Valeriy I Yudin
email: viyudin@mail.ru

Alexei V Taichenachev

Basalaev Yu

Thomas Willette

T E Mehlstäubler

Rodolphe Boudot

J W Pollock

M Shuker

E A Donley

J Kitching

Yu Basalaev

T Zanon-Willette

Combined error signal in Ramsey spectroscopy of clock transitions

Keywords: atomic clocks, light shift, Ramsey spectroscopy

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Atomic clocks based on high-precision spectroscopy of isolated quantum systems are currently the most precise scientific instruments, with fractional frequency instabilities and accuracies at the 10 -18 level [1,2,3,4,5]. Frequency measurements at this level enable improved tests of fundamental physics, as well as new applications like chronometric geodesy [6,7].

For many promising clock systems, probe-field-induced frequency shifts can limit the clock frequency instabilities and accuracies. In the case of magnetically induced spectroscopy [8,9], ac-Stark shifts can limit the achievable clock stability, and for ultranarrow electric octupole [10] and two-photon transitions [11,12], the large off-resonant ac-Stark shift can completely prevent high-accuracy clock performance. Similarly, the large number of off-resonant laser modes present in clocks based on direct frequency comb spectroscopy [13,14] induce large ac-Stark shifts. Probe-field-induced shifts also cause instability for microwave atomic clocks based on coherent population trapping (CPT) [15,16,17,18,19,20]. Compact microwave cold-atom clocks [21,22] and hot-cell devices like the pulsed optical pumping (POP) clock [23,24] that are based on direct microwave interrogation can also be affected by probe-induced frequency shifts.

Probe-induced shifts can be suppressed through the use of Ramsey spectroscopy [25] in combination with cleverly devised modifications. In contrast to continuous-wave spectroscopy, Ramsey spectroscopy has a large number of extra degrees of freedom associated with many parameters that can be precisely controlled: the durations of Ramsey pulses τ 1 and τ 2 , the dark time T , the phase composition of composite Ramsey pulses [START_REF] Levitt | Encyclopedia of Nuclear Magnetic Resonance[END_REF], variations in Ramsey sequences including the use of three or more Ramsey pulses, different error signal variants, and so on. Some modified Ramsey schemes for the suppression of the probe-field-induced shifts in atomic clocks were theoretically described in Ref. [START_REF] Yudin | [END_REF], which proposed the use of pulses of differing durations (τ 1 = τ 2 ) and the use of composite pulses instead of the standard Ramsey sequence with two equal π/2-pulses. This "hyper-Ramsey" scheme has been successfully realised in an ion clock based on an octupole transition in Yb + [28,5], where a suppression of the light shift by four orders of magnitude and an immunity against its fluctuations were demonstrated. Further developments in Ramsey spectroscopy resulted in additional suppression of probe-field induced frequency shifts. For example, the hyper-Ramsey approach uses new phase variants to construct error signals [29,30,31,32] to significantly suppress the probe-fieldinduced shifts in atomic clocks. However, as was shown in Ref. [33], all previous hyper-Ramsey methods [START_REF] Yudin | [END_REF]28,5,29,31,34] are sensitive to decoherence and spontaneous relaxation, which can prevent the achievement of state-of-the-art performance in some systems. To overcome the effect of decoherence, a more complicated construction of the error signal was recently proposed in Ref. [35], which requires four measurements for each frequency point (instead of two) combined with the use of the generalized hyper-Ramsey sequences presented in Ref. [31]. Nevertheless the method in Ref. [35] is not free from other disadvantages related to technical issues such as time dependent pulse area fluctuations and/or phase-jump modulation errors during the measurements.

The above approaches [START_REF] Yudin | [END_REF]28,5,29,31,34,35] are all one-loop methods, since they use one feedback loop and one error signal. However, frequency stabilization can also be realized with two feedback loops combined with Ramsey sequences with different dark periods T 1 and T 2 [33,36,37]. For example, the synthetic frequency protocol [33] in combination with the original hyper-Ramsey sequence [START_REF] Yudin | [END_REF] allows for substantial reduction in the sensitivity to decoherence and imperfections of the interrogation procedure. Auto-balanced Ramsey spectroscopy (ABRS) is another effective approach that was first experimentally demonstrated in a 171 Yb + ion clock [37]. This intuitive approach was rigorously substantiated and generalized theoretically in Ref. [38], and also recently realized in a CPT atomic clock [39]. For ABRS, in addition to the stabilization of the clock frequency ω, a second loop controls a variable second (concomitant) parameter ξ, which is an adjustable property of the first and/or second Ramsey pulses. While both of these two-loop methods [33,37,38] are robust and can perfectly suppress probe-induced shifts of the measurement of the clock frequency, their implementation can be complex due to the two-loop architecture.

A principal question remains: does a one-loop method exist that has comparable (or better) efficiency to ABRS? In this paper, we present a positive answer to this question. We have found a universal protocol to construct a combined error signal (CES), which allows for perfect suppression of probe-induced shifts with the use of only one feedback loop. The CES technique has exceptional robustness, in that it is independent of arbitrary relaxation processes and different non-idealities of the measurement procedure. This method can be considered as a preferred alternative to ABRS spectroscopy. Indeed, CES is technically simpler (because of one feedback loop) and can be more efficient when a hyper-Ramsey pulse sequence [START_REF] Yudin | [END_REF] is used. The CES protocol is applicable to optical atomic clocks as well as to microwave atomic clocks based on CPT Ramsey spectroscopy and POP clocks.

Theoretical model

In this section, we follow the analysis developed in Ref. [38], which we repeat here for completeness. We consider a two-level atom with unperturbed frequency ω 0 of the clock transition |g ↔ |e (see Fig. 1), which interacts with a Ramsey sequence of two completely arbitrary pulses (with durations τ 1 and τ 2 ) of the resonant probe field with frequency ω:

E(t) = Re{E(t)e -iϕ(t) e -iωt } . (1)
The pulses are separated by a free evolution interval (dark time) T , during which the atom-field interaction is absent (see Fig. 1). We emphasise that the Ramsey pulses with arbitrary durations τ 1 and τ 2 do not depend on the dark time T and can have an arbitrary shape and amplitude (i.e., during τ 1 and τ 2 an amplitude E(t) can be an arbitrary real function), and an arbitrary phase function ϕ(t) (e.g., the Ramsey pulses can be composite pulses, chirped pulses, and so on). In a given sequence of Ramsey measurements, the pulse shape and amplitude must be consistent from one measurement to another. We assume only one restriction: aside from phase jumps applied to generate the error signal (discussed below), the phase function should be constant during the dark time T , ϕ(t) = const, as is typical for Ramsey spectroscopy.

