with j ∼ 30 for a Gaussian pump. Optimizing f2 we found that a noticeable gap can be induced between the first and second gains, in this case Λ 11 /Λ 22 ≈ 1.43, to be compared with Λ 11 /Λ 22 ≈ 1.00 for a Gaussian pump.

We stress that this quantity is essentially unrelated to the Schmidt number. Optimizing f 2 or f2 leads to a higer Schmidt number than that obtained with an unshaped pump. The reason is that many modes after the first one have almost the same squeezing. This kind of configuration could be interesting for example to work in a regime in which only the first supermode is pumped slightly above threshold. The classical noise of the pump laser would then be conveyed in the first supermode, improving the purity of the others, which would still be squeezed [36]. The pump optimizing f1 carries about 30% of the power of the unshaped Gaussian pump, while this figure is about 40% for the pump optimizing f2 , meaning they may realistically be implemented in the lab. The opportunity of introducing the modifications in Eq. ( 45) and Eq. ( 46) is made more evident by comparison with the power of the pump profiles optimizing f 1 and f 2 (not shown here), which is of the order of 0.1% of the unshaped Gaussian.

The procedure outlined in this section can be carried out for any function that can be written in terms of the shaper's parameters u. For example maximizing the gain of the first supermode for a given maximum power, minimizing the spectral width of the first supermode, maximizing or minimizing the Schmidt number of the parametric gains as defined in [37,38], which gives a measure of the number of modes excited in the process. An example of interest for quantum information processing is treated in the following section.

B. Cluster states on frexels

We detail here how to optimize the profile of the pump to reduce the noise of the nullifiers of CV cluster states when the nodes of the graph are associated with a specific set of modes which have non-overlapping spectra.

Definining the detection modes: frexels

We turn our attention to a specific set of m orthogonal modes which are slices of a Gaussian pulse. We refer to these as frexel modes (from "frequency elements") and denote their annihilation operators by {π j }. frexels can be seen as a specific realization of the modes d in Eq. (36). First, we choose a set of frequency bands of limits (Ω 1 , Ω 2 ) , ..., (Ω m , Ω m+1 ). The frexel modes are then defined by the spectral amplitudes

π j (ω) = e iθ j √ Nj α (π) (ω) Ω j ≤ ω ≤ Ω j+1 π j (ω) = 0 otherwise (47)
where α (π) is a Gaussian pulse with a FWHM of 10 nm centered around 2λ 0 = 795 nm, θ j are arbitrary phases, which will turn out to be useful in the following, and [39]

N j = Ωj+1 Ωj dω α (π) (ω) 2 . (48)
An example with four frexels is depicted in Fig. 3a. The interest of these modes resides in the fact that, having non-overlapping spectra, they can be physically separated rather easily from one another using a prism or a grating [40]. It is worth noting that, in principle, modes with an arbitrary spectral profile could be separated from a bunch of co-propagating modes [41,42], but this would involve nonlinear interactions which would make it unpractical to separate more than one mode from all the others. MBQC with frequency or spatial pixel modes was also introduced in [43]. After being separated, frexels could be sent to different parties in a network or directly subject to independent homodyne measurements, for example. Indeed, the availability of multi-pixel homodyne detection schemes [44] is the main reason to introduce an overall Gaussian spectrum in the definition of frexel modes and an individual phase θ j for each of them. The latter could be adjusted simply changing the phase of the local oscillator in each frequency band. This is an important degree of freedom to consider, as a phase shift of the local oscillator implies the measurement of a different quadrature, which is at the heart of CV-MBQC [12]. Also, although a local phase-shift cannot change the amount of entanglement between frexels, it can change the kind of quantum correlations. As it will be clear from the next section, this is especially relevant for CV cluster states, which have a very specific type of correlations, resulting in the reduction of noise in the nullfier operators.

Finding the optimal frexel permutation

We consider, as an example, four frexels, and associate π j to the jth node of the 4-mode linear cluster state depicted in Fig. 3b, corresponding to the graph with adjacency matrix

G lin =    0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0    . (49)
This cluster state is universal for single-mode Gaussian MBQC [45]. We can compute the variance of nullifiers using the procedure explained in Sec. III B for a general set of modes. The choice of the local phases θ j defines which quadratures correspond to amplitude q π and which to phase p π . Since we assumed to be free to choose an independent phase reference for each pixel, we can use the θ j giving the lowest fluctuations for the nullifiers on average. For the numerical calculation, we assume that the unshaped pump is a Gaussian of amplitude α (g) (see

                                                                                                        

FIG. 2. (Color online) (a)

The normalized gain distributions obtained for a Gaussian pump and after optimizing f1 in Eq. ( 45), with a = 3, x (w) = 1/ (5w) 6 and k = 100 and f2 in Eq. ( 46), with b = 1, y (w) = 1/ (5w) 6 . (b) The first supermodes resulting from the pump optimizing f1 (top) and f2 (bottom). The solid blue line represents the amplitude, in arbitrary units, while the orange dashed line represents phase, in radiants (scale on the right). For clarity of representation we subtracted a linear phase of 260, 812 and 805 fs from the supermodes arising from the optimization of f1 and of 275, 275 and -390 fs from the supermodes arising from the optimization of f2. (c) and (d) show the pump profiles maximizing f1 and f2, respectively. The gray dashed line shows the original Gaussian, the solid blue line the optimal amplitude profile and the red dotted line the optimal phase.

