Fabio Pistolesi

Bistability of a slow mechanical oscillator coupled to a laser-driven two-level system

Keywords:

It has been recently proposed that single molecule spectroscopy could be employed to detect the motion of nano-mechanical resonators. Estimates of the coupling constant (g) between the molecular two-level system and the oscillator indicate that it can reach values much larger than the mechanical resonating pulsation (ωm) and the two-level system linewidth (Γ). Other experimental realization of the same system are also approching this strong coupling regim. In this paper we investigate the behavior of the system in the limit for slow mechanical oscillator ωm Γ. We find that, for sufficiently large coupling, the system undergoes a bistability reminiscent of that observed in optical cavities coupled to mechanical resonators.

I. INTRODUCTION

The rapid developpement of nano-electromechanics in the last decade has seen the proposal and the experimental realisation of several systems where in order to detect the displacement of a mechanical resonator it is coupled to a two-level system (TLS). This includes superconducting Qbits [1], nitrogen vacancy centers in diamonds [2][3][4][5][6][7][8][9], semiconductor TLS [10][11][12], spins [13][14][15][START_REF] Treutlein | Cavity Optomechanics[END_REF][17] or single molecules [18]. One of the interest in coupling a mechanical resonator to a TLS is that one can reach large couplings constants [2,13,18,19]. According for instance to the estimates of Ref. [18], the coupling constant can become larger than the mechanical pulsation ω m or the TLS linewidth Γ. Increasing the coupling not only improves the detection sensitivity of the mechanical displacement, but allows to reach new regimes, where the dynamics of the TLS and of the mechanical oscillator have to be considered on the same footing. This is well known for the case of mechanical resonators coupled to optical cavities [20][21][22], for which it is possible to reach the strong coupling limit since the effective opto-mechanical coupling constant is proportional to the square root of the number of photons present in the cavity. A striking effect is the onset of a static bistability, that was observed long time ago [21,23,24]. Reaching such a strong coupling limit with the bare coupling between the oscillator and the quantum system, in our case the TLS, is difficult, but currently at reach of the present technology.

In this paper we consider the case of a slow oscillator ω m Γ coupled to a laser-driven TLS and, by exploiting an adiabatic expansion, we obtain a description of the TLS-oscillator system in the strong coupling limit. We find that a static bistability is also expected, with a behaviour similar to that observed for optical cavities. In this case it is induced by the coupling to a single quantum degree of freedom of the TLS instead of the macroscopic condensate of photons. Note alo that the presence of the TLS renders the problem intrinsically non-linear. We calculate the luminescence that is the typical observed quantity in single-molecule spectroscopy experiments [START_REF] Tamarat | The Journal of Physical Chemistry A[END_REF]. We find that the interplay of the cooling-heating effect with the bistability gives rise to anomalous line-shapes. This bistability resembles the one observed in optical cavities, but with the notable difference that the quantum nature of the TLS has to be taken into account.

The paper is organized as follows. In Section II we present the model. In Section III the Born-Markov equations are derived. In Section IV we exploit the separation of time scales to simplify the Bork-Markov equations and obtain a description of the slow degree of freedom. In Section V we discuss the effective temperature induced by the coupling to the TLS. In Section VI the condition for the appearance of the mechanical bistability is discussed. In Section VII the effect of the stochastic fluctuations is considered by solving numerically the Fokker-Planck equation. Section VIII gives our conclusions.

II. SYSTEM

We consider a TLS coupled to a laser and to a mechanical oscillator as describled in Ref. 18 (see Fig. 1). We will focus on this sytem, but the model describles several systems, for instance the TLSs coupled by the strain to an oscillator [9]. The coupling of the TSL to the mechanical oscillator is due to the Stark effect and the difference of potential that is induced between the oscillator (for instance a suspended carbon nanotube) and the transparent and conducting substrate over which the molecules are dispersed. In experiments the light emitted by a single molecule is collected and detected as a function of the laser beam frequency and intensity. The system is describled by the following Hamiltonian:

H S = - ∆ 2 σ z +hΩσ x cos ω L t-hgσ z (b+b † )+hω m b † b . ( 1
)
Here ∆ is the TLS splitting, Ω the coupling intensity to the laser, g the electromechanical coupling, ω L is the laser frequency, and ω m the mechanical pulsation (h is the reduced Planck constant). The operators σ anent dipole e surrounding ibution to the than the quauch as in polycan be as large ane Shpol'skii ments, this gives an expected electric coupling constant 10 19 Hz=m, 4 orders of magnitude larger than the magnetic coupling constant observed in Ref. [8].

In order to show the capabilities of this detection technique, we provide explicit predictions for the luminescence excitation spectrum and the second order photon correlation FIG. 1: Left: Schematics of the system proposed in Ref. [18] and considered in this paper. Right: simplified view of the two-level system coupled to a mechanical oscillator via the Stark effect and to a laser.

with x z = h/2mω m the zero point displacement fluctuation amplitude for an oscillator of mass m. The TLS and the mechanical oscillator are coupled to the environnement that leads to a finite linewidth Γ of the TLS resonance and to a damping rate γ for the oscillator.

III. BORN-MARKOV EQUATIONS

We proceed by assuming a weak coupling with the environment of the system. By standard methods [START_REF] Schlosshauer | Decoherence: and the quantum-toclassical transition[END_REF] in the Born-Markov approximation the environnement can be traced out and an equation for the reduced density matrix ρ(t) for the system (oscillator plus TLS) can be derived: ρ = Lρ.

(

) 2
We define now ρ ij (x, x , t) = x, i|ρ(t)|x , j , where |x, i are the eigenstates of x and σ z with eigenvalues x and (σ z ) ii = ±1, respectively. It is convenient to introduce the variables x + = (x + x )/2 and x -= x -x . As discussed in the introduction we will consider in this paper the slow oscillator limit. In terms of the above introduced parameters the condition reads

γ ω m , g Ω, Γ ∆ . (3)
This implies a time scale separation between the TLS and the mechanical oscillator dynamics. It is thus convenient to write the Born-Markov operator as L = L f + L s with L f and L s the fast and slow component, respectively. Explicitly the fast component reads

L f =    0 iΩ/2 -iΩ/2 Γ iΩ/2 -iδ -Γ/2 0 -iΩ/2 -iΩ/2 0 iδ -Γ/2 iΩ/2 0 -iΩ/2 iΩ/2 -Γ    (4)
with δ = δ -2gx + /x z and δ = ω L -∆/h, the detuning. The components of the density matrix are {ρ 11 , ρ 12 , ρ 21 , ρ 22 }. The slow component reads

L s = L osc + x - x z L -, (5)
with

L osc = ih m ∂ + ∂ --i mω 2 m h x + x --2γx -∂ --Dx 2 -( 6
)
where L -= igdiag(1, 0, 0, -1), and the last two terms in Eq. ( 6) describe the coupling of the oscillator to an environment at temperature T . From the fluctuationdissipation theorem D = mγ(hω m /2)coth(hω m /2k B T ).

We use the notation ∂ ± = ∂/∂x ± . The operator L f describes the dynamics of the TLS, and implies a fast evolution of the density matrix on a scale of the maximum between Γ and Ω. Thus this evolution is much faster than that induced by the L s term, that takes place on the ω m time scale, for what concern L osc , and even slower for the dissipative part. We have included the term proportional to gx + in L f since we want to allow the possibility that this term becomes of the same order of Γ in the strong coupling limit. In order to check that the approximations are consistent what matters is the fluctuation of the oscillator position (∆x

+ ) 2 = x 2 + -x + 2 . If g∆x + /
x z is of the order or larger than ω m , then this coupling has to be included in the fast part. On the other side, it is typically correct to regard the term proportional to gx - as a slow contribution that can be included in L s . This second assumption is valid for g∆x -/x z Γ. We will come back to these two assumptions after the solution of the Eq. ( 2) has been obtained.