Our main goal is to develop a universal one-loop method, which allows us to stabilize the probe field frequency ω at the unperturbed frequency of the clock transition, ω = ω 0 , in the presence of decoherence, arbitrary relaxation and light shifts. For this purpose, we will use the formalism of the density matrix ρ, which has the following form

ρ(t) = j,k=g,e |j ρ jk (t) k| , (2)
in the basis of states |g and |e . In the resonance approximation, the density matrix components ρ jk (t) satisfy the following differential equations:

[∂ t + Γ -i δ(t)]ρ eg = iΩ(t)[ρ gg -ρ ee ]/2 ; ρ ge = ρ * eg ; [∂ t + γ e ]ρ ee -γ g→e ρ gg = i[Ω(t)ρ ge -ρ eg Ω * (t)]/2 , (3)
[∂ t + γ g ]ρ gg -γ e→g ρ ee = -i[Ω(t)ρ ge -ρ eg Ω * (t)]/2 .
Here the time dependencies Ω(t) and δ(t) are determined by the following: Ω(t) = d E(t)e -iϕ(t) and δ(t) = δ -∆ sh (t) during the action of the Ramsey pulses τ 1 and τ 2 , but Ω(t) = 0 and δ(t) = δ during the dark time T . d is a matrix element of the atomic dipole moment, δ = ω -ω 0 is the detuning of the probe field from the unperturbed atomic frequency ω 0 , and ∆ sh (t) is an actual probe-field-induced shift (see Fig. = γ e and γ g→e = γ g in the case of closed two-level system, while γ e→g < γ e and/or γ g→e < γ g in the case of open system. The constant Γ = (γ e + γ g )/2 + Γ describes the total rate of decoherence: spontaneous as well as all other processes, which are included in the parameter Γ (e.g., an influence of the nonzero spectral width of the probe field). Equations ( 3) can be rewritten in the vector form

∂ t ρ(t) = L(t) ρ(t) , (4)
where ρ(t) is a vector formed by the matrix components ρ jk (t),

ρ(t) =      ρ ee (t) ρ eg (t) ρ ge (t) ρ gg (t)      , (5)
and operator (Liouvillian) L(t) is 4 × 4 matrix determined by the coefficients of Eq. ( 3):

L(t) =      -γ e -iΩ * (t)/2 iΩ(t)/2 γ g→e -iΩ(t)/2 -Γ + i δ(t) 0 iΩ(t)/2 iΩ * (t)/2 0 -Γ -i δ(t) -iΩ * (t)/2 γ e→g iΩ * (t)/2 -iΩ(t)/2 -γ g      . (6)
In this case, a spectroscopic Ramsey signal can be presented in the following general form, which describes Ramsey fringes (as a function of δ),

A T (δ) = ( ρ obs , Ŵτ 2 ĜT Ŵτ 1 ρ in ) , (7)
where the scalar product is determined in the ordinary way: ( x, y) = m x * m y m . Operators Ŵτ 1 and Ŵτ 2 describe the evolution of an atom during the first (τ 1 ) and second (τ 2 ) Ramsey pulses, respectively, and the operator ĜT describes free evolution during the dark time T . Vectors ρ in and ρ obs are initial and observed states, respectively. For example, if an atom before the Ramsey sequence was in the ground state |g , and after the Ramsey sequence we detect the atom in the exited state |e , then vectors ρ in and ρ obs are determined, in accordance with definition (5), as

ρ in =      0 0 0 1      , ρ obs =      1 0 0 0      . ( 8
)
For stabilization of the frequency ω we need to form an error signal, which is obtained by using data from two Ramsey sequences, one with phase jump α + between the two pulses and one with phase jump α -(see Fig. 1), as was proposed in Ref. [40]. Note that the phase jumps of the laser field can be easily realized in experiments by the use of an acousto-optical modulator (AOM). We mathematically describe these phase jumps α + and α -(their precise timing does not matter) by the operators Φα + and Φα -, respectively. In this case, let us introduce the expression of the Ramsey signal in the presence of the phase jump α, described by the operator Φα ,

A T (δ, α) = ( ρ obs , Ŵτ 2 Φα ĜT Ŵτ 1 ρ in ) . (9)
As a result, the error signal can be presented as a difference,

S (err) T = A T (δ, α + ) -A T (δ, α -) = ( ρ obs , Ŵτ 2 DΦ ĜT Ŵτ 1 ρ in ) , (10)
with DΦ = Φα + -Φα -. To maximise this error signal, α ± = ±π/2 is typically used. However, in real experiments, we can have |α + | = |α -| due to various technical reasons (e.g., electronics) which will lead to a shift of the stabilised frequency ω in the case of standard Ramsey spectroscopy. Therefore, here we will consider the general case of arbitrary α + and α -to demonstrate the robustness of CES technique, where the condition |α + | = |α -| does not lead to an additional frequency shift.

Next we consider the structure of the following operators: ĜT , Φα + , Φα -, and DΦ .

The operator for the free evolution, ĜT , has the following general matrix form ĜT =

     G 11 (T ) 0 0 G 14 (T ) 0 e -(Γ-iδ)T 0 0 0 0 e -(Γ+iδ)T 0 G 41 (T ) 0 0 G 44 (T )      , (11)
which corresponds to Eq. ( 4), if Ω(t) = 0 and δ(t) = δ in the Liouvillian (6). The matrix elements G 11 (T ), G 14 (T ), G 41 (T ), and G 44 (T ) depend on four relaxation rates: {γ e , γ e→g , γ g , γ g→e }. In particular, for purely spontaneous relaxation of the exited state |e , when γ g = γ g→e = 0, we obtain ĜT =

     e -γ e T 0 0 0 0 e -(Γ-iδ)T 0 0 0 0 e -(Γ+iδ)T 0 γ e→g γ e (1 -e -γ e T ) 0 0 1      . ( 12
)
Operators for the phase jumps Φα + and Φαhave the forms

Φα ± =      1 0 0 0 0 e iα ± 0 0 0 0 e -iα ± 0 0 0 0 1      , (13)
which lead to the following expression for DΦ ,

DΦ = Φα + -Φα -=      0 0 0 0 0 (e iα + -e iα -) 0 0 0 0 (e -iα + -e -iα -) 0 0 0 0 0      . ( 14
)
As a result, taking into account Eq. ( 11), we obtain a formula for the matrix product ( DΦ ĜT ), DΦ ĜT = e -ΓT ΥδT ,

where the matrix ΥδT is defined as

ΥδT =      0 0 0 0 0 e iδT (e iα + -e iα -) 0 0 0 0 e -iδT (e -iα + -e -iα -) 0 0 0 0 0      . (16)
Note that ΥδT=0 = DΦ .

According to Eq. ( 15), the error signal ( 10) can be rewritten in the following form:

S (err) T (δ) = e -ΓT ( ρ obs , Ŵτ 2 ΥδT Ŵτ 1 ρ in ). ( 18
)
Note that this result is the same if we apply phase jumps α ± at any arbitrary point during the dark interval T . It is interesting to note that the expression of the error signal in the presence of relaxation is formally different from the the error signal in the absence of relaxation only due to the scalar multiplier e -ΓT , which primarily affects the amplitude, but not the overall shape of the error signal. This is one of the main specific properties of the phase jump technique for Ramsey spectroscopy that makes it robust against relaxation. Indeed, for other well-known methods of frequency stabilization, which use a frequency jump technique between alternating total periods of Ramsey interrogation (τ 1 + T + τ 2 ), relationship (10) does not exist. Thus, the phase jump technique has a fundamental advantage over the frequency jump technique in that it is less sensitive to relaxation. In addition, in the ideal case of α + = -α -= α, the error signal (10) can be expressed as

S (err) T (δ) = 2 sin(α)e -ΓT ( ρ obs , Ŵτ 2 ΘδT Ŵτ 1 ρ in ), (19)
where the matrix ΘδT ,

ΘδT =      0 0 0 0 0 ie iδT 0 0 0 0 -ie -iδT 0 0 0 0 0      , (20)
depends only on δT .