Eq. ( 25)) and that down conversion happens in a 0.5 mm BIBO crystal. We fix the pump power so that the squeezing in the leading supermode is 7 dB. We take 2πc/Ω 1 808 nm and 2πc/Ω 5 782 nm. The average of nullifiers' variances is found to be ∆ 2 avg δ 0.49 (vacuum is normalized to 0.5), which amounts to a noise reduction of about -0.08 dB. The same calculation may be carried out for any permutation σ of frexels, namely assigning π σ(j) to node j on the graph. It turns out that some permutations allow to sensibly reduce the average noise of nullifiers. For example ∆ 2 avg δ 0.29 for the permutation σ 2 = (π 1 , π 4 , π 2 , π 3 ), corresponding to about -2.35 dB. This may look surprising at first, since a simple relabeling of the modes cannot change the amount of entanglement. The point is that nullifiers' noise reduction is not just a signature of entanglement, but rather of very specific correlations among the nodes of the corresponding graph, and these may very well vary from one permutation to the other. In our example, the linear graph has a link between nodes 1 and 2, corresponding to frexels π 1 and π 2 if the trivial permutation is considered and to frexels π 1 and π 4 if one instead considers the σ 2 . Being symmetric with respect to the central frequency, we expect frexels π 1 and π 4 to be more entangled after the downconversion than frexels π 1 and π 2 whose spectra are on the same side of the central frequency of the downconverted field. We then expect a better noise reduction in the corresponding nullifier. The permutation σ 2 is actually the optimal for the conditions considered here. The second permutation σ2 leads to smaller nullifiers' noise for an appropriate choice of the global phase of each pixel (not shown in the drawing).

Optimal pump profiles

Starting from the best permutation in the previous section, we used numerical optimization to find the pump profiles minimizing the function [START_REF]Γδδ is a covariance matrix, so it is positive-semidefinite by construction[END_REF]

f 3 ( u) = Tr [Γδδ ( u)] (50)
with Γδδ defined as in Eq. (E4) for the four-modes linear cluster. For the optimization, we start from a reference Gaussian pump and assume the shaper is acting on a spectral window of ±2 standard deviations around the central frequency, corresponding to approximately 95% of the pump power. First, we fixed the squeezing of the leading supermode to 7 dB, which is consistent with the highest values measured in experiments with frequency combs [7]. Analogously to the case of the squeezing specthe algorithm converges to profiles which have a small overlap with the original pulse, so we also ran the optimization for the modified function

f3 ( u) = Tr [Γδδ ( u)] -h • w ( u) (51)
where h is a positive real number and w is defined as in Eq. ( 43). The results are shown in Fig. 4. Optimization of f 3 leads to a larger improvement of the nullifiers squeezing on average, but as shown in Fig. 4b the corresponding pump profile has a small overlap with the original Gaussian. As a consequence, the shaped pulse only contains ∼ 2% of the power of the unshaped pulse. Optimization of f3 leads to a profile (Fig. 4c) that still allows to reduce the average nullifiers' noise of about 0.5 dB with respect to the Gaussian profile while containing ∼ 80% of the Gaussian pulse's power. This could lead to a measurable improvement in realistic experimental conditions. The compromise between power in the shaped pump and noise reduction can be tuned changing the parameter h in Eq. ( 51) in order to adapt to specific experimental constraints. If more power is available, for example, the optimization could be performed for smaller values of h.

Relation between highest squeezing and nullifiers' noise

It is interesting to compute what happens when one changes the pump power keeping the shaper's configuration fixed. As long as the low-gain or below-threshold conditions are satisfied, this should just multiply the gains by a common factor. One could try and guess that more power, meaning a higher squeezing in all supermodes, would imply better noise reduction for the nullifiers. This is not actually the case, as can be seen from Fig. 5. In fact, the average nullifiers' noise is reduced from the shot noise until a certain value of the squeezing of the first supermode. If the power of the pump is further increased, the average nullifiers' noise starts increasing as well. One explanation could be that the number of squeezed modes in the system largely exceeds the number of frexels, so the contribution of all anti-squeezed quadratures to the nullifiers cannot be made arbitrarily small. The optimal configuration is found minimizing the contribution of the leading anti-squeezed quadratures. But even if the remaining anti-squeezed quadratures appear in the nullifiers with very small coefficients, at some point the corresponding noise will dominate, since it grows indefinitely with the gain.

Running the optimization with the squeezing of the leading supermode set to a different value results in a different optimal pump profile. With this different profile, the average nullifiers' noise will attain a minimum when the squeezing of the first supermode is close to the one chosen for the optimization. An example is shown in Fig. 5b, where the average nullifiers' noise as a function of the leading squeezing for a Gaussian pump and two profiles optimized at different leading squeezing are compared.

V. CONCLUSIONS

In summary, we showed that pump shaping can be used effectively to engineer the quantum state produced by the spontaneous parametric down-conversion of a frequency comb. To this end, we introduced a method, based on either Autonne-Takagi or, equivalently, Bloch-Messiah decomposition, that can be applied to the numerical study of any spectral profile, including pulses with a general frequency-dependent phase. As a first example, we used this method to study the effect of spectral chirping, which is commonly met in experiments with frequency combs, on the down conversion process. We found that the quadratic phase has a non trivial effect on the parametric gains and on the spectral profile of the supermodes. Furthermore, using an optimization algorithm we found optimal profiles for flattening the values of the parametric gains or creating a gap between the gain of the first Gaussian and second supermodes. In both cases we showed that the shape of the pump has a macroscopic effect on the output state which can lead to measurable improvements in realistic experimental conditions. We then applied the same technique to find the pump profiles which are optimal to produce CV cluster states when the nodes of the cluster correspond to spectral slices of a Gaussian pulse. We focused on a four-mode linear cluster state. This is universal for single-mode Gaussian CV-MBQC, so our results are directly applicable to CV-MBQC with frexel modes. Similar results can be obtained for different graphs, such as the six-mode centered pentagon used for CV secret sharing protocols in [START_REF] Van Loock | AIP Conference Proceedings[END_REF] and [8].

f 3 f 3 -6 -5 -4 -3 -2 -1 0 Δδ 1 Δδ 2 Δδ 3 Δδ 4 (dB)
We stress that our approach is very general and, besides the examples cited here, it can be applied with small modifications to optimize any property of the output state after the down-conversion, such as the squeezing of the leading supermode or the Schmidt number. The same approach was used, for example, in a recent work proposing the simulation of quantum complex networks with an all-optical setup [START_REF] Nokkala | [END_REF].