IV. ADIABATIC ELIMINATION OF FAST VARIABLES

We now exploit the separation of time scales, ω m Γ, using the method of adiabatic elimination of fast variable (see for instance [START_REF] Gardiner | Stochastic methods: a handbook for the natural and social sciences[END_REF]) to integrate out the TLS fast degrees of freedom and obtain an equation for the mechanical oscillator reduced density matrix. We begin by noting that L f is a function of x + only. One can then define the kernel of the operator L f by the equation:

L f (x + )ρ 0 (x + ) = 0 . ( 7
)
Physically ρ 0 (x + ) is the stationary state of the TLS for given value of x + . Since L f is not Hermitean left and right eigenvectors are different. Let's define w 0 as the left eigenvector of L f with vanishing eigenvalue. One can readily show that it has the form w 0 = {1, 0, 0, 1} and when projected on an arbitrary state ρ it gives its trace over the TLS states:

(w 0 , ρ) = ρ 11 + ρ 22 = Trρ . ( 8
)
Since probability is conserved by the time evolution 0 = dTr(ρ)/dt = Tr(L f ρ) = (w 0 , L f ρ) for any ρ, the relation w t 0 L f = 0 holds. We choose now the normalization of ρ 0 (x + ) such that (w 0 , ρ 0 ) = 1. Any ρ can then be written as a sum of its projection on the kernel of L f , the slow component, and on its othogonal complement, the fast component:

ρ ij (x + , x-, t) = ρ 0 ij (x + )R(x + , x -, t) + ρ f ij (x + , x -, t), (9)
where by construction (w 0 , ρ f ) = 0 and (w 0 , ρ) = R(x + , x -). Let us define P as the projector on ρ 0 , and Q = 1-P as the projector on the complement. Note that P and Q depend on x + , they are the projectors on the 4×4 phase space of the TLS. Again for given x + , the following properties hold : P L f = L f P = 0. We substitute now this expression in the master Eq. ( 2) and by applying alternatively P and Q we obtain the two equations:

Ṙρ 0 = P L s (Rρ 0 + ρ f ), (10)
Q ρs = QL f ρ f + QL s (Rρ 0 + ρ f ) . (11)
Up to now these equations are exact. Let's now use the fact that L f is very large to solve the second equation (a systematic expansion could be derived using the Laplace transform, we consider here only the leading term)

QL f ρ f = -QL s (Rρ 0 ) . ( 12
)
Since L f acts on the subspace orthogonal to the kernel and the solution is seeked in the same space QL f Q is actually invertible in this subspace (while L f has one vanishing eigenvalue in the full space of ρ). We thus substitute the solution into Eq. ( 10) and obtain:

Ṙρ 0 = P L s (Rρ 0 ) -P L s (QL f Q) -1 QL s (Rρ 0 ). ( 13
)
We begin by evaluating

QL s (Rρ 0 ) = x - x z RQL -ρ 0 + ih m ∂ -RQ∂ + ρ 0 . (14)
By projecting Eq. ( 13) on w 0 we obtain

Ṙ = L osc + x - x z α 1 - ihx - mx z α 2 ∂ -+ x 2 - x 2 z α 3 R (15)
where

α 1 = (w 0 , L -ρ 0 ), α 2 = (w 0 , L -(QL f Q) -1 Q∂ + ρ 0 ), and
α 3 = (w 0 , L -(QL f Q) -1 QL -ρ 0 ).
Using the explicit form of L f one can readily find the three matrix elements:

β 1 = - iα 1 g = Γ 2 + 4δ 2 Γ 2 + 4δ 2 + 2Ω 2 , ( 16
)
β 2 = ix z Γ 2 α 2 2g 2 = - 32δ ΓΩ 2 2Γ 2 + Ω 2 Γ 2 + 4δ 2 + 2Ω 2 3 , (17)
β 3 = α 3 Γ g 2 = 4Ω 2 Γ 2 + 4δ 2 2Γ 2 + Ω 2 Γ 2 + 4δ 2 + 2Ω 2 3 . (18)
Substituting these expressions into Eq. ( 15) and introducing the Wigner transform W (x + , p) =

dx -e -ipx-R(x + , x -) we have x -→ ih∂ p and ∂ -→ ip/h in Eq. (13). We thus find for W the following equation:

Ẇ = - p m ∂ + + (mω 2 m x + -F )∂ p + 2γ t ∂ p p + D t ∂ 2 p W ( 19
)
where F is the average force acting on the oscillator: Equation (19) with the relations ( 11) and ( 9), allow to obtain the behaviour of the mechanical oscillator and the optical response of the TLS. In practice ρ f is very important for the derivation of the equation of motion of R, but its contribution to ρ is small (of the order ω m /Γ), and it can be neglected in the calculation of the averages in the following.

F = hg x z σ z = - gβ 1 x z , (20)

V. EFFECTIVE TEMPERATURE

We begin the study of Eq. ( 19) by defining an effective temperature in analogy with the fluctuation-dissipation relation:

coth hω m 2k B T eff = 2D t γ t hω m . (21)
In the case of γ γ 0 one finds

coth hω m 2k B T eff = - Γ 2 + δ 2 4δ ω m . ( 22
)
For positive values of δ the system is unstable (negative damping term). For negative values of δ the function in Eq. ( 22) has a minimum value of Γ/4ω m 1 at δ = -Γ. This means that when the coupling to the TLS dominates over the coupling to the environment, the system reaches a classical stationary state (k B T eff hω m ). Thus for γ = 0 one can write

k B T eff = h Γ 2 + δ 2 8δ . ( 23
)
We note that the form of the effective temperature is the same that is found for an oscillator coupled to a cavity [28]. This follows for the similarity of the spectrum of fluctuation of σ z and that of the number of photons in a cavity. The result implies clearly that it is not possible to use the TLS in the Γ ω m limit to cool the oscillator in the quantum regime [18,29].

We are now in the position to check the conditions on ∆x + and ∆x -assumed for the solution of the problem.

From the equipartition theorem mω 2 m (∆x + ) 2 = k B T eff and ∆x -∼ h/∆p with (∆p) 2 /m = k B T eff . The condition on ∆x -reads then

g Γ k B T eff hω m 1/2 . ( 24
)
Since k B T eff > hΓ, a sufficient condition for the validity of the approximations is g Γ(Γ/ω m ) 1/2 , that gives a large window of validity of the theory.

For finite value of γ a part of the region for which δ > 0 becomes stable, but before discussing this point we need to take into account the fact that δ = δ -2gx + /x z and that this induces a mechanical bistability.

VI. MECHANICAL BISTABILITY

The value of x + entering δ is a stochastic variable whose statistics can be obtained from the solution of Eq. (19). In general the distribution function is strongly peaked around the equilibrium position x e that satisfies the equation

mω 2 m x e = F (x e ) . (25)
We thus begin by solving Eq. ( 25). The equation can be studied more conveniently by eliminating

x e = (δ -δ )x z /2g (26)
from the definition of δ (x e ). The equation then reads

δ -δ = λΓβ 1 (δ ) , (27)
where we defined λ = 4g 2 /ω m Γ that is the relevant dimensionless coupling constant also known as cooperativity.

A relevant interpretation of λ for our problem is also the following: When the TLS is in the excited state an additional force F o = hg/x z acts on the oscillator modifying its equilibrium position ∆x e = F 0 /mω 2 m . Let us call P = F o ∆x e = hg 2 /ω m the (classical) energy scale that corresponds to (twice) the variation of the potential energy of the oscillator. Comparing it with the relevant energy scale of the TLS hΓ we have P /hΓ = λ/2, that measures the relevance of the TLS on the oscillator dynamics. This phenomenon resembles to the bistability expected in suspeded carbon nanotubes forming a singleelectron transistor [30,31]. There the two-level system is the empy or filled suspended quantum dot, and the role of the laser driving is played by the electrons entering the quantum dot for transport.

We come now back to Eq. ( 27). Comparing the derivatives with respect to δ of the right-and lefthand sides of Eq. ( 27) one finds that for λ < λ c = 2(Γ 2 + 2Ω 2 ) 3/2 /Ω 2 3 √ 3 there is a single solution for the equation for any value of δ. For λ > λ c three solutions exist, two stable and one unstable. The two stable solutions for δ correspond to two stable (or metastable) equilibrium positions given by Eq. ( 26). From the expression of λ c one can see that λ c takes the minimum value of 2 for Γ = Ω. In terms of the bare coupling g this implies that the minimum value required to observe the bistability is g min = (ω m Γ/2) 1/2 , the geometric mean of the mechanical frequency and of the TLS inverse lifetime. Equivalently, the requirement is that the cooperativity λ > 1.

In order to find the region of bistability as a function of λ and the physical laser detuning δ, we begin by recognizing that the bistable behaviour upon increasing λ begins when the d/dδ of the left and right-hand side of Eq. ( 27) coincide:

-1 = λΓ dβ 1 dδ . ( 28
)
This is an equation for δ and λ. Its solution is shown in Fig. 2 for Ω = Γ. One can see that the bistable region takes place for negative values of δ , for which the TLS generate a standard positive dissipative term. One should however be cautious when converting this result to the externally tunable δ. This can be done by using Eq. ( 27). The result is shown in Fig. 3 The dashed line is the region δ = 0. As one can see a part of the bistable phase is now apparently in the region of dynamical instability (δ > 0). This is more subtle, since actually the transformation ( 27) is not bijective, to the same value of δ two values of δ may be associated. We will discuss later the consequences of this fact in more details.