CES protocol

In this section we demonstrate the universality and robustness of the CES technique. We use the Ramsey interrogation of the clock transition for two different, fixed intervals of free evolution T 1 and T 2 , where we have two error signals S

(err)

T 1 (δ) and S (err)

T 2 (δ) described by Eq. (18). However, for frequency stabilization we introduce the combined error signal (CES) as the following superposition,

S (err) CES (δ) = S (err) T 1 (δ) -β cal S (err) T 2 (δ) , (21)
where a calibration coefficient β cal is to account for decay of the Ramsey fringe amplitude and will be defined below. Thus, the shift of the stabilized frequency δclock is determined as a solution of the equation S (err) CES (δ) = 0 in relation to the unknown δ. In accordance with Eq. ( 18), the expression (21) can be written in the form

S (err) CES (δ) = e -ΓT 1 ( ρ obs , Ŵτ 2 ΥδT 1 Ŵτ 1 ρ in ) -β cal e Γ(T 1 -T 2 ) ( ρ obs , Ŵτ 2 ΥδT 2 Ŵτ 1 ρ in ) . ( 22
)
If we assume that

β cal = e -Γ(T 1 -T 2 ) , (23)
then we obtain

S (err) CES (δ) = e -ΓT 1 ( ρ obs , Ŵτ 2 ΥδT 1 Ŵτ 1 ρ in ) -( ρ obs , Ŵτ 2 ΥδT 2 Ŵτ 1 ρ in ) . (24)
If we apply δ = 0 for operators ΥδT 1 and ΥδT 2 , then due to Eq. ( 17) we have ΥδT 1 =0 = ΥδT 2 =0 = DΦ . In this case, we obtain from Eq. ( 24):

S (err) CES (0) = 0 , (25)
since the two terms inside of the square brackets cancel in Eq. ( 24). Thus, we have analytically shown that the CES method always leads to zero field-induced shift of the stabilized frequency ω in an atomic clock, δclock = 0. From a practical viewpoint, it is most important that the calibration coefficient β cal [Eq. (23)] does not depend on the values of the phase jumps α ± used for error signals, or other parameters (such as: amplitude, shape, duration, phase structure ϕ(t), shift ∆ sh (t), etc.) of the two Ramsey pulses τ 1 and τ 2 . Thus, β cal can be considered as a phenomenological parameter, which is fixed for given setup (via the relaxation constant Γ) and for given T 1,2 (via the difference T 1 -T 2 ). In the ideal case with no relaxation (Γ = 0), we obtain β cal = 1 for arbitrary T 1,2 . However, in the general case, the value of β cal should be empirically determined before long-term frequency stabilization.

As we see from Eq. ( 24), to maximize the slope of S

CES (δ) it is necessary to use the condition T 2 ≪ T 1 . Formally we can even use T 2 = 0 (with the phase jumps α ± in the virtual point between pulses τ 1 and τ 2 ). However, due to technical transient regimes (i.e., in acousto-optic modulators) under switching-off/on of Ramsey pulses in real experiments, we believe that it is necessary to keep some nonzero dark time, T 2 = 0, which significantly exceeds any various transient times. For example, in the case of magnetically-induced spectroscopy [8,9], the transient processes, associated with switching-off/on of magnetic field, can be relatively slow.

In Fig. 1, we use some abstract shapes of the Ramsey pulses to stress that our exact analytical result (25) is valid for arbitrary Ramsey pulses, and, therefore, the CES method is very robust to different technical non-idealities, which can exist in real experiments. Indeed, in real clock experiments, the Ramsey pulses have rectangular shapes (e.g., see Fig. 2), which, however, can be deformed due to various technical reasons (e.g., the transient regime in an AOM during switching off/on of the Ramsey pulses, some phase chirping, and so on). In the case of usual Ramsey spectroscopy and hyper-Ramsey methods [START_REF] Yudin | [END_REF]28,5,29,31,34,35], these technical causes can lead to an additional (technical) shift of the stabilized clock frequency ω, while the CES method is insensitive to these non-idealities.

Note that the CES approach has some formal similarity to the two-loop methods in Refs. [33,37,38], because of the use of two different dark times T 1 and T 2 . However, the CES technique requires only one feed-back loop for frequency stabilization.

CES for different Ramsey sequences

We assume that the main reason for the shift of the stabilized frequency ω arises from the probe-induced shift ∆ sh during Ramsey pulses. All calculations are done for the ideal case of the phase jumps: α + = -α -= π/2, to maximize the error signal. Also for simplicity, we take into account (for presented calculations) only one relaxation constant Γ (rate of decoherence), while all other relaxation constants are negligible: γ e = γ e→g = γ g = γ g→e = 0, as is typical for high-precision modern atomic clocks based on strongly forbidden optical transition 1 S 0 → 3 P 0 in neutral atoms (such as Mg, Ca, Sr, Yb, Hg) and ions (e.g., Al + , In + ), or for the octupole transition in the ion Yb + .

In this section, we compare CES spectroscopy for two different pulse sequences: the usual Ramsey sequence with two equal rectangular π/2-pulses (see Fig. 2a), and the hyper-Ramsey sequence proposed in Ref. [START_REF] Yudin | [END_REF] (see Fig. 2b). If we use the exact calibration coefficient (23), then both sequences have the identical ideal result, δclock = 0. However, in real experiments, we can know the value of β cal with only limited accuracy. In this case, any deviation from the ideal value (23) will lead to some residual shift of the stabilized frequency, δclock = 0, which depends on the type of Ramsey sequence. Thus, there is a problem for the optimal Ramsey sequence with minimal sensitivity to the deviations of β cal in Eq. ( 21) from the ideal value (23).

Therefore, in our calculations we will use the following expression for the calibration coefficient,

β cal = χe -Γ(T 1 -T 2 ) , (26)
Figure 3. Comparison of a nonideal CES method for two different pulse sequences: the dashed lines are for a standard Ramsey sequence with two equal pulses (see Fig. 2a), the solid lines are for a hyper-Ramsey sequence using the composite pulse (see Fig. 2b). These graphs show the shift of the clock frequency δclock (∆ sh ) versus the probe-induced shift ∆ sh during the Ramsey pulses. In the calculations, we assumed in Eq. ( 27) a fivepercent deviation of β cal from the ideal value ( 23 where the parameter χ determines the deviation of β cal from the ideal value (23). In this case, instead of Eq. ( 24) we obtain another formula for the CES, S

CES (δ) = e -ΓT 1 ( ρ obs , Ŵτ 2 ΥδT 1 Ŵτ 1 ρ in ) -χ( ρ obs , Ŵτ 2 ΥδT 2 Ŵτ 1 ρ in ) , (err)
where the solution of the equation S In Fig. 3 we present a comparison of the CES method for two different pulse sequences: a standard Ramsey sequence with two equal pulses (see Fig. 2a) and the original hyper-Ramsey sequence [START_REF] Yudin | [END_REF] using a composite pulse (see Fig. 2b). These graphs show the shift of the clock frequency δclock (∆ sh ) versus the probe-induced shift ∆ sh during the Ramsey pulses. In the calculations, we have assumed a five-percent deviation of β cal from the ideal value (23), i.e., 0.95 χ 1.05 in Eq. ( 27). As we see, the hyper-Ramsey sequence is more robust and persistent, because the use of this scheme leads to a significant reduction of the residual shift δclock in comparison with the usual Ramsey scheme.