Finally, we note that our results rely on the use of a non-deterministic optimization routine. Our goal was to show the effectiveness of the overall approach but we did not compare the performances of this specific algorithm with others. On the other hand, the general procedure is the same if a different routine is used. The results may then potentially be improved using a different optimization algorithm. Also, conceptually the same approach can be used in closed-loop experiments in which the fitness function is replaced by a measured quantity.

the state is mixed in this coarse grained desription, but our approximation turns out to be very good as long as the number of frequencies we take into account is large enough to represent all the supermodes which are significantly squeezed. Throughout this work, frequency modes will be identified with the coarse grained frequency pixels, although analytical calculations rigorously hold only for the teeth of the comb. variances and covariances of the normalized nullifier operators, even if the corresponding modes, defined by the rows of W in Eq. (E3), are not always orthogonal.

Appendix F: Linear combinations of quasi-degenerate supermodes

Consider the first supermodes resulting from the optimization of f1 and the associated gains λ j ≡ Λ jj . Since λ 20 > 0.99λ 1 and λ 30 > 0.97λ 1 , when the first supermode has 5 dB of squeezing, the difference of squeezing with the thirtieth supermode is about ≈ 0.13 dB, while the squeezing of the twentieth supermode differs by less than 0.05 dB from that of the first. This difference would hardly be detectable in experiments. As a consequence, it is reasonable to ask whether a linear combination of the first supermodes could by approximated by a simpler shape. Note that the coefficients in these linear combinations need to be real if one wants the resulting mode to be squeezed. One finds that a real Gaussian mode of about 37 nm FWHM has more than 92% overlap with a real combination of the 30 first supermodes. If one allows for a linear phase, which only amounts to a delay, as we noted earlier, the overlap is about 98% for a Gaussian amplitude of about 24 nm FWHM considering a real combination of 22 supermodes. The state of the modes resulting from these superpositions of supermodes would be slighly inpure but would nonetheless display squeezing in one quadrature which would exceed that of the least squeezed supermode in the linear combination.

FIG. 1 .

 1 FIG. 1. (Color online) Comparison between the effect of the quadratic phase and simply increasing the pulse duration. (a) Largest parametric gain for a chirped (dotted green curve) and non-chirped (solid blue curve) as a function of the increase in pulse duration ∆t /∆t. The gains are all divided by the largest parametric gain for φ2 = 0 and ∆t /∆t = 1, denoted by Λ (0) 11 . (b). First one hundred gains for increasing pulse duration for chirped (dark blue) and non-chirped (light green) pulses (normalized as in (a)). (c) Spectral amplitude (blue, solid line) and phase (orange, dotted line) of the first supermode obtained with ∆t = 2∆t (φ2 ≈ 2700 fs -2 ) compared to the first supermode for φ2 = 0 (gray, dashed line).

                        

FIG. 3 .

 3 FIG. 3. (Color online) (a) Spectral amplitude of four frexels within 3 standard deviations around the central frequency of the downconverted comb. The amplitudes are not normalized for clarity of representation. (b) A linear four-modes cluster state and two possible mappings of frexels onto its nodes. The second permutation σ2 leads to smaller nullifiers' noise for an appropriate choice of the global phase of each pixel (not shown in the drawing).

FIG. 4 .FIG. 5 .

 45 FIG. 4. (Color online)Results of the optimization of the pump shape to reduce the average noise of the nullifiers of a four-modes linear cluster. (a) shows the nullifiers' noise reduction in dB for a Gaussian pump and for the optimal profiles found optimizing f3 (Eq. (50)) and f3 (Eq. (51)) with c = 1.35. The squeezing of the leading supermode was fixed to 7 dB. The bar on the left of each triplet (dark blue) corresponds to the Gaussian case, the central bar (dark green) to f3, the bar on the right (light green) to f3. The horizontal lines show the average squeezing in each case (The Gaussian case, f3 and f3 correspond to the top, middle and bottom lines, respectively). The pump profiles optimizing f3 and f3 are shown in (b) and (c) respectively. The scale on the left refers to amplitude, while that for the phase is on the right.

ACKNOWLEDGMENTS

We thank Valentina Parigi, Adrien Dufour, Giuseppe Patera, Valerian Thiel, Dmitri Horoshko and Mikhail Kolobov for useful discussions. We thank Tobias Lipfert for interesting discussions and for pointing out ref. [49] to us. We are grateful to the anonymous referee for the careful review and for providing constructive inputs on earlier versions of the manuscript. This work is supported by the European Union Grant QCUMbER (No. 665148) and the French National Research Agency projects COMB and SPOCQ. N. T. is a member of the Institut Universitaire de France.

Appendix A: Phase-matching SPDC in BiBO

This appendix briefly reviews how phase matching is achieved for SPDC in BiB 3 O 6 crystals, commonly known as BiBO [50]. In practice birefringence is exploited to match the propagation velocity of pump and signal/idler fields.

BiBO is a biaxial crystal. The dispersion relations for polarized light propagating along one of the axis x, y or z can be computed using Sellmeier's equations

where i = x, y, z and λ is the wavelength. The Sellmeier's coefficients are

n x 3.07403 0.03231 0.03163 0.013376 n y 3.16940 0.03717 0.03483 0.01827 n z 3.6545 0.05112 0.03713 0.02261 Consider a plane wave of wave vector k propagating in the medium. We denote by Π the plane perpendicular to k and containing the origin of the ellipsoid E of indices. For historical reasons, the phase mathing angles θ and φ describing the rotation of E with respect to its axes is described with geographical coordinates, so the triad of axis is left-handed. φ is the angle from the xz plane to the yz plane and θ is the angle from y to z. The refractive index for given wavelength and propagation direction is determined through

.