Let us begin by studying the consequences of the bistability on the luminescence. In order to obtain the luminescence we only need the probability of occupation of the excited state, P e , as a function of the externally fixed detuning δ. From the solution of L f ρ 0 = 0 one finds the familiar result: We plot parametrically P e and δ(δ ) from Eq. ( 27) in Fig. 4. One can see that the lineshape is strongly modified, with the apperance of a part with three values possibles, corresponding to the two stable and one unstable equilibrium state. Clearly the unstable state cannot realized, but the oscillator spends a sizable part of the time on the two (meta-)stable states. In order to evaluate the actual form of the expected luminescence line we need to solve the Fokker-Planck Eq. (19).

P e = Ω 2 Γ 2 + 2Ω 2 + 4δ 2 . ( 29

VII. EFFECT OF FLUCTUATIONS

We need to consider the effect of fluctuations. Since the system may be unstable for positive values of δ , one cannot discard anymore the intrinsic dissipation due to the coupling to the environnement at temperature T . The region of dynamical instability is defined by the condition γ t < 0, where γ t is defined after Eq. ( 20). This gives the equation for the critical line:

1 + 2λQ ω m Γ β 2 (δ ) = 0 , (30)
where we introduced the quality factor Q = ω m /γ. We show in Figs. 2 and3 the regions of dynamical instability as a function of λ and δ or δ, respectivity. It is important to realize that the dynamical instability takes place very close to the static bistability, we will see that this has consequences in the expected lineshapes.

Let us now discuss the interplay of the heating/cooling effect and the bistable behaviour. In Fig. 5 we show P e as a function of δ and δ. One can follow how the states indicated by the letters A, B, and C are mapped in the δ plot. The point A is in the cooling part of the line (δ < 0) but in the δ space it appears at value of δ larger than the value of δ corresponding to B (δ = 0), that defines the border between cooling and heating. The cooling state A is thus bistable with the heating state C. Due to the fluctuations the system will spend some time in both states, with a probability that is determined by Fokker-Planck equation. This also explains the shape of the static bistability region, that apparently leaks on the dynamically unstable region, as shown in Fig. 3.

In order to find the contribution of the fluctuations quantitatively we solve numerically the Fokker-Planck Eq. ( 19) by discretizing the phase space x-p in N x and N p points, respectively. The operator entering Eq. ( 19) and acting on W becomes thus a matrix of dimension N = N x N p . The stationary solution of Eq. ( 19) is found by solving the equation with the constraint of the normalization of W . We find that typically N x = N y = 100 is already sufficient to obtain a solution of the equation in the range of interest of the parameters. The Fokker-Planck equation can be rewritten in terms of the dimensionless variables x = x/∆x e , p = pω m /F o , and t = ω m t. This gives

∂W ∂ t = -p∂ x + [x -β 1 ]∂ p + γt ∂ p p + Dt ∂ 2 p W (31)
where Results of the numerical solutions for the luminescence are shown in Fig. 6 for k B T / P = 0.01, ω m /Γ = 10 -3 , λ = 2 and for different values of Q. For small Q the luminescence follows closely the mean field result (shown dashed) apart for the bistable region. Increasing Q one enters the region of dynamical instability, as expected from Eq. ( 30) and Fig. 3, and fluctuations increase dramatically, with the system spending a sizable time on the heating region. This is confirmed by the dependence of the effective temperature on δ for given value of Q, as shown in Fig. 7. One can note that large values of Q the effective temperature increase for positive values of δ reaching values of the order of P , and thus washing out the bistable behavior. The result is a very smooth and asymmetric lineshape.

γt = ω m Γ λβ 2 2 + Γ Qω m , Dt = - ω m Γ β 3 + k B T P Q . ( 32

VIII. CONCLUSIONS

In this paper we studied the behavior of a slow mechanical resonator (Γ ω m ) coupled to a laser driven TLS in the strong coupling regime. We began by the Born-Markov description of the system [cf. Eq. ( 2)], and then, by eliminating the TLS fast variables we obtained Eq. ( 19)] for the Wigner function of the mechanical oscillator.

By analyzing this equation we showed that the oscillator effective temperature can be controlled by the laser detuning and the coupling (cooling or heating). We found that, when the coupling of the TLS dominates over the coupling to the environment, the mechanical oscillator can only be in a classical regime (k B T eff hω m ). We showed than that for sufficiently strong coupling the mechanical system can undergo a bistability. The conditions on the cooperativity and detuning for its observation are resumed in Fig. 3. Contrary to optomechanical cavities, the coupling controlling the instability cannot be tuned by the laser intensity (that here is parametrized by Ω). As a rule of the thumb, we find that a coupling constant of the order of (ω m Γ) 1/2 is necessary for the bistability to take place. The bistability allows the possibility that the two stable states are one in the cooling and the other in the heating regime. This leads to a peculiar shape of the luminescence linewidth (cf. Fig. 6) , that is broaden mainly due to the increase of the oscillator fluctuations induced by the heating effect. A promising experimental system where this effect could be observed is the one proposed in Ref. [18] of single molecules coupled to a carbon nanotubes. The typical values of the parameters are m = 10 -21 Kg, Γ > 10 7 Hz, g < 10 9 Hz, with a mechanical frequency of the carbon nanotube that depending on its length can vary from kHz to MHz. With these numbers it should be possible to reach large values of λ, for instance for g = 10 6 Hz, ω m = 2π • 10 5 Hz, Γ = 10 7 Hz one finds λ ≈ 0.5. For these couplings the effective mechanical quality factor Q o = ω m /γ o induced by the coupling to the oscillator is (Γ/g) 2 β 2 /2 ≈ 10 2 . Thus even if the scale of the energy barrier between the two stable states ( P ) is very small, when the coupling to the environnement is sufficiently weak (Q = ω m /γ Q o ) the effective temperature is controlled uniquely by the TLS and k B T eff P . This should lead to the observation of a luminescence line similar to the one predicted in Fig. 6.

Concerning the approximations used in the paper, it turns out that one of the main condition is that the spread of the quantum variable x -is small: ∆x -/x z Γ/g. This is necessary to include the term gx -in the slow component (L s ) of the master equation, leaving the fast operator independent on x -. A less technical way of stating this condition is to say that it fixes a limit on the quantum nature of the mechanical degree of freedom. In terms of the coupling constant the condition is given by Eq. (24). One finds that the scale of validity is set by g Γ(Γ/ω m ) 1/2 [or λ (Γ/ω m ) 2 ]. The bistable be-havior as describled above is thus well inside the limit of validity of the approximation. For g (Γ/ω m ) 1/2 the coupling is so strong, that quantum coherence cannot be neglected anymore between two transition events, even if Γ ω m . This regime if of course relevant for investigatigations on quantum manipulation of the mechanical states and constitutes an interesting perspective for future work.

  x and σ z are Pauli matrices, and b and b † are the destruction and creation operators for the mechanical oscillator excitations. The displacement operator reads x = x z (b + b † )

1 FIG. 1 (

 11 FIG. 1 (color online). Schematic representation of the proposed experiment. (a) Standard far-field confocal microscopy setup for single molecule detection with a carbon nanotube suspended from an AFM tip. (b) Modulation of molecular level splitting during the nanotube oscillations.

γ

  t = γ + γ o and D t = D + D o , are the total dissipation and diffusion coefficients, respectively, with γ o = hg 2 β 2 /mx 2 z Γ 2 and D o = h2 g 2 β 3 /x 2 z Γ the dissipation and the diffusion coefficients induced by the coupling to the driven TLS.

FIG. 2 :

 2 FIG.2: Region of static (shaded blue region for negative δ ) and dynamical (shaded regions for positive δ ) instabilities in the plane λ-δ for Ω = Γ and for Qωm/Γ =4 (left most brown region), 2 (grey), 1 (green). The dashed line corresponds to the value δ = 0, separatrix between cooling and heating in the case γ = 0 (or Q = ∞).

FIG. 3 :

 3 FIG.3: Same as Fig.2in the plane δ-λ. Due to the bistability the contour of the bistability region can be in coincidence with the dynamically unstable region of the phase space.

FIG. 4 :

 4 FIG.4: Probability of occupation of the excited state (Pe proportional to the luminescence) as a function of the detuning δ for Ω = Γ and λ = 0.1, 0.5, 1, 2 and 3 (from left to right). The bistable behavior begins at λ = 2.

FIG. 5 :

 5 FIG.5: Mapping of the luminescence line from δ to δ. The points A, B, and C, correspond to cooling, neutral, and heating values, respectively. One can see that for δ = 2 both cooling and heating are possible, depending on value of x that determines if the point A or C is actually occupied.