In addition, Fig. 4 shows that the combination of the CES technique with a hyper-Ramsey sequence significantly exceeds the possibilities of standard hyper-Ramsey spectroscopy [START_REF] Yudin | [END_REF], even for imperfect determination of the calibration coefficient β cal . For example, typical experimental conditions used in current lattice optical clocks are T 1 ∼ 1 s, τ ∼ 20 ms and π/2 excitation pulse (Ω 0 τ = π/2). Fig. 4(a) shows that if imperfect CES is used and β cal is offset from its optimal value by 5%, a light shift of ∆ sh ∼ 2π × 2.5 Hz during the interrogation period (∆ sh /Ω 0 = 0.2) results in a total clock frequency shift of only δclock ∼ 2π × 200 µHz that corresponds to a fractional clock uncertainty for optical range at the level of | δclock /ω 0 | ∼ 10 -18 . This is about 20 times smaller than what would be obtained using simple hyper-Ramsey spectroscopy (at the same interrogation light shift level) that corresponds to the level of | δclock /ω 0 | ∼ 10 -17 . Of course, for the perfect CES spectroscopy (23), this clock shift will be reduced to zero, δclock = 0.

Generalized CES and the procedure for frequency stabilization

The calibration coefficient β cal can be estimated as a ratio of the amplitudes of the central Ramsey fringes related to the interrogation procedures with T 1 and T 2 dark times. However, in this section we describe a more precise method to determine β cal . For this purpose, we will consider a generalized combined error signal (GCES)

S (err) GCES (δ) = S (err) T 1 (δ) -β(δ)S (err) T 2 (δ) , ( 28
)
where the generalized calibration coefficient β(δ) is a function of δ, which satisfies the following condition,

β(0) = β cal = e -Γ(T 1 -T 2 ) . (29)
In this case, the stabilized frequency [with the use of GCES (28)] will also always be unshifted, δclock = 0. There are many different variants of the function β(δ). For example, β(δ) can be constructed as following functions

β(δ) = A T 1 (δ, α + ) -A T 1 (δ, α = 0) A T 2 (δ, α + ) -A T 2 (δ, α = 0) ; β(δ) = A T 1 (δ, α -) -A T 1 (δ, α = 0) A T 2 (δ, α -) -A T 2 (δ, α = 0) , (30)
where we use an additional measurement in the absence of a phase jump (α = 0) before the second Ramsey pulse, A T (δ, α = 0) = ( ρ obs , Ŵτ 2 ĜT Ŵτ 1 ρ in ). However, another definition,

β(δ) = A T 1 (δ, α + ) + A T 1 (δ, α -) -2A T 1 (δ, α = 0) A T 2 (δ, α + ) + A T 2 (δ, α -) -2A T 2 (δ, α = 0) , (31)
Figure 5. Comparison of CES and GCES spectroscopies for two different pulse sequences: left panels are for standard Ramsey sequence with two equal pulses (see Fig. 2a), right panels are for hyper-Ramsey sequence (see Fig. 2b). Graphics are presented in the presence of a shift ∆ sh during the Ramsey pulses τ 1 and τ 2 : ∆ sh = 0 (blue solid lines), ∆ sh /Ω 0 = 0.3 (red dashed lines), ∆ sh /Ω 0 = 0.5 (green dashed lines). All calculations are done with the following values:

T 1 /τ = 50, Ω 0 τ = π/2, Γ = 0.25/T 1 , and T 1 /T 2 = 20. (a) signals S (err)
CES (δ) calculated by the use of Eq. ( 21) for ideal value of β cal [see Eq. ( 23 is preferable because of "symmetry" in relation to the phase jumps α ± .

In Fig. 5, we compare signals of CES (21) and GCES (28) for two different pulse sequences (see Fig. 2) in the presence of the field-induced shift ∆ sh (during Ramsey pulses). As we see from Fig. 5a, as ∆ sh increases the lineshape S (err) CES (δ) becomes significantly non-antisymmetrical, while the lineshape S (err) GCES (δ) (see Fig. 5b) maintains its antisymmetry (especially for the hyper-Ramsey scheme, see the right panel in Fig. 5b). Fig. 5c shows the dependencies of β(δ) calculated by the use of Eq. (31).

The procedure of frequency stabilization can be organized in conformity with several scenarios. First, we can continually apply GCES (28) together with Eq. ( 31) using six measurements for each frequency point (three different phase jumps, α = ±π/2, 0, and two different dark times, T 1,2 However, the use of six measurements can reduce the efficiency of the frequency stabilization, because it increases the length of the interrogation procedure. From our viewpoint, more optimal scenario is the following. In the initial period of frequency stabilization, we use GCES with Eq. (31). It allows us to determine the calibration coefficient β cal [see Eq. ( 29)] with satisfactory accuracy, because during measurements we will have the information about the value β(δ) under δ ≈ 0. Then the procedure of long-term frequency stabilization can be done with the CES technique (21), using only four measurements for each frequency point (two phase jumps, α = ±π/2, and two dark times, T 1,2 ). Moreover, we can regularly (but rarely) use GCES again. Indeed, on the one hand, it allows us to do a regular adjustment of the coefficient β cal [to eliminate, for example, an influence of possible slow variations of the parameter Γ in Eq. ( 23)]. On the other hand, such intermittent application of GCES will not lead to the significant slowing-down of the process of long-term frequency stabilization.

In addition, as we see from Figs. 345, the CES or GCES technique works better if the ratio |∆ sh /Ω 0 | becomes smaller. Distortions in the error signals arising from this problem can be largely reduced by the use of an additional and well-controllable frequency step ∆ step only during the Ramsey pulses τ 1 and τ 2 [41,[START_REF] Yudin | [END_REF]. In this case, all dependencies presented in Figs. 5-4 will be the same if we will replace ∆ sh → ∆ eff = (∆ sh -∆ step ). Thus, we can always apply a frequency step ∆ step ≈ ∆ sh (e.g., with an acousto-optic modulator) excitation to achieve the condition |∆ eff /Ω 0 | ≪ 1 for an effective shift ∆ eff , as it was in [28,5,29,37], where the required value of ∆ step was determined empirically.

CES technique for CPT Ramsey spectroscopy

In this section, we describe the CES technique for Ramsey spectroscopy of the resonances based on coherent population trapping (CPT). As a model, we consider rf CPT resonances that are formed in a three-level Λ system under interaction with a resonant bichromatic field,

E(t) = E 1 e -iω 1 t + E 2 e -iω 2 t + c.c. . (32)
The CPT resonance is formed when the difference between optical frequencies (ω 1 -ω 2 ) is varied near the low-frequency rf transition between lower energy levels |1 and |2 : ω 2 -ω 1 ≈ ∆ hfs [see Fig. 6(a)]. For example, these energy levels can correspond to the hyper-fine structure of alkali atoms (Rb, Cs, etc.) In this case, the stabilized rf frequency difference (ω 2 -ω 1 ) is the operating frequency for CPT based clocks, where the hyper-fine splitting ∆ hfs plays role of the unperturbed frequency ω 0 from previous sections.