(A2) According to [50], BiBO can phase-match Type I (e+e → o) processes with φ = π/2 for signal and idler and θ varying depending on the fundamental wavelength. For SPDC, this means that we can take the pump field polarized along x (θ = 0) and the polarization of signal and idler in the yz plane. Eq. (A2) gives for the the refraction index of signal and idler

We consider a collinear configuration and denote by 2ω 0 the central frequency of the pump. The down-converted field will then be centered around ω 0 . The phase matching condition requires that the phase mistmatch (Eq. ( 3)) is zero for the central frequencies

Eq. (A4) is then satisfied if n e (2πc/ω 0 , θ) = n x (πc/ω 0 ). Assuming that the central wavelength of the pump is 2πc/2ω 0 = 397.5 nm, this is achieved for θ = 2.63214 (θ = 150.811 • ).

Appendix B: Details of the numerical simulation

We are mainly concerned with optical frequency combs, in which case the number of frequency modes involved is of the order of 10 5 . Using the full comb to describe the system would make the problem numerically intractable. We adopt then a coarse-grained description of the system, treating first the comb as a continuum and then discretizing the problem. This may also be motivated by the fact that the free spectral range is too small for the single teeth of the comb to be resolved in the experiments. Moreover, the average photon number for the single frequency is too low to display quantum features, each frequency mode being essentially in the vacuum state. We took about 500 points for the discretization, which is close to the number of the physical pixels of commercially available pulse shapers. Obviously, ideally The shaper is modeled as a function interpolating the values u of amplitude and phase at 32 frequencies within a spectral window centered at the central frequency of the Gaussian pump comb ω 0 . For our calculations we chose the half width of the window to be two or three times the standard deviation of the Gaussian. The Gaussian comb is then multiplied by the function I ( u) (ω) to obtain the shaped pump. The number of frequencies independently controlled by the shaper is taken to be between 20 and 40, which is compatible with the spectral resolution of the shaper in a 4-f configuration [28]. As a consequence, the complex amplitude of the pump pulse after shaping, as well as the spectrum of the supermodes and the respective gains, will depend on the vector of real parameters u. Interpolation is needed to smooth the solution for the output comb and obtain pulse shapes which could be practically realized in experiments. In fact, despite the large number of pixels in the shaper, the configuration of neighbouring pixels is correlated due to electromagnetic interactions, which makes, for example, a π phase between neighbouring pixels practically impossible to realize. We chose to interpolate the function with cubic polynomials.

Appendix D: Optimization algorithm

The algorithm mimicks Darwinian evolution to stochastically explore the parameter space and uses statistical analysis to find the direction of fastest ascent of a fitness function. It goes as follows: first, a point in the parameter space is chosen at random. A new generation, that is a number of mutations (which grows logarithmically with the dimension of the parameter space) is generated around the first point. At the first iteration, the mutations are generated according to an isotropic Gaussian distribution. The fitness function is evaluated for each mutant. The best half of the mutants are linearly combined to generate a new starting point for the algorithm. Since the algorithm was initially developed for applications in experiments, to mitigate the effect of experimental noise the new point is actually a combination of the mutants and the starting points of previous generations. Statistical analysis is then performed on the current generation to find the axes corresponding to greater improvement of the fitness function. The covariance matrix for the next generation is modified accordingly, stretching the corresponding axes. A general step size parameter is also adjusted as follows: if the direction of fastest ascent was roughly the same in the last generations, then the algorithm is travelling in the good direction and the step-size is increased. If the direction changed many times over the last generations, the algorithm is probably close to an optimum and the step-size is decreased to accelerate convergence.

In our case the parameters of the optimization will be the vector u of amplitude and phase parameters of the shaper introduced in Sec. II D and the fitness functions will be derived from properties of the output state obtained pumping the down-conversion process with the corresponding shaped pulse.

Appendix E: Nullifiers's fluctuations through homodyne detection

In fact, even though each δ j in Eq. ( 40) is not the quadrature of a mode, its normalized version is. Let us define δj ≡ r j δ j where r j is a real number such that

if the field is in the vacuum state. Then it is possible to find a mode whose amplitude quadrature is precisely δj . The normalization r j is readily computed as r j = 1/ 1 + N (j), with N (j) the number of nearest neighbours of node j.

Using the definition of quadratures for the d modes

2 and Eq. ( 36), δj may be rewritten as

) where A j is the annihilation operator associated with the mode defined by the spectral amplitudes

These are the amplitudes of the electic field to print on the local oscillator in order to measure δj . They may as well be used to define a transformation R W analogous to R D in Eq. ( 38). Accordingly, one finds the covariance matrix associated with the nullifiers, which contains their squeezing as well as correlations between them and the conjugated operators ζj

For an ideal cluster state Γδδ → 0 [START_REF] Menicucci | [END_REF]. Note that Γδ is a legitimate covariance matrix, in the sense that it contains


Anna Maria Vegni
email: annamaria.vegni@uniroma3.it

Valeria Loscrì
email: valeria.loscri@inria.fr

Pietro Manzoni
email: pmanzoni@disca.upv.es

Valeria Loscrí

Data Forwarding Techniques Based on Graph Theory Metrics in Vehicular Social Networks

whether they are published or not. The documents may come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

Research on vehicular networks is a challenging area due to the rapid topological changes and the high variability of the data links. Nevertheless, several automobile manufactures are investing on research on smart intervehicle communication (IVC) approaches, and a further push towards such direction is also coming by the advent of Internet of Vehicles (IoV) where different kinds of interactions among vehicles (V2V), vehicle and infrastructures (V2I) or vehicle and pedestrian (V2P) can be envisaged [START_REF] Maglaras | Social internet of vehicles for smart cities[END_REF]. On a parallel (research) lane, wireless social networks have gained their place in the context of the Internet of Things (IoT) paradigm by creating straightforward social interactions among objects [START_REF] Atzori | The social internet of things (siot) -when social networks meet the internet of things: Concept, architecture and network characterization[END_REF].