FIG. 6 :

 6 FIG.6: Luminescence: λ = 2, ωm/Γ = 10 -3 , kBT / P = 0.01, Q =10, 10 2 , 10 3 , and 10 4 for the curves from the stepest to the smoothest, respectively. Increasing Q increases the fluctuations and smothen the lineshape. The parameter Qωm/Γ entering Eq. (30) takes thus the values 10 -2 , 10 -1 , 1, and 10.

FIG. 7 :

 7 FIG.7: Effective temperature kBT eff / P as a function of the detuning δ for the same value of the quality factors of Fig.4. The higher values of the temperature are of course obtained for the highest Q.

Acknowledgments

I thank the Conseil régional de Nouvelle-Aquitaine for financial support. I also thank V. Puller, R. Avriller, and C. Dutreix, for discussions.

Appendix A: Relation to the quantum noise method

Let us briefly comment on the physical interpretation of the fluctuation and dissipation terms. For weak coupling it is known [32] that the dissipation and fluctuation can be derived directly from the quantum correlation function of the force operator. In our case this is defined as:

where the time evolution is ruled by only the TLS part of the Hamiltonian (g = 0). The quantity S zz can be obtained explicitly:

where M z = (L -+ L + )/(ig) is the superoperator for σ z and we introduced L + = igdiag(0, 1, -1, 0) by analogy with the definition of L -. Introducing the Laplace transform S zz (s) = +∞ 0 dte st S zz (t) (with Res < 0) we have

Using the projectors Q and P one can readily show that

For s = 0 the inverse has to be performed in the subspace defined by the projector Q. The power spectrum S zz (ω) = +∞ -∞ dte iωt S zz (t) can then be related directly to the Laplace transform by using the property S * zz (t) = S zz (-t) that leads to

We note that w 0 L + = 0. By explicit calculation one can verify that the term proportianal to L + coming from the M z on the right matrix element of Eq. (A4) gives an imaginary term for s = 0, and thus it does not contribute to S zz (ω → 0). One can thus substitute L -into the definition of M z = L -/(ig) in this case. For ω m Γ we can obtain the diffusion constant of the Fokker-Plack equation from the vanishing frequency value of the force fluctuation spectrum:

This value coincides with the adiabatic approach result. One can also verifiy by direct calculation that the derivative with respect to ω of S F F (ω) gives also correctly the damping term entering Eq. (19). Thus the weak coupling calculation allows to find the form of the coefficients entering the Fokker-Plack equation, but it does not allow to prove the validity of the approach. According to the derivation presented in the main text, actually it is not the weak-coupling condition that allows to obtain the Fokker-Plack description, but the separation of time scales.


Anthony Moussa
email: anthony.moussa@iae-aix.com

Virginie De Barnier

2017), three studies isolated the effect

of branded longevity on consumers' perception (Zhang et al., 2017;[START_REF] Beck | Discovering the "since effect" -how communicating the company age affects consumers' brand perception[END_REF][START_REF] Desai | When old is gold: the role of business longevity in risky situations[END_REF]. [START_REF] Desai | When old is gold: the role of business longevity in risky situations[END_REF] demonstrate that branded longevity reduces consumers' perceived risk toward the brand, only when consumers have low levels of involvement in the task. This is because, when consumers are in low involvement conditions, they are not likely to be motivated to search extensively for information (Petty and Cacioppo, 1986), and thus branded longevity can be a diagnostic cue regarding the brand's expertise [START_REF] Purohit | Effect of manufacturer reputation, retailer reputation, and product warranty on consumer judgments of product quality: A cue diagnosticity framework[END_REF][START_REF] Desai | When old is gold: the role of business longevity in risky situations[END_REF]. [START_REF] Beck | Discovering the "since effect" -how communicating the company age affects consumers' brand perception[END_REF] challenged this notion by measuring the effect branded longevity on highly involved participants. They found that branded longevity had significant effects on consumers' evaluation of the brand in conditions of high involvement. Even though neither a theoretical framework nor an explanation for these results were provided, they challenged the theoretical framework previously provided by [START_REF] Desai | When old is gold: the role of business longevity in risky situations[END_REF]. This creates a gap in the literature and leads us to believe that further studies are needed to determine consumers' consumers' response to branded longevity.

In order to fill this gap, this study provides a theoretical explanation based on processing fluency theory [START_REF] Lee | The Effect of Conceptual and Perceptual Fluency on Brand Evaluation[END_REF]. Processing fluency can be defined as the metacognitive experience surrounding the ease of performing a mental action [START_REF] Herrmann | The Power of Simplicity: Processing Fluency and the Effects of Olfactory Cues on Retail Sales[END_REF]. Hence, this approach steps away from the tradition information integration approach usually employed to understand consumers' response to branded longevity [START_REF] Winkielman | The hedonic marking of processing fluency: Implications for evaluative judgment. The psychology of evaluation: Affective processes in cognition and emotion[END_REF]. Processing fluency can take different forms (perceptual fluency, conceptual fluency, and retrieval fluency). However, it exerts the same influence on judgments independently of its form (Alter & Oppenheimer, 2009). The core assumption of this theory is that consumers internally monitor the effort expended on performing a mental process and that this subjectively perceived ease of processing manifests itself as an accessible feeling [START_REF] Schwarz | Metacognitive experiences in consumer judgment and decision making[END_REF]). This feeling can then have an effect on subsequent judgmental tasks through elicited affect [START_REF] Winkielman | The hedonic marking of processing fluency: Implications for evaluative judgment. The psychology of evaluation: Affective processes in cognition and emotion[END_REF]. While subjective feelings of processing ease can be elicited in a variety of ways (e.g., retrieval of stored memories), the focus of the current research is on the perception and processing of an encountered external visual stimulus [START_REF] Schwarz | Metacognitive experiences in consumer judgment and decision making[END_REF]): Branded longevity. Branded longevity -Processing fluency: Processing fluency is enhanced by repeated stimuli [START_REF] Fang | An Examination of Different Explanations for the Mere Exposure Effect[END_REF]. This is because they require fewer neural resources (Reber et al., 1998) because the existing memory representation of the stimulus will facilitate the encoding and processing of the stimulus on later exposures and make processing more fluent (Jacoby et al., 1989;[START_REF] Mandler | Nonspecific effects of exposure on stimuli that cannot be recognized[END_REF][START_REF] Janiszewski | Effects of brand logo complexity, repetition, and spacing on processing fluency and judgment[END_REF]. We assume that consumers are exposed to multiple accounts of branded longevity in everyday environments (e.g product packages & ads). Pecot and de Barnier, (2017b) demonstrate that consumers' recall that they have been exposed to branded longevity even though they do not remember the exact dates they were exposed to. Berger & Fitzsimons (2008) found that frequent incidental exposure to stimuli in everyday life can influence product evaluation and choice. This is because some stimuli that has meaning for the consumer such as branded longevity can trigger automatically representations in memory, rendering them more accessible (Higgis, Rholes and Jones, 1977). Meaning memory associations can be activated automatically by corresponding stimuli. Moreover, other associations that are part of the associative network of the consumers' memory can be activated automatically (Anderson, 1983;Collins & Loftus, 1975). Pecot & de Barnier (2017a) demonstrate that branded longevity is usually communicated in product categories known for their traditional aspects of production (alcohol, biscuits, jam). The consumer has generic associations in memory regarding product categories [START_REF] Chakravarti | Product category perceptions, elaborative processing and brand name extension strategies[END_REF]. So in product categories where brands communicate their longevity, the consumer may have it as an association. This is due to consumers being often exposed to branded longevity whether on product packages or in advertising. Thus, branded longevity increases consumers' processing fluency because it renders the brand more prototypical of the product category. H1: Branded longevity in product categories associated with traditional aspects of production, increases the processing fluency of the brand Branded longevity -Brand familiarity: Experiments have already established that familiarity and processing fluency are two different mechanisms [START_REF] Rugg | Event-related potentials and recognition memory[END_REF]. Brand familiarity is a desirable objective for marketers [START_REF] Bogart | What makes a brand name familiar?[END_REF]. It "captures consumers' brand knowledge structures, that is, brand associations that exist within a consumer's memory" (Campbell and Keller, 2003, p. 293). Hence, it can facilitate brand consideration in consumer choice [START_REF] Holden | Know the name, forget the exposure: Brand familiarity versus memory of exposure context[END_REF]. However, an unfamiliar brand with previously encountered similarities to another brand can create an illusion of brand familiarity [START_REF] Brown | Creating Illusions of Past Encounter Through Brief Exposure[END_REF]. In our case, the previously encountered similarity is branded longevity. Indeed, features that facilitate processing fluency (such as branded longevity) may also enhance consumers' evaluation of the brand [START_REF] Chang | When new commercials do not meet expectations[END_REF] such as its familiarity. H2: Branded longevity positively influence brand familiarity. Involvement in the task: Involvement in the task can vary in shopping situations according to the shopping motivation of the consumer [START_REF] Clarke | The effects of product involvement and task definition on anticipated consumer effort[END_REF]. It is distinct from involvement with the product category, and can be understood as the effort that the consumer exerts in the task [START_REF] Clarke | The effects of product involvement and task definition on anticipated consumer effort[END_REF]. [START_REF] Desai | When old is gold: the role of business longevity in risky situations[END_REF] found that low involvement conditions allow for branded longevity to have an influence on the consumer because it provides a diagnostic cue regarding the brands' expertise. Recently, Zhang et al., (2017) found that branded longevity has an effect for the opposite reason: people in high involvement situations use extensive processing of information and extensive search for the information, therefore branded longevity allow consumers to buffer negative information regarding the brand. Involvement in the task is an important variable because it reflects consumers' processing of the information [START_REF] Petty | Involvement and persuasion: Tradition versus integration[END_REF]. Indeed, the more the consumer is involved with the task, the more direct he or she is likely to process brand claims, hence more likely to attribute branded longevity the reason of his fluency, ultimately diminishing its effect (Hawkins & Hoch, 1992). Recent results in processing fluency literature imply that consumers are more likely to rely on their fluency experience for their evaluations, in low involvement conditions [START_REF] Fang | An Examination of Different Explanations for the Mere Exposure Effect[END_REF][START_REF] Torelli | Extending Culturally Symbolic Brands: A Blessing or a Curse[END_REF]. H3: Consumers' involvement in the task moderates the effect of branded longevity on processing fluency : the more (less) the consumer is involved in the task, the less (more) branded longevity has an effect on processing fluency. H4: Consumers' involvement in the task moderates the effect of branded longevity on brand familiarity : the more (less) the consumer is involved in the task, the less (more) branded longevity has an effect on brand familiarity.