The dynamics of the Λ system in the rotating wave approximation are described by the differential equation system for the density matrix components,

[∂ t + γ opt -iδ 1ph ]ρ 31 = iΩ 1 (ρ 11 -ρ 33 ) + iΩ 2 ρ 21 , [∂ t + γ opt -iδ 1ph ]ρ 32 = iΩ 2 (ρ 22 -ρ 33 ) + iΩ 1 ρ 12 , [∂ t + Γ 12 -iδ R ]ρ 12 = i(Ω * 1 ρ 32 -ρ 13 Ω 2 ), [∂ t + Γ 12 ]ρ 11 = γ 1 ρ 33 + Γ 12 Tr{ρ}/2 + i(Ω * 1 ρ 31 -ρ 13 Ω 1 ), (33)
[∂ t + Γ 12 ]ρ 22 = γ 2 ρ 33 + Γ 12 Tr{ρ}/2 + i(Ω * 2 ρ 32 -ρ 23 Ω 2 ), [∂ t + Γ 12 + γ]ρ 33 = i(Ω 1 ρ 13 -ρ 31 Ω * 1 ) + i(Ω 2 ρ 23 -ρ 32 Ω * 2 ), ρ jk = ρ * kj (j, k = 1, 2, 3); Tr{ρ} = 11 + ρ 22 + ρ 33 = 1.
Here δ 1ph is the one-photon detuning of frequency components ω 1 and ω from optical transitions (see Fig. 6); δ R = ω 2 -ω 1 -∆ hfs -∆ sh (t) is the two-photon (Raman) detuning; Ω 1 (t)=d 31 E 1 (t)/ and Ω 2 (t)=d 32 E 2 (t)/ are the Rabi frequencies for the transitions |1 ↔|3 and |2 ↔|3 (d 31 and d 32 are reduced matrix elements of dipole moment for these transitions); γ is the spontaneous decay rate of upper level |3 ; γ opt is rate of decoherence (spontaneous, collisional, etc.) of the optical transitions |1 ↔|3 and |2 ↔|3 the case of pure spontaneous relaxation γ opt = γ/2); γ 1 and γ 2 are corresponding spontaneous decay rates for different channels (γ 1 + γ 2 = γ in the case of closed Λ system); Γ 12 is the relatively slow (Γ 12 ≪ γ, γ opt ) rate of relaxation to the equilibrium isotropic ground state: ρ0 = (|1 1| + |2 2|)/2. Note that ∆ sh (t) is an additional actual shift (AC Stark shift) between levels |1 and |2 during the pulses, which results from off-resonant interactions of components of the laser field with different hyperfine states (e.g., see Ref. [START_REF] Pollock | AC Stark Shifts of Dark Resonances Probed with Ramsey Spectroscopy[END_REF]).

In the case of Ramsey excitation, the scheme of the time dependencies Ω 1 (t) and Ω 2 (t) is shown in Fig. 6(b), where the first pulse (with duration τ 1 ) prepares an atomic coherence between lower levels |1 and |2 , T is the free evolution interval, and the second pulse (with duration τ 2 ) is the detecting pulse, which forms a spectroscopic Ramsey signal. The time dependence ∆ sh (t) is also shown. If τ 1 is much longer than the time for the atoms to enter the dark state, then at the end of first pulse (before the free evolution interval) we have a steady-state condition. In this case, the transient frequency shift, described in [15], becomes equal to zero. As a result, the residual shift of the central Ramsey fringe δclock = ω 2 -ω 1 -∆ hfs results from the off-resonant shift ∆ sh , which is present only during Ramsey pulses (τ 1 and τ 2 ) [see Fig. 6(b)]. ∆ sh is the well known AC Stark shift, which is proportional to the total light field intensity.

In pulsed CPT spectroscopy, the error signal is usually determined from the absorption of the CPT light by the atoms during the second CPT pulse, which is proportional to the integral value

A (CPT) T (δ, α) = t d +τ 2 t d ρ 33 (t ′ )dt ′ , (34)
where the time integration interval is over the second (detecting) pulse τ 2 , which starts at the time t d [see Fig. 6(b)]. The value α corresponds to the phase jump during the corresponds to Eq. ( 36). Also we show non-ideal CES with a five-percent deviation of β (CPT) cal from the ideal value (36): χ = 0.95 (red colored dashed lines) and χ = 1.05 (green colored dashed lines). For comparison, we show δclock (∆ sh ) for usual Ramsey spectroscopy (T = T 1 , see blue dashed line). All calculations are done with the following values: dark time T [see Fig. 6(b)]. In the case of CPT spectroscopy, this phase jump describes a phase difference of the product (E 1 E * 2 ) τ 1 during the first Ramsey pulse τ 1 and the product (E 1 E * 2 ) τ 2 during the second pulse τ 2 (e.g., see in Ref. [START_REF] Guerandel | [END_REF]):

γ 1 = γ 2 = γ opt , Ω (0) 1 = Ω (0) 2 = 0.1γ opt , τ 1 = ∞, τ 2 = 200γ -1 opt , T 1 = 10 4 γ -1 opt , T 1 /T 2 =
(E 1 E * 2 ) τ 2 = e -iα (E 1 E * 2 ) τ 1 . (35)
Using the replacement A T (δ, α) ⇒ A (CPT) T (δ, α) in the formulas (10) and ( 21)-( 31) from the previous sections, we describe a realization of the CES/GCES techniques for CPT Ramsey spectroscopy. However, in this case, it is necessary to use Γ 12 [instead of Γ in Eq. ( 23)] to determine the calibration coefficient

β (CPT) cal = e -Γ 12 (T 1 -T 2 ) . (36)
Also we assume that T 1,2 ≫ (γ 1 + γ 2 ) -1 , i.e., the dark time between Ramsey pulses should significantly exceed the lifetime of the exited state |3 . Calculations presented in Fig. 7 clearly demonstrate that the CES/GCES technique can be very effective for an rf clock based on CPT Ramsey spectroscopy. Indeed, for an ideal calibration coefficient (36) we see a total absence of the clock shift, δclock = 0 (see horizontal line in Fig. 7). But even for non-ideal calibration coefficient β (CPT) cal , the residual shift for the CES method (see green and red colored dashed lines in Fig. 7) is much less than for usual Ramsey spectroscopy (see blue colored dashed line in Fig. 7).