By considering that individuals also play a central role in vehicular networks, the concept of Vehicular Social Networks (VSNs) has appeared [START_REF] Vegni | A survey on vehicular social networks[END_REF], [START_REF] Alam | Toward social internet of vehicles: Concept, architecture, and applications[END_REF], considered as a special class of VANETs based on the integration of social aspects and features. On the other hand, the topological evolution of vehicular networks has been investigated in several works (e.g., [START_REF] Pallis | On the structure and evolution of vehicular networks[END_REF], [START_REF] Fiore | The networking shape of vehicular mobility[END_REF]), exploiting structural analysis and statistical measures for graph analysis. A key challenge to design efficient data forwarding approaches in vehicular environments based on graphs properties is the knowledge and update of the topological characteristics of the underlying VSNs, that can be achieved through service-oriented architecture and cloud computing [START_REF] Alam | Workload model based dynamic adaptation of social internet of vehicles[END_REF].

For this purpose, in this paper we present a fogbased architecture based on the results obtained in [START_REF] Manzoni | A proposal for a publish/subscribe, disruption tolerant content island for fog computing[END_REF], where information can assist in the acquisition of the necessary information to gather specific metrics such as central betweenness, local clustering, node degree, etc. Information about such graph metrics can be then exploited to analyze the social behavior of the network nodes. Knowing how social is a node allows to select a next-hop forwarder that is able to well disseminate a message. Indeed, if a node is "socially active" it is expected to be willing to retransmit a message within the network, while a no social node does not represent an optimal choice for data retransmission. Determining how social a node is, represents an important challenge. Indeed, if a node is elected as "too social", then we can encounter multiple copies of a message then affecting the network load; on the other side, a node assumed as "poor social" will not be selected as next hop forwarder, thus providing further delay in the transmission phase.

In this paper, we consider two main contributions:

• We present a fog-based architecture in order to represent a real-world based solution to gather the needed information for the forwarding approaches proposed; • We integrate graph properties and metrics in the forwarding decision design and we analyze the different metrics in two representative vehicular network scenarios;

In particular, we evaluate two message forwarding techniques based one on the betweenness metric, called "GrAph MEtrics based forwarding technique (GAME)", and another one on the betweenness and local clustering coefficient, called "Small-WORlD based ForwardIng technique for Social Hub (SWORDFISH)". By analyzing the vehicular social degrees derived from such techniques, we can infer interesting features of both the approaches. Betweenness and local clustering are associated with a kind of "willingness" of a node to play the relay role. In practice, higher values of betweenness of nodes in a network are associated with a higher aptitude to forward information to other nodes. Following the simulation results in this paper, we can conclude that GAME seems to be more "node-selective" by choosing very few nodes (i.e., hubs) with high social activities, while SWORDFISH allows the selection of a pool of nodes to be eligible as forwarders.

The rest of the paper is organized as follows. Section II describes a fog-based architecture that allows to gather necessary information about graph theory metrics associated to the vehicular network. Such information is taken into account in order to compute the vehicular social degrees. In Section III we present two message forwarding techniques (i.e., GAME and SWORDFISH) and the analysis of their graph-based features in two different traffic scenarios (i.e., sparse and dense traffic scenarios). In Section IV we describe the behavior of GAME and SWORDFISH in terms of forwarding probability. Finally, conclusions are drawn at the end of this paper.

II. FOG-BASED ARCHITECTURE

The concept of "node" representing a vehicle is here extended using the idea of fog computing. Following the results in [START_REF] Manzoni | A proposal for a publish/subscribe, disruption tolerant content island for fog computing[END_REF], in this paper we consider each vehicle as a "content island" capable of interconnecting groups of "things" packed-up together to interchange data and processing among themselves and with other content islands. A content island is a publish/subscribe based system where a set of "things", capable of some type of processing and sensing, interchange messages inside the island and with other islands, talking MQTT. The island core allows to directly handle the flow of messages inside the island and among the islands.

In this context a message can represent: (i) a collection of data, (ii) a processing or data analysis request, or (iii) an action trigger, and can be interchanged among nodes that either belong to the same island or to other islands. A collection of data is clearly a set of values obtained from a source inside an island, like the evolution of the battery charge in an electric vehicle during a day. A processing or data analysis request can indicate the need for data intensive processing over a data-set, like for example computing the linear relationship between a response (humidity) and one or more predictive terms (temperature). Finally, an action trigger can be a request to, if certain conditions hold, to turn off a device, or publish a message to Twitter. Three elements i.e., (i) the MQTT broker, (ii) the DTN daemon, and (iii) the MQTT near-user edge gateway (MQTT-NEG), constitute the island core. These elements will typically be executing in the same specific node of the island.

Two classes of messages are used: local messages and global messages. Global messages are both distributed inside the island and forwarded to other islands by the DTN daemon. The basic connectivity inside the island can take place using any common standard technology, like Ethernet, Wi-Fi, etc.

Inter-islands connectivity is where the VSN and the forwarding techniques described in the following sections come into play, giving the possibility to distribute information, create "conversations" among the members of specific groups of islands, and deploy applications allowing users to cooperate.

III. PROPOSED APPROACHES

Mobile social networks make use of the often periodic physical locality patterns of the users to improve their knowledge about opportunities to interact and take advantage of the proximity among nodes to form spontaneous temporal communities.

Nodes localization can be performed using dedicated low-cost HW capable of tracking information about other close-by devices, sensing their physical layer activities, e.g., BT beaconing, WiFi SSID beaconing and correlate these information with GPS coordinates. All these data can be used to accumulate knowledge of the neighborhood status and how it evolves and how it organizes in time, according to short term patterns (i.e., minutes, hours) to longer term patterns (i.e., days or weeks). Moreover, nodes can periodically broadcast profiling data related with the topics they are interested in, using a predefined format.