Research Method

To answer our research questions, we conducted a two-group experimental design with only one group being exposed to branded longevity. The participants were randomly assigned to one out of the two experimental conditions. A vignette displaying an advertisement for a fictitious beer brand (Steigal) was used following [START_REF] Lei | Base-rate information in consumer attributions of product-harm crises[END_REF]. Both vignettes included the same picture and description of the beer brand. Manipulations concerned the branded longevity that is displayed under the brand name. To manipulate branded longevity, we added 'since 1952' (vs. no indication of a founding date) as a tagline to the logo. The manipulation of branded longevity was done conforming with Zhang et al., (2017). We measured involvement in the task using a trick question that was preceded by a 7-line paragraph that asked participants not to answer a question regarding their hobbies. Before starting with the main data collection, we conducted a pretest (N=24; mean age= 22 years; 51% female) in order to test the applied scales and to assure the understanding of the vignettes. Afterwards, we collected data using student participation in a large French University. Scales were translated from English to French using retro-translation methods [START_REF] Bartikowski | Calibration internationale des échelles sémantiques[END_REF]. After conducting manipulation checks for branded longevity (does this brand communicate its date of creation?) and for involvement in the task (rate your effort in the task), we assessed the processing fluency through the usage of conceptual fluency (3 items from Srianni et al., 2013) and brand familiarity (3 items from Simonin & Ruth, 1998) using 7-point Likert scales. Participants who failed to answer the manipulation check question regarding longevity correctly were excluded from the data analysis. This lead to a final sample of 100 participants (mean age = 22.5 years; 60% female), so that 50 participants were included in each condition. This sample size provided an acceptable level of statistical power with an effective size of 0.50 at a two-tailed 5% significance level [START_REF] Sawyer | Statistical power and effect size in marketing research[END_REF].

Results

We controlled for confounding variables by including variables measuring the mood and attitude toward the ad [START_REF] Lee | The Effect of Conceptual and Perceptual Fluency on Brand Evaluation[END_REF] as covariates to extract a purer effect of the experimental manipulation. Similar to [START_REF] Lee | The Effect of Conceptual and Perceptual Fluency on Brand Evaluation[END_REF], these variables are not significantly different in both groups. Both Chronbach's alpha for processing fluency (.826) and for brand familiarity (.820) were satisfactory. This ensures the internal validity of the scales. To analyze our hypotheses, we used factorial analyses of variance (ANOVA) with branded longevity (1952 vs. none) being a fixed factor. Manipulation checks proved that participants did perceive the brand's founding date. Regarding the first hypothesis, results demonstrate significant difference between processing fluency between the two experimental conditions, F(1, 99) = 4.94, P=0.022. This indicates that branded longevity significantly influences processing fluency. The effect is positive (The means of conceptual fluency : M control =4.02 ;SD=3.15 ; M Branded longevity =4.75 ; SD=2.39). Therefore H1 is corroborated. Analysis for the second hypothesis demonstrates that brand familiarity is significantly different between the two experimental conditions, F(1, 99) = 7,65, P= 0.006. This indicates that branded longevity significantly influences brand familiarity. The effect is positive (The means of brand familiarity : M control =1.86 ; SD = .84 ; M Branded longevity = 2.51 ;SD =1.81). This leads to the corroboration of H2. Involvement in the task does moderate the relationship between branded longevity and brand familiarity, and processing fluency. However, branded longevity only has a significant effect on brand familiarity F(1,50) = 15,85, P<.01, as well as processing fluency, F(1,50) = 3.9, P=0.05, when the consumer has low involvement in the task. This leads to the corroboration H3 and H4.

Discussion, Implications, and Limitations

Our experiment reveals two main findings: first, our results communicate that branded longevity has a positive effect on brand familiarity and on processing fluency. Second, the results demonstrate that branded longevity can have this effect, only when the consumer has a low involvement in the task.

On a theoretical level, this research contributes to both literature on brand heritage and fluency theory. Brand heritage can benefit from the conceptualization of branded longevity as a standalone construct. We contribute to brand longevity literature [START_REF] Beck | Discovering the "since effect" -how communicating the company age affects consumers' brand perception[END_REF]Zhang et al., 2017 ;[START_REF] Desai | When old is gold: the role of business longevity in risky situations[END_REF] by providing a theoretical framework that provides an explanation of how does branded longevity influence consumers' evaluation of the brand. It addition it allows the examining how do consumers perceive the chronological age of the brand. And under which conditions (task involvement) does branded longevity has an effect on the consumer. Fluency theory can benefit from this research by empirically examining the effect of processing fluency on familiarity. In addition, in determining if visual cues representing the past can increase fluency and if fluency is a mediator of the relationship between branded longevity and brand familiarity. In addition, we contribute to fluency theory by determining the moderating role of task involvement on the relationship between branded longevity & fluency. Indeed, this allows for a counter-intuitive conclusion: more information (branded longevity) leads to less neural effort. On a managerial level, brand managers can benefit from understanding how does branded longevity influence consumers' evaluation of the brand. Brand managers can communicate branded longevity on the packages or ads of new brands. New brands can benefit from communicating their brand longevity because it will allow them to appear more familiar [START_REF] Holden | Know the name, forget the exposure: Brand familiarity versus memory of exposure context[END_REF]. Authors identify three main vocations for future research. First, differences in putting the brand name and branded longevity on the left or on the right following conceptual metaphor theory can be analyzed (Chae & Hoegg 2013). Second, future research can examine the two other modalities of branded longevity such as anniversaries and the age of the brand in order to identify if branded longevity can have other effects on the consumer such as nostalgia (as mentioned in [START_REF] Morhart | Brand authenticity: An integrative framework and measurement scale[END_REF]. Third, future research can examine other conditions that can influence consumers' response to branded longevity such as product involvement and congruency. Indeed, literature on consumers' response to branded longevity has demonstrated confusing results concerning these two conditions. While some authors believe that high product involvement leads to greater impact of branded longevity on the consumer [START_REF] Beck | Discovering the "since effect" -how communicating the company age affects consumers' brand perception[END_REF], others suggest contradicting results [START_REF] Desai | When old is gold: the role of business longevity in risky situations[END_REF]. Moreover, while it is believed that branded longevity can have a positive effect on the consumer for all sectors [START_REF] Rose | Emphasizing brand heritage: Does it work? And how?[END_REF][START_REF] Desai | When old is gold: the role of business longevity in risky situations[END_REF][START_REF] Beck | Discovering the "since effect" -how communicating the company age affects consumers' brand perception[END_REF]Zhang et al., 2017), recent studies demonstrate that it might be more congruent to communicate the brand's longevity in sectors perceived as traditional [START_REF] Pecot | Brands using historical references: a consumers' perspective[END_REF].