Conclusion

We have developed a universal one-loop method to form the reference signal for stabilization of arbitrary atomic clocks based on Ramsey spectroscopy. This method uses the interrogation of an atomic system for two different Ramsey periods and a specially constructed combined error signal (CES) [see Eq. ( 21)]. The CES technique requires four measurements for each frequency point as well as a preliminary measurement (or estimation) of the calibration coefficient β cal . It was shown that the highest robustness is achieved with the combination of the CES protocol and a hyper-Ramsey pulse sequence (see in Ref. [START_REF] Yudin | [END_REF]). Also a method of generalized combined error signal (GCES) was developed [see Eq. ( 28)], which requires six measurements for each frequency point and has an exceptional robustness. The CES/GCES spectroscopy allows for perfect elimination of probe-induced light shifts and does not suffer from the effects of relaxation, time-dependent pulse fluctuations and phase-jump modulation errors and other nonidealities of the interrogation procedure. A variant of the frequency stabilization using CES with intermittent GCES protocols has been proposed. In addition, the applicability of CES/GCES techniques for CPT atomic clocks has been described. The implementation of this approach can lead to significant improvement of the accuracy and long-term stability for a variety of types of atomic clocks.

Also, it will be interesting to experimentally compare the one-loop CES/GCES method with the two-loop auto-balanced Ramsey spectroscopy (ABRS) [37,38,39]. We believe that both methods have comparable efficiency for frequency stabilization, but CES/GCES is technically simpler because only one feedback loop is required. Moreover, the case of optical transitions, the CES/GCES protocol with the use of hyper-Ramsey pulse sequence (see in Ref. [START_REF] Yudin | [END_REF]) can be even more efficient in comparison with ABRS.

Figure

  Figure1. Left part: schematic illustration of a sequence of two arbitrary Ramsey pulses (with durations τ 1 and τ 2 ) separated by the dark time T , during which the phase jumps α ± are applied to create an error signal(10). Aside from phase jumps α ± , the phase ϕ(t) [see in Eq. (1)] should not be changed during the dark time T (i.e., ϕ(t) = const), while during the Ramsey pulses τ 1 and τ 2 the function ϕ(t) can have an arbitrary behavior. Right part: scheme of the clock transition |g ↔ |e (with unperturbed frequency ω 0 ) interacting with the probe field at the frequency ω, where ∆ sh (t) is an actual probe-field-induced shift during Ramsey pulses τ 1 and τ 2 .

Figure 2 .

 2 Figure 2. Two different Ramsey sequences: (a) standard Ramsey sequence [25] with two equal pulses; (b) original hyper-Ramsey sequence [27] using the composite pulse with a π phase jump.

  Figure3. Comparison of a nonideal CES method for two different pulse sequences: the dashed lines are for a standard Ramsey sequence with two equal pulses (see Fig.2a), the solid lines are for a hyper-Ramsey sequence using the composite pulse (see Fig.2b). These graphs show the shift of the clock frequency δclock (∆ sh ) versus the probe-induced shift ∆ sh during the Ramsey pulses. In the calculations, we assumed in Eq. (27) a fivepercent deviation of β cal from the ideal value (23): χ = 0.95 (red colored lines) and χ = 1.05 (green colored lines), for two different values of τ : T 1 /τ = 50 (left figure) and T 1 /τ = 10 (right figure). All calculations are done with the following values: Ω 0 τ = π/2, Γ = 0.5/T 1 , and T 1 /T 2 = 20.

  ) = 0 (in relation to the unknown δ) determines the residual shift δclock for the stabilized frequency ω.

Figure 4 .

 4 Figure 4. Comparison between the usual hyper-Ramsey method without CES[START_REF] Yudin | [END_REF] (black dashed lines, T = T 1 ) and non-ideal CES using a hyper-Ramsey sequence (see Fig.2b) for two different values of τ : T 1 /τ = 50 (left figure) and T 1 /τ = 10 (right figure). These graphs show the shift of the clock frequency δclock (∆ sh ) versus the probe-induced shift ∆ sh during the Ramsey pulses. In the calculations, we assumed in Eq. (27) a five-percent deviation of β cal from the ideal value (23): χ = 0.95 (red solid lines) and χ = 1.05 (green solid lines). All calculations are done with the following values: Ω 0 τ = π/2, Γ = 0.5/T 1 , and T 1 /T 2 = 20.

  )]; (b) signals S (err) GCES (δ) calculated by the use of Eqs. (28) and (31) for β(δ); (c) dependencies β(δ) calculated by the use of Eq. (31).

Figure 6 .

 6 Figure 6. (a) Atomic three-level Λ system. (b) Schematic time dependencies of the Rabi frequencies Ω 1,2 (t) and probe-induced shift ∆ sh (t), where t d is the beginning of the second (detecting) Ramsey pulse and α denotes the value of the phase jump [see Eq. (35)] during the dark time T .

Figure 7 .

 7 Figure 7. These graphs show the shift of the clock frequency δclock (∆ sh ) versus the probe-induced shift ∆ sh during the CPT Ramsey pulses. The case of ideal CES (horizontal black solid line) when the calibration coefficient β (CPT) cal

  20, and Γ 12 = 0.5/T 1 .

  1. Left part: schematic illustration of a sequence of two arbitrary Ramsey pulses (with durations τ 1 and τ 2 ) separated by the dark time T , during which the phase jumps α ± are applied to create an error signal(10). Aside from phase jumps α ± , the phase ϕ(t) [see in Eq. (1)] should not be changed during the dark time T (i.e., ϕ(t) = const), while during the Ramsey pulses τ 1 and τ 2 the function ϕ(t) can have an arbitrary behavior. Right part: scheme of the clock transition |g ↔ |e (with unperturbed frequency ω 0 ) interacting with the probe field at the frequency ω, where ∆ sh (t) is an actual probe-field-induced shift during Ramsey pulses τ 1 and τ 2 .

  {γ e , γ e→g , γ g , γ g→e , Γ}: γ e is a decay rate (e.g., spontaneous) of the exited state |e ; γ e→g is a transition rate (e.g., spontaneous) to the ground state |g ; γ g is a decay rate of the ground state |g (e.g., due to black-body radiation and/or collisions); γ g→e is a transition rate from the ground state |g to the exited state |e . Note that γ e→g

	1) of the clock transition during the Ramsey pulses [e.g., it can be the ac-Stark
	shift, which is proportional to the |E(t)| 2 and does not depend on the phase ϕ(t)]. Also
	Eq. (3) contains five relaxation constants,

Acknowledgments

We thank E. Ivanov, J. Elgin, C. Oates, M. Lombardi, Ch. Sanner, Ch. Tamm, E. Peik, and N. Huntemann for useful discussions and comments. This work was supported by the Russian Science Foundation (No. 16-12-10147). M. Yu. Basalaev was also supported by the Ministry of Education and Science of the Russian Federation (No. 3.1326.2017/4.6), and Russian Foundation for Basic Research (No. 17-02-00570 and 16-32-60050 mol a dk). T. E. Mehlstäubler was supported by the German Research Foundation (DFG) through grant ME 3648/1-3. R. Boudot was supported by NIST Guest Researcher Program and Délégation Générale de l'Armement (DGA). Contributions to this article by workers at NIST, an agency of the U.S. Government, not subject to U.S. copyright.