In this paper, we describe two multi-hop forwarding probabilistic techniques, based on some graph theory metrics, namely: distance, betweenness and clustering coefficient. The forwarding decision mechanism occurs when a source vehicle needs to re-transmit a message within its transmission range. Potential forwarders distinguish based on vehicular social degree, defined according to specific graph metric. Higher the vehicular social degree is, higher the vehicle's forwarding probability is expected.

The first approach, called GAME ("GrAph MEtrics based forwarding technique") exploits the information about node betweenness. Let us assume that for the i-th node in the network it is available the information about its betweenness i.e., B i , since a fog-based architecture as explained in Section II is considered. The probability that the i-th node is defined as "social active" i.e., P GAM E s,i , since it shows a high value of betweenness B i , is given as:

P GAM E s,i = log 1 + B i max (B) , ( 1
)
where B is the betweenness vector of all the N vehicles in the network, i.e., B = [B 1 , B 2 , . . . , B N ]. Mathematically, the betweenness represents the measure of centrality of a node in a graph G = (E, V ), based on shortest paths i.e.,

B i = s =i =t s,i,t∈V ρ st (i) ρ st , (2)
where ρ st is the number of shortest paths from vehicle s to t, and ρ st (i) is the number of shortest paths from s to t passing through node i. Notice that B i is a measure of the global importance of vehicle i; as a consequence, a vehicle with high betweenness centrality plays a crucial role in the connectivity of the network, since the higher B i , more the number of shortest paths in the network passing through the vehicle i. In our scenario, for every pair of vehicles, there exists at least one shortest path between them such that the number of hops that the path passes through is minimized. Moreover, if a node is more central, then it is expected to be able to forward a message to a larger number of neighbors. However, information about betweenness only can be limitative in order to detect a social vehicle, specially if a node is not so central but its neighbors are central. Another important metric that needs to be considered is the local clustering coefficient (LCC), which captures the degree to which the neighbors of a given node link to each other i.e.,

LCC i = 2L i k i (k i -1) , (3)
where L i represents the number of links between the k i neighbors of the i-th node. The local clustering coefficient gives information if two neighbors of a node link to each other, and it is one of the most important metrics for the social network analysis. Taking into account both betweenness and local clustering allows detecting those social vehicles whose neighbors are expected to be social as well.

In the second technique, SWORDFISH ("Small-WORlD based ForwardIng technique for Social Hub"), considers both the betweenness and the local cluster coefficient for each node in the vehicular network graph.

The proposed approach computes the social degree parameter for the i-th node, defined as:

P SW ORDF ISH s,i = log [1 + λ] = = log 1 + LCC i • B i LCC i + B i , (4)
where λ is a parameter that considers both LCC i and B i , which are the local clustering coefficient and the betweenness of the i-th vehicle, respectively. According to GAME approach, for low betweenness the vehicular social degree is low, while it increases for higher values of betweenness. Moreover, as expected, high values of social degree are only for a limited number of vehicles. This aspect allows the GAME approach to distinguish social vehicles, which are those with higher values of social degree.

In SWORDFISH the vehicular social degree tends to higher values for higher λ, while only poor nodes present lower social degree. Indeed, since SWORDFISH vehicular social degree is based not only on betweenness, but also on local clustering coefficient, when the network size increases the number of vehicles elected as social ones is higher. On the other side, for a reduced network size, SWORDFISH will distinguish vehicles with a higher social degree for a fixed λ. This behavior allows SWORDFISH to select more social vehicles for an increasing number of nodes in the network. Again, this is due since the SWORDFISH social degree considers both LCC and B; therefore, if a node has low betweenness but high local clustering coefficient, it is expected to still being a social node. The LCC parameter contains information about the small-world structure of the network graph. Indeed, as known, a small-world structure occurs generally when nodes tend forming clusters and the average distance is low, as well as when nodes have low betweenness and higher centrality.

In Fig. 2 we show the SWORDFISH vehicular social degree versus λ for different network sizes. On the other side, in Fig. 3 we depict the GAME vehicular social degree versus the betweenness for different network sizes. We observe that GAME tends to select a poor number of social vehicles, specifically it selects only hub nodes with higher betweenness. Indeed, for higher network size, there is expected a limited number of nodes with high betweenness, while the majority will be with a low betweenness. In general, for low betweenness the social degree is low, while it increases for higher values of betweenness. Moreover, as expected, high values of social degree are only for a limited number of vehicles. This aspect allows the GAME approach to distinguish social vehicles, which are those with higher values of social degree.

Finally, the GAME/SWORDFISH vehicular social degrees are used in the expression of the forwarding probability of the i-th vehicle, respectively as follows:

P GAM E f,i = exp -ρ z -d i P GAM E s,i , (5)
P SW ORDF ISH f,i = exp -ρ z -d i P SW ORDF ISH s,i , (6)

IV. PERFORMANCE EVALUATION

Let us now consider two different scenarios representing a sparse and a dense traffic scenario, comprised of 21 and 41 nodes, respectively. In the first scenario, node ID 21 is a source vehicle that needs to re-transmit a message to one of its neighboring nodes in its transmission range. Similarly, node ID 41 is the source node in the dense traffic scenario. Fig. 4 shows the network graphs that represent the two different network scenarios, respectively.

The network distance computed for each node in the graph with respect to the source vehicle (i.e., node 21 for the sparse traffic scenario and node 41 for the dense traffic scenario) is depicted in Fig. 5 respectively. For both two scenarios, we also compute the betweenness metric for each node in the networks (see Fig. 6).

For the sparse traffic scenario, the potential next-hop forwarders are only the nodes at 1-hop distance from the source. Therefore, from Fig. 5 (a) the potential eligible forwarders are only nodes 13, 16, 17 and 18.