Diane Duchemin

Jean-Marie Gorce

Claire Goursaud

Code Domain Non Orthogonal Multiple Access versus ALOHA: a simulation based study

Keywords: Internet of Things, Random access, Non Orthogonal Multiple Access, Compressive sensing, ALOHA

Non Orthogonal Multiple Access (NOMA) is expected to play an important role for IoT networks, allowing to reduce signaling overheads and to maximize the capacity of dense networks with multiple packets simultaneous transmission. In the uplink, NOMA can improve significantly the performance of an ALOHA random access if the receiver implements a multi-user detection algorithm. In this paper we compare the performance of a code domain NOMA with a classical ALOHA protocol, through simulations. The code domain NOMA uses random Gaussian codes at the transmitters and exploits compressive sensing at the receiver to maximize users detection and to minimize symbol error rates.

I. INTRODUCTION

The context of this study is the transmission of short and independent information quantities from a dense population of nodes to a single base station (BS) under high reliability, low latency and low power consumption constraints. This transmission scenario is crucial for massive Internet of Things (IoT) Low Power Wide Area Network (LPWAN) [START_REF] Xiong | Low power wide area machine-to-machine networks: techniques and prototype[END_REF] with Ultra Reliable and Low Latency Communications (URLLC) constraints [START_REF] Schulz | Latency Critical IoT Applications in 5G: Perspective on the Design of Radio Interface and Network Architecture[END_REF].

In order to provide the nodes with quasi-instantaneous and on-demand communications, the hand-shake protocol has to be minimal and a direct random network access without primary connection is seducing. Indeed, on account of latency constraint and energy efficiency, high overhead and delay induced in classic resource allocation protocols prevent from using a connection-oriented mode in this context. This problem is one of the main target of the future 5G NB-IoT protocol, but the investigated solutions are based on a three-step handshake mechanism at the minimum (instead of five in usual 4G or 5G protocols).

The most classical protocol-free multiple access scheme is ALOHA [START_REF] Abramson | THE ALOHA SYSTEM: another alternative for computer communications[END_REF] but it suffers from poor performance when the number of transmitters and the transmission probability grow simultaneously due to a high collision probability. At the protocol level, the collision probability can be kept low by This work has been partly supported by the INSA Lyon -SPIE chair on the Internet of Things and by the French National Agency for Research (ANR) under grant n • ANR-16-CE25-0002 -EPHYL increasing the frequency bandwidth allocated to the random access channel and multi-slotted ALOHA techniques, extensively studied in the past [START_REF] Ghez | Stability properties of slotted Aloha with multipacket reception capability[END_REF], can be chosen.

An alternative, investigated at the Physical layer relies on the use of Non-Orthogonal Multiple Access (NOMA) techniques associated with a powerful multi-user detection algorithm at the receiver [START_REF] Shirvanimoghaddam | Massive Non-Orthogonal Multiple Access for Cellular IoT: Potentials and Limitations[END_REF]. NOMA techniques can be designed in the power, code or time-frequency domains. Power-domain NOMA relies on a classic simultaneous multiple access (MA) using a Superposition Coding (SC) strategy in order to allow more users to access the same resource. On the receiver side, Successive Interference Cancellation (SIC) is employed for recovering each communication. A perfect recovery is possible only if the transmission powers or the individual rates are tuned as a function of the channel states. Another category consists in multiplexing the competing users in the code domain by using user-specific spreading sequences, having a sparse, low-density and low inter-correlation properties. One of the advantages of such MA techniques is the grant-free access provided by the non-orthogonality feature, but at the cost of introducing users interference. This kind of technique can be seen as a step back to the Code Division Multiple Access (CDMA) technology extensively studied in the 90s. The most important difference relies in the fact that in the context of IoT, the receiver doesn't know in advance the active codes and has to detect them and to estimate their data. The recent contribution in this field relies on the use of the sparsity assumption allowing to use algorithms based on compressive sensing (CS) [START_REF] Candès | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[END_REF]- [START_REF] Xie | Many Access for Small Packets Based on Precoding and Sparsity-Aware Recovery[END_REF]. Last but not least, we should mention that another popularized technique is the use of SCMA [START_REF] Nikopour | Sparse code multiple access[END_REF] which redefines modulation and code multiplexing as a whole.

In this paper, we focus on the Code domain NOMA following [START_REF] Fletcher | On-off random access channels: A compressed sensing framework[END_REF]- [START_REF] Xie | Many Access for Small Packets Based on Precoding and Sparsity-Aware Recovery[END_REF], where Gaussian random codes and CS are used. Following Xie et al. [START_REF] Xie | Many Access for Small Packets Based on Precoding and Sparsity-Aware Recovery[END_REF] we use their proposed algorithm for the decoding task.

Our work compares the performance of such approach with the performance achievable with a standard ALOHA technique, taken as a reference in terms of uncoordinated random access scheme. As Choi in [START_REF] Choi | NOMA-Based Random Access With Multichannel ALOHA[END_REF], we show that NOMA allow a performance improvement compared to a slotted ALOHA access. While [START_REF] Choi | NOMA-Based Random Access With Multichannel ALOHA[END_REF] is dedicated to Power domain NOMA with an analysis of the throughput in multichannel ALOHA, our study deals with Code domain NOMA in a single channel ALOHA scheme. Based on the implementation of the proposition of [START_REF] Xie | Many Access for Small Packets Based on Precoding and Sparsity-Aware Recovery[END_REF], we evaluate the gains of NOMA with different metrics, covering both the users detection and the data estimation tasks.

The rest of the paper is organised as follows: section II describes our model, with the two confronted access types. Section III details the metrics used for the performance comparison. A theoretical analysis of detection performance in a simple scenario is realized in section IV while the actual detection and estimation are jointly retrieved and evaluated in section V under a more realistic scenario. Section VI concludes the paper.

II. MODEL DESCRIPTION

A. Scenario

The scenario herein proposed follows [START_REF] Xie | Many Access for Small Packets Based on Precoding and Sparsity-Aware Recovery[END_REF]. The reference topology is a single cell in the uplink, with a base station (BS) made of a unique receiver equipped with multiple antennas. A very dense population of users (herein called nodes) is spread over the cell and generates random and bursty transmissions. The nodes set is referred to as N , whose size is N = card(N ).

The BS emits a synchronization frame periodically, allowing the nodes to work in a slotted manner. It is assumed that an unknown and small subset of nodes N a ⊂ N is active simultaneously in a given time slot. Their number is noted N a . Thus the medium is a single resource, shared among users.

The wireless channel is modeled as follows. The pathloss is assumed to be known by the nodes which use power control to compensate it for. Therefore, the BS then receives all packets at the same power in average, but due to block fading the different signals are received at the BS with different powers. More precisely, the channel between each node and the BS, equipped with α antennas, is characterized by identically and independently distributed (i.i.d.) random variables. Note that all channel coefficients remain constant in a time slot. An Additive White Gaussian Noise (AWGN) is also considered.

The signal received at the BS can be expressed as follows:

Y = √ ρ 0 Na n=1 h n x n T + Z (1)
where Y ∈ C α×m is the received signal, ρ 0 is the Signal to Noise Ratio (SNR). The vector h n denotes the channel response of node n, whose each of the α elements is the fading coefficient for one of the BS antennas during the m channel uses of the message x n . Z denotes the AWGN.

[.] T is the transpose operator. All elements of Z and h n follow a standard complex normal law. The specific expressions of x n for each multiple access type are given hereunder.

B. ALOHA-type system 1) Slotted ALOHA transmissions scheme: For a fair comparison, and as the assumptions in II-A hold for both NOMA and ALOHA transmissions, we consider a slotted ALOHA scheme.

Our ALOHA-based system does not use any coding strategy. The data stream is send as is, nevertheless, it is preambled by its node unique identifier w n . A unique bit sequence is simply mapped to a node, the length L of the identifier bit sequences is related to the nodes set size: L = log 2 (N ) bits. This sequence only requires a modulation before being preambled to the data: w n is the modulated symbols vector of the node n identifier bit sequence. Let s n ∈ S d denote the modulated symbols of data streams, where S ⊂ C is the set of modulated symbols. A message is thus obtained as follows:

x n = [w n T s n T ] T (2)
An active node, i.e. willing to send a message, will transmit on the next time slot whose length is d + L/M where M is the chosen modulation's rate, and . the ceiling function.

Note that when the messages are dedicated to the nodes activity notification, only the identifiers are sent.