Keywords: Représentations sociales, noyau central, médicament, groupes sociaux, psychologie de la santé Social representations, central core, medicine, social groups, health psychology

. Dans une étude exploratoire (N = 54) nous avons collecté, à l'aide de la technique d'associations verbales, les principaux éléments associés au médicament. Dans une seconde étude (N = 166), le test d'indépendance au contexte révèle des différences de représentations sociales en fonction de l'âge, le genre, et le niveau d'éducation. Il apparait que chaque groupe social à une relation spécifique au médicament, ce qui doit être pris en compte dans les recommandations de santé.

Introduction

The average Frenchman purchases 48 boxes of medicines a year, which makes him the biggest consumer in the whole Europe. According to a report from the French National Health Service Online (Ameli, 2011), pharmaceutical industries earned 21 billion euros in 2009. The increasing of longevity and the decreasing of death rate could partly explain that phenomenon. However, in accordance to many authors (Mishara and Légaut, 2000;[START_REF] Pérodeau | Consommation de psychotropes par les personnes âgées : réflexions et recommandations de médecins et de pharmaciens[END_REF], we believe that this high consumption of medicines cannot be explained by biological factors only. Throughout the theory of social representations [START_REF] Moscovici | La psychanalyse, son image, son public[END_REF] we have tried to find out how people consider medication, not only relying on the actual knowledge of the medicine, but also on beliefs, hopes, and opinions, causing the theoretical framework to be based on the social representations [START_REF] Moscovici | La psychanalyse, son image, son public[END_REF].

Background

The theory of social representations can be defined as a set of information, opinions, beliefs and values shared among a social group to describe a social object. Throughout their representations, social groups organize their social environment and simplify it. Afterwards, objects become understandable, so people can interact with it, and talk about it. A structural approach of this theory has been developed by French authors (see, among others, the works of [START_REF] Abric | La recherche du noyau central et la zone muette des représentations sociales[END_REF][START_REF] Moliner | Les méthodes de repérage et d'identification du noyau des représentations[END_REF][START_REF] Rateau | Social Representations Theory[END_REF] who suggest a double system: the central core, which contains the strongest items of the representation, shared by the vast majority of the social group, and the peripheral system, more individualistic, and depending of the context. Amongst the different ways to comprehend the element of the central core, the most recent is the test of context independence (TCI) by Lo [START_REF] Lo Monaco | Le test d'indépendance au contexte (TIC) : une nouvelle technique d'étude de la structure représentationnelle[END_REF]. According to the authors: "the TCI aims to specify the trans-situational or contingent character of representational elements" (2008, p.119). Moreover, it provides the advantage of a reduced cognitive load perceived, compared to the calling into question technique [START_REF] Moliner | Validation expérimentale de l'hypothèse du noyau central des représentations sociales[END_REF].

The 19 th century marked the beginning of the modern medicine. The pharmaceutical industry started to develop synthetic molecules which reduced the high level of mortality. But in those days, research is stagnating. Hence, a significant focus has been set on the different forms a same medicine may take. In France for instance, the leader trademark of paracetamol, Doliprane, might be consumed as capsule, syrup, powder sachet or even suppository; sharing however the same molecule. [START_REF] Niquette | Représentations sociales et relations au médicament : le cas de l'hormonothérapie de substitution[END_REF] references the main relationships between the medicine and the users, among them; utilitarian: the medicine is perceived by users as the materialized part of the pain's answer. So, in this context, this object has to be efficient and relieving; affective: Gittelman and Squires (2011) showed the difficulty to discriminate a medicine from a candy, within two groups, both children and adults. The medicine has a strong symbolism power which refers to the sweetness of the childhood, cognitive, the medications require techniques, but they have been constructed, learned and acquired.

The sale of medication increased, and so the longevity. As we get older, as we take more and more medicines. 50% of the refund of the healthcare refunds is devoted to the 55 of age and older. 1.2 billion of elderly people consumes at least 7 medicines and more a day [START_REF] Hordé | Patients, tout ce qu'on vous cache : Médecine[END_REF]. Last but not least, the consumption of psychotropic substances also increases with age [START_REF] Collin | La problématique de la consommation de psychotropes chez les personnes âgées en France et au Québec[END_REF]. Aïach and al (2001) evoke a health inequality in regard of gender. Women consult the doctor more often than men but their working conditions are less painful. Also, they live much longer but their quality of life is relative, composed by an important part of widowhood and other losses. Women are also considered as the pharmacist in their house. [START_REF] Boltanski | Les usages sociaux du corps[END_REF], made the connection between health behaviors and social class.

Indeed, the perception of a treatment is different, according to the patient's social class.

Higher social tend to transform a bad corporal sensation in a symptom. According to the author, it would be the result of the reduction of the patient / doctor hierarchy.

Elderly, gender, social class. Those three variables may show differences. In order to verify this hypothesis, analyses must be done.

Method Study 1

Participants, material and procedure

The present study mobilized 54 French males and females recruited online (M= 23.6,SD= 3.5). None of them was remunerated. Participants completed individually a verbal association task (VAT) online based on the word "medicine", containing blanks to fill in.

Consumption of medicine in France: a social groups approach Revue Internationale sur le Médicament, hors-série (1), 2015 29

They could write from one to sixteen words, sentences, adjectives, ideas associated to this word.

Results

The population produced 573 words (M=10.61, SD= 4.08) that we analyzed and split into nine categories: medical frame, which contains items related to medical staff, medical structures as pharmacy or hospital, disease, related to symptoms, bad use consequences are the secondary effects of medicine, esthetic / taste, are shape and flavor's medicine, appearance, the container, care, the relieved after the medication is taken, molecule / brand, the medications when they are named according to the active principle or the brand, economy are the items related to the price of medicine and various. The main items of those categories became the material of the questionnaire of the second study.

Graphic 1: repartition of the categories

The analysis of the distribution reveals the importance of the medical framework when the word medicine is thought (27%), followed by care and disease, which are also two important associations (respectively 18% and 17%). Then, the molecules or brands of the medicines are quoted (12%). The appearance of the container are named in the fifth instance (9%), and eventually, in approximately the same proportions, the esthetic or taste of the medicine itself (6%), the bad use consequences (5%), and various (5%). The minor category is related to the dimension of economy (1%).

Study 2

Participants, material and procedure 166 participants were divided into age bracket (from 18-24, 25-34, 35-44, 45-54, to 55 and more), gender (male, female), and level of education (no diploma, BEP or CAP (technical school certificates), High School Certificate, Bachelor, or Master's degree). Halimi-Falcowicz (2008), each item was presented as an affirmation. For the word "pharmacy" for example, the sentence was "a medicine is, always, in any case, sold in pharmacy". Participants responded to the items on a 4-point Likert scale, ranging from strongly disagree (1) to strongly agree (4). A large part of the social group analyzed has to be strongly agreed on an item to consider it as central. According to the TCI, the threshold of consensually in each group is defined by the Kolmogorov-Smirnov test [START_REF] Kanji | 100 statistical tests[END_REF].

Results

In order to identify the homogeneous groups of variables, a hierarchical cluster analyze has been done. Based on the participants' answers, this statistical test aims to understand the links between the items. Three main dimensions have been isolated:

"knowledge of the processes of fabrication and selling of medicine" containing the items from the medical and pharmaceutical research, contains an active substance, has a defined function, associated to a symptom, sold in pharmacy and sold into a box, "legitimacy of medicine use" (prescribed by a doctor, associated to a disease, covered by social welfare and distributed in hospital). The last dimension is the "impact of medicine" (presents a danger, consumed in an abusive way, provides care for the patient, relieves the patient, sold under a brand, expensive ,has an unpleasant taste and sold in a big quantity).