The forwarding probabilities of the potential next-hop nodes in case of GAME and SWORDFISH approaches are reported in Fig. 7 (a) in case of a sparse traffic scenario with 21 nodes. Here we show the dependence of the forwarding probability for varying distances within the transmission range. As expected for increasing distances the probability will increase as well. Fig. 7 compares GAME to SWORDFISH approach, for different values of 1-hop neighbor social degree P s . We observe that GAME has detected one neighbor with highest social degree (i.e., P s = 1), while other curves Again, these insights are confirmed when we analyze the forwarding probabilities of the two approaches for the higher node density scenario (i.e., dense traffic scenario), as depicted in Fig. 7 (b). We observe that GAME detects one vehicle with a high social degree, while the other neighbors have lower degree. Consequently, the vehicles with very low social degree will present a very low forwarding probability, and will be likely not selected as next-hop forwarders. On the other side, SWORDFISH has detected neighbors with higher social degree. Likely, for SWORDFISH all these vehicles can be eligible as next-hop forwarders.

V. CONCLUSIONS

In this paper we presented two next-hop forwarding techniques based on vehicle social features, taking into account graph theory metric. Betweenness and local clustering coefficient are two main features that highlight how important (i.e., social in the context of social networks) a node is. More social a node is, more effective the data dissemination with the network will be. GAME approach considers the betweenness of a node, so that the node more central in a transmission range will be selected as next-hop forwarder. On the other side, SWORDFISH considers not only the betweenness but also the local clustering coefficient. In this way, SWORDFISH will detect not only central vehicles in a transmission range, but also nodes whose neighbors are social. When traffic data becomes high and the node with the highest sociality is already managing a lot of data to be forwarded, it is better to have an approach as SWORDFISH allowing the selections of more social nodes and then more options.

Fig. 1 .

 1 Fig. 1. Content island overall architecture as part of a vehicle.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Normalized values of SWORDFISH vehicular social degree versus λ for different network sizes.

  where d i [m] is the inter-vehicle distance lower than the source transmission range z [m] (i.e., d i ≤ z [m]), and ρ is the vehicular density [veh/m].

Fig. 4 .

 4 Fig. 4. Network graphs with (a) 21 nodes (sparse traffic scenario), and (b) 41 nodes (dense traffic scenario).

Fig. 5 .

 5 Fig. 5. Network distance of each node with respect to a source vehicle, specifically (a) node 21, and (b) node 41.

Fig. 6 .

 6 Fig. 6. Betweenness metric of each node in the network graph with (a) 21 nodes, and (b) 41 nodes.

Fig. 7 .

 7 Fig. 7. GAME/SWORDFISH forwarding probabilities versus distance, for different social degree values associated to 1-hop neighbors of a source vehicle in (a) sparse traffic scenario, and (b) dense traffic scenario.

  are for very low vehicular social degrees (i.e., P s = [0.037, 0.0041]) associated to other 1-hop nodes. On the other side, SWORDFISH associates different values of vehicular social degree (i.e., P s = [0.66, 0.82, 0.35]) for each potential 1-hop neighbor. This allows concluding that SWORDFISH is able to select more forwarders with acceptable social degree, while GAME detects that neighbor with highest social degree. SWORDFISH can be well suitable when we need to choose among several social vehicles, while GAME presents its applicability when we need to select the highest social neighbor.


Gilles Bailly
email: gbailly@mpi-inf.mpg.de

Thomas Pietrzak
email: pietrzak@lifl.fr

Jonathan Deber
email: jdeber@dgp.toronto.edu

Daniel J Wigdor
email: dwigdor@dgp.toronto.edu

Métamorphe: Applications for Shape-Changing Keyboards

Keywords: Height-changing keys; augmented keyboard ACM Classification H, 5, 2, Information interfaces and presentation (e, g, , HCI): User Interfaces -Interaction Styles

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Introduction

We investigate how shape-changing interfaces can modify the way users use traditional input devices such as keyboards and mice. We designed and built Métamorphe (Figure 1), a novel shape-changing keyboard. While Métamorphe retains the form factor, layout, and tactile qualities of a traditional keyboard, it embeds solenoids that can raise individual keys, and adds force sensors to the sides of the keys, allowing a variety of novel gestures to be performed. We first present the system. We then discuss five potential scenarios for the Métamorphe keyboard. In particular, we show how Métamorphe can favor the guessability, learnability, and eyes-free selection of hotkeys. Finally, we report results of two user studies before concluding.

Related Work

Métamorphe is related to shape-changing input and output devices such as mice [START_REF] Kim | Inflatable Mouse: Volume-adjustable Mouse with Air-pressure-sensitive Input and Haptic Feedback[END_REF], keyboards [5], interactive surfaces [START_REF] Richter | EdgeMatrix: Remote Tactile Feedback on Interactive Surfaces Using a Shape Display[END_REF], and displays [START_REF] Poupyrev | Lumen: Interactive Visual and Shape Display for Calm Computing[END_REF]. However the Métamorphe's shape-changing keyboard differs from these earlier works, as its rising keys and force sensors provide a unique modality of shape changes.

Métamorphe is also related to augmented keyboards such as the Optimus [START_REF]Optimus Maximus Keyboard[END_REF], which contains a small screen on each key, or TypeRight [START_REF] Hoffman | TypeRight: A Keyboard With Tactile Error Prevention[END_REF], which provides haptic feedback to prevent errors during text entry.

Métamorphe not only provides visual and haptic feedback, but also enables novel key gestures.

The Métamorphe Prototype

The Métamorphe prototype [START_REF] Bailly | Métamorphe: Augmenting Hotkey Usage with Actuated Keys[END_REF] is shown in Figure 2bottom. It is built atop a regular HP KU-0316 USB keyboard. It reuses both the mechanical and the electronic components of the original keyboard. However, it also embeds BLP Series 134 6V (30Ω) push-type solenoids in each key (Figure 1-b), so that they can be individually raised. Each Métamorphe key has four functional positions. One dimension is set by the computer: raised (the key is higher than the rest of the keys) or unraised (the key is in its regular position). The other dimension (pushed or not pushed) is controlled by the user's input. The magnitude of the height change for actuated keys posed a critical design factor for Métamorphe. Pilot studies showed that a key height of 4 mm is easy to perceive but does not impede ease of use. Finally, dynamic Métamorphe keys allow users to perform a large inventory of key gestures, as shown in Figure 2-top. Gestures are detected by force sensors attached to the four side faces of the keys.