2) SIC decoder: The decoder of the slotted ALOHA-based transmission scheme exploits Successive Interference Cancellation (SIC). We assume that the decoder have prior information about the channel response for each node, their identifiers w n and the SNR. A predefined number of iterations K is set. During each iteration, a node is detected before its data are estimated. The detection criteria is the lowest distance between the node's identifier sequence and its estimated symbols sequence (obtained with a hard decision on the result of a Least Square algorithm, given the channel response of the node, the SNR and the received signal Y ). The data estimation is performed on the data part of the received signal with a Least Square (LS) algorithm as well, considering the SNR and the channel response of the detected node. The estimated contribution of the detected node is then canceled and another iteration starts.

C. NOMA-type system

The random code based approach proposed by Xie et al. in [START_REF] Xie | Many Access for Small Packets Based on Precoding and Sparsity-Aware Recovery[END_REF] is now described; the reader is refer to their paper for more details.

1) Code domain NOMA transmissions scheme: The use of Gaussian codes comes from Compressive Sensing (CS) theory. In CS theory, an initial signal of dimension N is projected into a new signal composed of very few (k) non-zero components among N , referred to as a k-sparse signal (or having a sparsity level k). The set of indices of these k components constitutes the sparsity pattern. One then reconstructs this version of the signal from only m measurements obtained by a specific m * N measurement matrix with m < N . This under-determined problem can be solved thanks to an algorithm which performs a l1-minimization or a sparse approximation. The dimension's reduction of the observed signal (from N to m) gives the compressive characteristic to this sensing technique. In [START_REF] Fletcher | On-off random access channels: A compressed sensing framework[END_REF], the authors establish the parallel between users detection and the sparsity pattern detection in CS, and refer to the On-Off Random Access channel for modeling a communication scheme where the signal sent by the N users can be represented by a vector whose each entry corresponds to a user: if the user is active, its entry will be a non-zero value whereas an inactive user will be assigned a zero in the vector. The goal of the receiver is to detect the unknown active nodes rather than retrieving the actual value of the signal vector. Xie et al. [START_REF] Xie | Many Access for Small Packets Based on Precoding and Sparsity-Aware Recovery[END_REF] introduced a framework dedicated to simultaneous detectionestimation.

Similarly, each node n is associated to a unique i.i.d. complex Gaussian coding normalized vector c n or column-normalized matrix C n , which is used to encode the data s n . x n ∈ C m denotes an encoded message. For the general case:

x n = C n s n , (3)
the received signal is then:

Y = √ ρ 0 Na n=1 h n (C n s n ) T + Z . (4)
(For d = 1, the equation 3 reduces to x n = c n s n and s n can be considered as an activity notification bit.)

2) NBOMP algorithm decoder: The Normalized Block Orthogonal Matching Pursuit is an iterative algorithm which, at each iteration, selects an additional potentially active node and jointly estimates the signals of previously selected nodes. Since it defines the number of detections, the iterations number K should be at least as large as the expected number of active nodes to be able to detect all of them. The detection and estimation are respectively realized by means of correlation and Least Square (LS) operations and require the codes, SNR and channels knowledge. The word "block" in NBOMP refers to the block sparsity structure of the unknown signal and the block nodes signatures which appear when vectorizing equation 4:

y = √ ρ 0 Na n=1 (C n ⊗ h n )s n + z = √ ρ 0 Na n=1 B n s n + z (5)
where vec(Y ) = y, ⊗ denotes the Kronecker product, B n corresponds to the block signature matrix of node n.

For the detection task, at iteration i, a node n is detected if its correlation coefficient c n,i is the largest of all remaining (undetected) nodes. c n,i is defined as follows:

c n,i = B H n * res i 2 ||h n || 2 (6)
where [.] H denotes the hermitian operator and res i is the residual signal, i.e. the rest of the received signal y once the estimated symbols contributions (from the LS computation at iteration i-1) are retrieved. When the max(c n,i ) is selected, the index n is added to the set of detected nodes λ i . The estimation is performed by a LS operation. Indeed, once the additional node is detected, the algorithm is intended to solve sλi = arg min

s0∈S d.i ||y - √ ρ 0 B λ,i s 0 || 2 (7)
where s λ i is the concatenation of the estimated data streams of the i detected nodes and B λ,i denotes the concatenation of the i signature matrices B n with n ∈ λ i . The Moore-Penrose pseudo-inverse can be computed but, for the sake of complexity, our implementation of the NBOMP rather employs a QR factorization. A decision is taken afterward to map the estimated values to the closest modulated symbols, as such, sλi is the solution to [START_REF] Foucart | A Mathematical Introduction to Compressive Sensing[END_REF]. The last iteration provides the data stream estimates for each detected node.

III. PERFORMANCE METRICS

A. Detection evaluation 1) Detection Error Rate: This metric is used for the first approach focusing on user detection only. The number of message arrivals follows a Poisson law with parameter µm. µ = µ 0 N where µ 0 is the message arrival rate per node. Let X denote the random variable of message arrivals number and P m (X = k) the probability of k message arrivals in a slot of length m symbols. According to the Poisson law:

P m (X = k) = e -µm (µm) k k! (8)
The Detection Error Rate (DER) depends on the expected messages error number, such as:

DER = ∞ k=κ+1 (kP m (X = k)) ∞ k=0 (kP m (X = k)) = ∞ k=κ+1 (kP m (X = k)) E[X] (9
) where E[X] is known to be equal to the parameter µm.

2) Detection Success Rate: In the second approach, the detection efficiency is quantized by the Detection Success Rate (DSR). Let λ Na denote the detected set,

DSR = card(λ Na ) N a = card(λ K ∩ N a ) N a (10)
3) Missed Detection and False Alarm: In order to have more details about the detection performance, we use two additional metrics: the Missed Detection (MD) and False Alarm (FA) ratios. Let λ Na denote the FA (also known as the False Positive) set, MD and FA ratios are thus:

M D = card(N a ∩ λ Na ) N a = 1 -DSR (11)
F A = card(λ Na ) card(N ∩ N a ) = K -card(λ Na ) N -N a (12)
B. Estimation evaluation 1) Symbol Error Rate: The estimation performance is evaluated by means of the classic Symbol Error Rate (SER) metric. Err n denotes the erroneously estimated symbols of node n, SER can be expressed by:

SER = d * card(λ Na ) + n∈λ Na card(Err n ) d * N a (13)
C. Latency evaluation 1) Latency: The latency can be defined as the sum of the waiting time (the maximum channel use number that a node would have to wait before being allowed to send its message), the transmission time and the decoding time. However, the decoding time can be neglected since, in this work, we consider that the BS is not limited from computation and power resources viewpoints. Moreover, as the transmission time varies according to the distance between the node and the BS, we will only take the static part, i.e. the message duration, into account. As a result, the latency should be approximately twice the message duration. As it is directly proportional to the message length, the latency of both access scheme can therefore be evaluated by the message length m.

IV. DETECTION PERFORMANCE

We firstly focus on the detection performance evaluation. We consider an ideal scenario and a simple use case with no data transmission but where the nodes aim to notify of their activity. In this kind of application [START_REF] Fletcher | On-off random access channels: A compressed sensing framework[END_REF], the decoder only has to detect the active nodes. We simplify the model, assuming that the transmissions occur in a high SNR regime (noise is neglected). We also assume that both the transmitters and the receiver have one antenna and that the channel is not subject to fading. As such, only the presence of competing communications makes the transmissions succeed or fail, according to their number. Especially, it is assumed that an ALOHA-based transmission is correctly detected if no other communications occur within the same time slot, i.e. if no collision occurs. NOMA can handle a few simultaneous transmissions, according to the code length.

Theorem 6 of [START_REF] Tropp | Signal recovery from random measurements via orthogonal matching pursuit[END_REF] defines the length of measurement vectors (in relation to the sparsity level k of the signal, its dimension N and some positive constants δ and β) as the necessary length for a successful reconstruction of the signal via OMP using a predefined measurement matrix, with a probability of success above 1 -δ. The length m is defined as βk log(N/δ). Further, the authors observe for a success probability P s of 99%, that m can be approximated by 2k log(N ). Moreover, [START_REF] Fletcher | Orthogonal matching pursuit from noisy measurements: A new analysis[END_REF] confirms this estimation by showing that m = 2k log(N -k) is sufficient for reconstruction under asymptotic conditions. The authors clarify that, when k is unknown but between k min and k max , having m = 2k max log(N -k min ) measurements is sufficient. We are therefore using the similar formula for the Gaussian codes length with the assumption that k min = 0, leading to m = 2k max log(N ). We assume that this code length allows to detect up to k max simultaneous transmissions with 99% of success, according to the numerical model from [START_REF] Tropp | Signal recovery from random measurements via orthogonal matching pursuit[END_REF]. Let κ denote the maximal detected message number in a time slot: κ aloha = 1 and κ noma = k max .