Social representations and age

Quantitatively, the number of elements in the central core increase with age: one in the 18-24 age group, four In the 25-34 age group, thirteen for the 35-44 age group, sixteen for the 45-54 age group and the older group, the 55 and over, has also sixteen central items. The relieving associated to the medicine seems to be an important part of any representation. But, as the age bracket is older, as the medicine contains various dimensions. For the 25-34, a medicine has to be the product of the research but has to be associated to a symptom to relieve, and contained in a box, which is not the case with the youngest. For the 35-44 age bracket, we can notice the superiority of items linked to the misuse of medicine but at the same time the care provided by it. The cost is considered as higher, considered as the previous age brackets. The 45-54 age bracket includes almost all the items in its central core but the fact that a medicine is necessary covered by the social welfare, the abuse, the taste and the drug administration in hospital. Eventually, the oldest has almost the same representation, except they don't consider that the medicine has to be associated to a disease, and it does not represent a constraint. But on the other hand, they consider that the medicine is always covered by social welfare and has an unpleasant taste.

The hierarchical cluster analyses shows a statistically significant difference for the "legitimacy of medicine use" dimension F(4,161) = 3.54, p< .01 among the different age brackets but the other dimensions did not show significant differences among the groups.

Social representations and gender

None of the elements submitted to the men group reached the threshold of consensually, but on the other hand, women have two items in their central core: the research and the relief associated to the medication.

The hierarchical cluster analyses showed a significant difference regarding the dimension "knowledge of the processes of fabrication and selling of medicine" (U=2057,5, p<.03). It appears indeed that there is a difference in the perceived importance of the research (U=1739, p<.001), the active substance (U=1961, p<.01), and the selling place (the pharmacy) (U=1980, p<.01). But there are not significant differences in the items defined function (U=2242 p=.168), symptom (U=2420, p=.509) nor sold in the box (U=2115, p=.061).

Figure 1. Central core per gender (on the left: women, on the right: men).

Social representation and level of education

Participants had to choose a level of education in a list of six levels: no diploma, BEP or CAP (technical school certificates), High School Certificate, 2 nd year of Bachelor, Bachelor, or Master. Then, their answers were analyzed regarding the 20 items.

social welfare, associated to a disease, be a constraint, consumed in an abusive way or distributed in hospital to be considered as a medicine. The BEP / CAP group has a similar central core, except that it does not considers the medicine as a danger, a care and it does not has to have an unpleasant taste. But it considers that a medicine is always consumed in an abusive way. The only few elements that the high school certificate group does not consider as representative of medicine are the prescription by the doctor, the covering by social welfare, the unpleasant taste and the distribution in hospital. But we observe an opposite trend from the 2 nd year of bachelor to the master groups. Indeed; the 2 nd year of bachelor perceives the medicine in its functional way: it has to be from the research, associated to a symptom, to relieve, to have a defined function and only an element refers to its appearance; sold in a box.

The bachelor has the same construction but the necessity of the sell into a box. The Master group has only one item: the medicine has to relieve.

We hypothesized that the highest levels of education should have a better knowledge of medicine than the lowest ones. They should also develop much more elements in their central core. But, there is not a statistically significant difference between the different levels F(5,160) = 2.14 p> .05 and(5,160) = 1,026, p> .05. The other dimensions did not show significant differences.

Discussion

Since the exploratory study, we saw that the social object medicine is polymorph.

Indeed, the participants were very expressive about this term. The difficulty was more about finding a consensus. At every step of the study, it has been felt that the population had an individual link to the medicine products. We regret that we could not develop the discourse of any of those people.

Concerning the elderly, the oldest population has more interest on this social object with a record of sixteen elements in the central core. Older people have very specific central items that the other groups did not share, for example the necessity for the medicine to be covered by social welfare or to have an unpleasant taste. The rising number of central items in regard of age brackets indicates that there is an age effect on the social representation of medicine, probably associated to the consumption.

The legitimacy of medicine use has also its importance as the age bracket is older. We can hypothesize here that this population members, very influenced by doctor's prescription, is not interested in making their own research, which became accessible via Internet and is done by a large part of the whole population. According to the French Medical Association's website, seven out of ten people in France search for medical advices online. The only consensual item for every age bracket is the relief.

About the gender, it can be noted that only the women have central items in the elements presented in the TCI. Again, an explanation could come from the relation between the doctor and the patient. In 1991, women consulted the doctor 9.3 times a year, versus 6.7 times a year for the men (Aïach, 2001). It's 37% more. That means that women may have a close link to medicine due to this consultation frequency. These results tend to accentuate the differences already shown in the literature about the gender discrimination in regard of health.

The level of education shows a reversed tendency compared to age brackets: the lower the diploma is, the higher the number of central items is. Following the observation of [START_REF] Boltanski | Les usages sociaux du corps[END_REF] we can infer that the higher diploma tend to describe in a better way their symptom; and possibly get rid of it easily, whereas the lower diplomas groups need to mark the whole symbolic relations in order to feel a well-being. Here, the relief is, again, shared by all the groups, but the structure around this sensation helps us to understand that it is not the same perception according to the group: the relief has to come from a symptom and the medicine has to cure for the lower diplomas, whereas for the highest ones, it comes with the knowledge of the processes of fabrication and selling of medicine.

The results of this research tend to verify our hypothesis. We assume that age, gender, and social class have an impact on the social representation structure. Elderly people, women, lower diplomas are the most concerned by medicine. We think that this is closely related to practices.

There is not one, but many medicine's consumers, with many social representations associated. This idea must be shared by the health actors, who contribute to the fabrication, the conception of the boxes, medical instructions, and form of the medicine and also the doctors who prescribe it, the pharmacists… in order to adapt the product to the consumers' representations.

Table 1 .

 1 Number of participants per age, gender or level of education.

		18-24	25-34	35-44	45-54	55 and more
	Age bracket
		92	44	10	11	8
		Males	Females
	Gender
		43	123
	Level of	No diploma	BEP/CAP	High school certificate	2 nd year of Bachelor	Bachelor's degree	Master
	education
		8	8	10	45	39	56

The more the category was overriding among the whole entity, the more items were extracted from it. A total of twenty items were presented: from the medical and pharmaceutical research, prescribed by a doctor, sold in pharmacy, covered by social welfare, associated to a disease, associated to a symptom, presents a danger, presents a constraint for the patient, consumed in an abusive way, provides care for the patient, cures, relieves the patient, contains an active substance, sold under a brand, has a defined function, expensive, has an unpleasant taste, sold in a big quantity, sold into a box, and distributed in hospital. According to the test of context independence (TCI) of Lo Monaco, Lheureux and

Table 2 .

 2 Centrality per age bracket.

	Age bracket	18-24	25-34	35-44	45-54	55 and
	Items

Concerning the number of elements in the central core, the more the diploma is high, the less the items are considered as a necessity. The no diploma group has the highest rank with fifteen central words associated to medicine. The product does not have to be covered by