We equipped our prototype with eight functioning dynamic keys, two of which embed force sensors; this is sufficient to allow a thorough exploration of the Métamorphe design space.

Output: Haptic and Visual Feedback

Métamorphe transforms the keyboard into a haptic display with its height-changing keys. When a key is raised, the height difference compared to other keys provides a tactile or kinesthetic cue that can be used by users to confirm that the desired key has been reached. A raised key can also be used as an eyes-free landmark to help locate a nearby unraised target key. Users reach the raised haptic landmark, and then perform a quick ballistic movement towards the target location.

In addition to its haptic feedback, a raised key also provides visual feedback, since it looks distinct from the surrounding keys. These visual cues can be useful, particularly for novice users. Top: the physical manipulations that can be detected on each key. Bottom: our Métamorphe prototype.

Input: Novel Gesture Vocabulary

Métamorphe's raised keys introduce a novel design space for key input. On a standard keyboard, a key can be pushed, held, and released. However, Métamorphe's raised keys allow users to interact with several faces of the raised keys, enabling the 9 new gestures shown in Figure 3. Six of these new gestures have been implemented in our current prototype (Figure 2).

Users can directionally push (or side push) any of the four sides of a key, as illustrated in Figure 3-c-f. Directional pushes require only one finger, and allow for varying placement of the finger on the side of the raised key. Users can also squeeze a key with two fingers along the X-axis (a Left-Right Squeeze, Figure 3-g) or the Y-axis (a Front-Back Squeeze, Figure 3-h). Raised keys can also be rotated Clockwise and Counterclockwise (Figure 3-i-j). Finally, users can Pull a raised key a small distance upwards (Figure 3-b). This novel vocabulary of gestures dramatically increases the expressiveness of the keyboard's input.

Applications

We envision several applications for Métamorphe involving hotkey usage, text entry, games, etc.

Hotkey Usage

Hotkeys are an efficient way to select commands with a keyboard. However, many users -even experienced ones -do not use them. Novice users may be not aware of hotkeys, and intermediate users may find them a challenge to learn due to non-intuitive mappings. Expert users may still have difficulties accurately reaching keys without looking at the keyboard. While Métamorphe can act as a traditional keyboard for text entry, it can also improve hotkey usage. For instance, it can favor the discoverability of hotkeys; pressing the Control key could raise a subset of hotkeys, revealing the most frequent ones. Some keys could also be raised when the cursor hovers over an object to indicate commands applicable for this object. Métamorphe can also increase the guessability of hotkeys, since the novel gesture input vocabulary offers increased flexibility to map commands to hotkeys. For example, semantically-related commands such as "Save" and "Save As" could be assigned to different gestures on the same key 'S', and symmetrical commands like "Previous" and "Next" could be assigned to opposing push gestures on the same key. Finally, it can make hotkey selection easier, since expert users can first use their spatial memory to develop a rough idea of the physical location of the desired hotkey on the keyboard, and then use Métamorphe's haptic feedback to accurately relocalize the hotkey without looking at the keyboard.

Text Input

Métamorphe could be adapted to predict the next character in a string of text and automatically raise that key to (subtly) prompt the user and avoid spelling errors. This capability would be most appropriate with short and constrained text entry. For instance, if users must enter the name of a city in a form, and begin entering "Pari", the system could suggest 'S' (for "Paris") by raising that key. In addition, rising keys could provide tactile hints when learning typing skills.

Gaming

Raised keys could indicate a ready to use action (e.g., a spell), and the increased vocabulary could provide additional game controls.

Security

Métamorphe could provide additional security features for password entry, since passwords could consist of both characters and gestures. This would increase the keyspace of inputs to the system, and could make passwords easier to remember because users could incorporate spatial memory.

Tangible Interfaces

Physical controls could be provided on-demand for a wide variety of tasks. For instance, an on-screen dialog with three slider widgets could raise three keys to provide tangible controls for each widget.

User Studies

We conducted two user studies in the context of hotkey usage. The first was a user-defined interface study that provided information about the mental models that users associate with raised keys, and demonstrated that users readily incorporate the novel input capabilities of raised keys to highlight semantic relationships between commands. The second study evaluated the impact of raised keys on the eyes-free selection of hotkeys, and showed that 1) raised keys are easier to locate on a keyboard than unraised keys, and 2) raised keys act as haptic landmarks, making it easier to locate neighboring, unraised keys.

Conclusion and Future Work

We have presented Métamorphe, a novel keyboard with height-changing keys that provides haptic and visual feedback. As an input device, Métamorphe also offers a novel vocabulary of gestures, which increases the number of available actions on each key. We presented several applications for Métamorphe, and demonstrate how Métamorphe can improve hotkey usage.

Solenoids are an appropriate technology for shapechanging keyboards. However, there are also several challenges that need to be investigated further.

Because solenoids require continuous power to maintain a key in the raised position, they necessitate heat dissipation. In addition, because they must be able to lift the key cap, their own weight, and the sensors, they may requiring overvolting to generate sufficient mechanical power (19V for 50ms). From a scalability point of view, each key has 4 sensors and 1 actuator, resulting in a large number of connecting wires for the entire keyboard; additional work would be required to miniaturize the system. Finally, we aim to integrate screens into each key, combining the advantages of the Optimus and Métamorphe.

Figure 1 .

 1 Figure 1. a) The Métamorphe keyboard raises a subset of hotkeys when users press the CTRL key. b) Each key can be individually raised with an embedded solenoid, and contains force sensors that allow a variety of novel gestures to be performed on the key.

Figure 2 .

 2 Figure 2.Top: the physical manipulations that can be detected on each key. Bottom: our Métamorphe prototype.

Figure 3 .

 3 Figure 3. Design space of raised key gestures.