Recalling equation 9, the Detection Error Rates are:

DER aloha = ∞ k=2 (kP m (X = k)) E[X] = 1 -e -µ0N L/M ( 14
)
DER noma = P s ∞ k=kmax+1 (kP m (X = k)) E[X] + 1 -P s = 1 -P s .e -2µ0N.kmax log(N ) kmax k=0 (k (2µ0N.kmax log(N )) k k! ) 2µ 0 N.k max log(N ) (15)
The DER of the two access schemes are shown in Fig. 1 for a range of node set size from 1.10 3 to 25.10 3 . The results are given between 1 and 10 -2 due to the upper bound approximation. The Poisson law parameter µ 0 is set to 1/3.6.10 5 message . For larger N , the message length of NOMA becomes too large to maintain a number of simultaneous active nodes below k max in a time slot with a high probability, thus the fast degradation up to the point where the ALOHA system takes advantage of shortest time slots even if only one active node is allowed for a successful detection. Fig. 1 shows the benefit of NOMA with Gaussian codes compared to a slotted ALOHA for a range of large N . But this analysis suffers from several weak points: the model is dedicated to a network-level performance evaluation, showing the effect of collisions on both access techniques. However, it does not consider channel distortions, neither fading nor noise. Secondly, it also relies on approximations. The real performance of decoding algorithms are not exploited. Furthermore, this analysis is not adapted to data transmission evaluation. Therefore, a more sophisticated model, taking physical layer aspects into account, is presented in the next section.

V. DETECTION AND ESTIMATION PERFORMANCE

In order to evaluate node detection probability and data estimation error rates, we implemented code domain NOMA and ALOHA-based systems with simulations under the more realistic transmission scenario described in section II-A.

a) Comparison of NOMA to ALOHA transmissions performance: Fig. 2 and3 represent the detection and estimation performance of the confronted systems. The results are drawn as function of the SNR in the range [-4; 20]dB, for different active nodes numbers (from N a = 8 to 30). The nodes set size is fixed to 80, and the packets are all made of d = 100 QPSK symbols. NOMA messages length m is 500 symbols, while ALOHA messages are transmitted over 104 symbol time (or channel use). To keep a quasi-equivalent channel occupancy, The BS possesses α = 8 antennas and runs K = 30 iterations of either the SIC-based decoder for the Aloha system or the NBOMP decoder for the NOMA transmissions. Fig. 2 shows that the DSR of NOMA is subject to little variations w.r.t. N a and to the SNR while the detection performance of ALOHA degrades when N a rises, and shows a local maximum when SN R ≈ 4dB. For the whole range of represented SNR and active nodes number, NOMA outperforms ALOHA in terms of DSR. Similarly, in Fig. 3, the performance divergence of ALOHA and NOMA can be observed. Indeed, the SER of NOMA remains quasi-insensitive to the active node number whereas the impact of the worse detection with ALOHA is clear. For SNR in [-4,4]dB, we take for granted that, for both access schemes, when the SNR increases, the SER decreases. For SNR values above 4dB, the degradation of the SER of ALOHA is related to the decreasing DSR whereas the SER of NOMA continues to improve beyond 10 -2 . Even though the NOMA performance are better than the ALOHA as a general rule, one can nevertheless notice the superiority of the data estimation of the ALOHA system when N a = 8 for the lowest SNR values, although the detection probability of ALOHA is lower than that of NOMA. For larger N a , NOMA takes advantage of the redundancy introduced in the coding scheme despites of the collisions. However, if the Missed Detections are not taken into account, i.e. when the SER is restricted to the active nodes that have been detected, the SER of NOMA and the worse SER of ALOHA even coincide for the range of lowest SNR values. In this case, NOMA also outperforms Aloha for any active nodes number as soon as SNR>4dB.

Despites of the inter-dependence of the detection and the estimation tasks of the NBOMP or SIC-based algorithms, one can not state on a better efficiency of an access scheme compared to the other based on only one of the previous evaluation metrics. Indeed, the behaviors of the transmission techniques on the detection side don't directly predict the performance on the estimation side therefore the two metrics must be considered as a whole. Moreover, the choice of parameters such as m or K can be beneficial to a particular metric while degrading the results of another performance metric, as pointed out in the following paragraphs.

b) Latency versus SER: Let consider that a constant number of active nodes has access to the channel within a frame length, or channel use number, in order to transmit the same amount of data. In Fig. 4, N a = 8 nodes among N = 80 transmit d = 100 data symbols. The messages with NOMA occupy the full frame whereas for ALOHA, the nodes keep the same time slot length (i.e. m = d + L/M ) but their message arrivals number in a slot decreases with the increase of the available channel use number.

The results of the SER w.r.t. the available frame length are shown at the 16 th and the 30 th iteration, with α=8, SN R=10dB. Fig. 4 shows how NOMA SER drops sharply around 200 or 300 channel use (resp. for 16 and 30 iterations) whereas the ALOHA SER decreases slowly with the lengthening of the frame. This illustrates how latency trades with reliability for each access scheme: a longer m in NOMA, allowing more redundancy, improves the estimation capability at the cost of an increase of the latency of the same order. On the opposite, the ALOHA system benefits from a short latency, independent of the frame length and allowing fewer concurrent accesses during a transmission, however not enough to compensate for the lower performance, dropped down by a low DSR. Another difference stems from the SER of NOMA that significantly The vertical gap between red and blue curves of NOMA is representative of this SER degradation. On ALOHA side, the slight difference is due to the loss in the DSR. Additional iterations allow to increase the detection probability thus the SER also improves. c) Detection Error Trade-off: Fig. 5 represents the Detection Error Trade-off (DET) of NBOMP and ALOHA SIC-based decoder. The MD ratio is represented as a function of the FA rate for N = 80, N a = 24, α = 8, SN R = -10dB, d = 100, m N OM A = 500, m ALOHA = 104 and K varying between 10 and 40 iterations. One can make two observations: firstly, again NOMA has better MD and FA ratios than ALOHA for the same set of parameters. Moreover, one can note the efficient decrease of MD for NBOMP with a few iterations, but once K reaches N a this tendency is inversed and the detection of a few more active nodes costs much more False Alarms. This makes it clear that the choice of the iterations number for the decoding algorithm also has a great impact on the detection performance. The simulation studies performed in this paper highlight the performance gain that can be achieved thanks to a code domain NOMA random access. Despite longer packets resulting in a higher channel occupancy rate, the NOMA outperforms a classical ALOHA scheme, thanks to a robust multi-user decoder technique. Our future work will consist in optimizing the coding scheme to maximize the system performance, which may be measured by the nodes density that is achievable for a fixed bandwidth and under some error rate constraint.

It is worth mentioning that the multi-user receiver used in this paper implements a multi-user LS receiver which suffers from a high complexity. Even if the algorithmic complexity at the BS can be considered negligible compared to the nodes complexity, this technique cannot directly scales for very dense networks with hundreds and even thousands nodes. Sub-optimal algorithms will be necessary.

Fig. 3 .

 3 Fig. 3. ALOHA and NOMA Symbol Error Rate

Fig. 5 .

 5 Fig. 5. MD vs FA ratios

  Fig. 1. Detection Error Rate of NOMA and ALOHA systems according to the node set size N per symbol time, which corresponds to a mean activity of one message per node per hour with a modulation rate of 100bps. The NOMA DER are bounded by the success probability threshold P s , related to the approximation of the sufficient length from the observations of Tropp et al. and the curves might be shaped by this 10 -2 bound for low node set size or message arrival rate. Nevertheless, we can observe that NOMA outperforms the slotted ALOHA system up to N ≈ 17.10 3

					Message arrival rate (µ)
		10 0	0.01	0.02	0.03	0.04	0.05	0.06
	Detection Error Rate	10 -1						slotted aloha NOMA, kmax=10 NOMA, kmax=20 NOMA, kmax=30 NOMA, kmax=40
								NOMA, kmax=50

  Fig. 4. SER versus available channel use frame lengthdegrades by an iteration number increase while ALOHA is subject to an almost negligible estimation improvement. Indeed, at N a = 8, additional NBOMP iterations introduce FA. Since the estimation of the K node signals is performed jointly, the estimated signals of active nodes are corrupted by the FA.

	10 0
	Symbol error rate	ALOHA, K=16 NOMA, K=16
		ALOHA, K=30
		NOMA, K=30
	100 10 -5	200	300	400	500	600	700	800
				Available frame length