Soumya Bera

Harold U Baranger

Serge Florens

Dynamics of a Qubit in a High-Impedance Transmission Line from a Bath Perspective

We investigate the quantum dynamics of a generic model of light-matter interaction in the context of high impedance waveguides, focusing on the behavior of the photonic states generated in the waveguide. The model treated consists simply of a two-level system coupled to a bosonic bath (the ohmic spin-boson model). Quantum quenches as well as scattering of an incident coherent pulse are studied using two complementary methods. First, we develop an approximate ansatz for the electromagnetic waves based on a single multimode coherent state wavefunction; formally, this approach combines in a single framework ideas from adiabatic renormalization, the Born-Markov approximation, and input-output theory. Second, we present numerically exact results for scattering of a weak intensity pulse by using Numerical Renormalization Group (NRG) calculations. NRG provides a benchmark for any linear response property throughout the ultra-strong coupling regime. We find that in a sudden quantum quench, the coherent state approach produces physical artifacts, such as improper relaxation to the steady state. These previously unnoticed problems are related to the simplified form of the ansatz that generates spurious correlations within the bath. In the scattering problem, NRG is used to find the transmission and reflection of a single photon, as well as the inelastic scattering of that single photon. Simple analytical formulas are established and tested against the NRG data that predict quantitatively the transport coefficients for up to moderate environmental impedance. These formulas resolve pending issues regarding the presence of inelastic losses in the spin-boson model near absorption resonances, and could be used for comparison to experiments in Josephson waveguide quantum electrodynamics (QED). Finally, the scattering results using the coherent state wavefunction approach are compared favorably to the NRG results for very weak incident intensity. We end our study by presenting results at higher power where the response of the system is nonlinear.

I. INTRODUCTION

Quantum optics deals with the interaction of matter and electromagnetic waves in regimes where the granularity of light comes into play. Typically, because both the fine structure constant and atomic dipoles are small, matter-light coupling is weak, allowing treatment by simple and controlled methodologies, such as the rotating wave approximation (RWA), Born-Markov schemes, and Fock space truncation [START_REF] Meystre | Elements of quantum optics[END_REF]. However, recent artificial systems such as superconducting waveguides [START_REF] Niemczyk | [END_REF][3][4][5][6][7], especially the ones using Josephson junction elements to boost the circuit impedance [8][9][10][11][12], constitute metamaterials where charge density fluctuations mimic an optical-like medium in which the coupling to matter (namely superconducting qubits) can be ultra strong. In this situation, a plethora of interesting phenomena, such as wide-band frequency conversion [13], large nonlinearities [14], and non-trivial many-body vacua [13,15], have been theoretically predicted.

In this context, the standard theoretical approaches mentioned above are insufficient: Born-Markov schemes are clearly violated by the large coupling constant, and counter-rotating terms cannot be neglected anymore. In addition, a brute force Fock truncation of the full Hilbert space becomes numerically prohibitive due to the large number of photons generated in the environment, unless one can target physical states using an optimal variational basis, such as matrix product states (MPS) [16][17][18][19][20] or within the systematic coherent state expansion (CSE) pioneered in Ref. 21 and susequently extended to a variety of dissipative models [22][23][24][25].

In this article, we probe the idea that multi-mode coherent states are a well-adapted tool to deal with ultrastrong coupling quantum optics by investigating in depth several dynamical properties that are experimentally relevant, such as population decay, coherence buildup, and photon scattering in a large-impedance superconducting waveguide. Historically, this approximate single-coherent state approach was devised for the ground state of the spin-boson model by Luther and Emery [26] and subsequently by Silbey and Harris [27,28] and other authors [29,30], before the demonstration that it can be turned into a systematic expansion that allows one to reach the exact many-body ground state for any coupling strength [21][22][23] and an arbitrary bath spectral density [24]. Starting down this same route, we study quantum dynamics here only at the simplest Silbey-Harris level, thus using a single variational coherent state as a lowest order approximation for the time-dependent problem (this is also called the Davidov ansatz). Such an approach has been previously applied to study quantum dynamics for a variety of physical problems [31][32][33][34][35], and it can be adapted to address scattering of Fock states [36]. The present study allows us to thoroughly assess the merits and drawbacks of this very economical approach; the numerically-exact full generalization of the systematic CSE to the time domain will be addressed in subsequent work.

We will demonstrate here that this simple-minded single coherent state approach already contains most of the physics at play and naturally bridges from the weak to ultra-strong coupling regimes. Indeed, population decay is found to cross over from underdamped to overdamped relaxation at increasing environmental impedance, as expected physically [37,[START_REF] Weiss | Quantum Dissipative Systems[END_REF]. In addition, this singlecoherent state approximation predicts elastic transmission lineshapes beyond the RWA that in the linear response regime match precisely our non-perturbative calculations using the Numerical Renormalization Group (NRG). These simulations are used to derive simple and accurate analytical formulas for the transmission coefficients and total inelastic deficits, that can then be used to compare to experiments in waveguide QED.

However, we point out some shortcomings of the single coherent state dynamics that were not reported so far in the literature. Artifacts are indeed generically found whenever the two-level system is subject to a strong temporal perturbation, such as a quantum quench or a strong irradiation pulse (beyond the linear response regime). In all these cases, relaxation is found towards an incorrect steady state where long range correlations are spuriously maintained between the two-level system and modes propagating away from it. These artifacts are a consequence of the constrained form of the single coherent state ansatz, which neglects entanglement within the bath states at all times, and are likely to be cured by extending the present technique to a dynamical version of the systematic coherent state expansion [21,22].

The paper is structured as follows. In Sec. II we introduce the spin-boson model and establish the dynamical equations based on the single coherent state ansatz. Then, in Sec. III we study a simple quantum quench where the two-level system is subject to the temporal oscillation of a polarization field. We show that the population decays as physically expected, with a crossover from underdamped to overdamped behavior at increasing coupling strength. However, the dynamics of the quantum coherences is incorrect: it builds up to a value that does not match the expected steady state. We relate this to artifacts in the bath dynamics. When the switching of the perturbation is made adiabatic, however, proper relaxation is finally recovered, a behavior that we relate to a factorization property of the emitted wavepacket. A different physical setup is then considered in Secs. IV and V, whereupon the environment subjects the two-level system to a train of incoming photons. Transport coefficients are extracted from the quantum dynamics and compared favorably to NRG calculations. Simple and accurate formulas are also extracted from these simulations. Some perspectives are given as a conclusion to the paper.

II. SINGLE COHERENT STATE DYNAMICS

A. Model

We start with the standard spin-boson model [37,[START_REF] Weiss | Quantum Dissipative Systems[END_REF], where a two-level system (for instance, a superconducting qubit in the context of circuit QED) is coupled to a set of quantized harmonic oscillators describing propagating modes in a transmission waveguide. We assume in what follows a geometry where the qubit is side-coupled to the waveguide, but our results can be straightforwardly extended to the case of inline coupling. The initial Hamiltonian reads:

H = ∆ 2 σ x - σ z 2 k∈R g k (a k + a † k ) + k∈R ω k a † k a k . ( 1
)
Here ∆ is the splitting of the two-level system (typically set by the Josephson energy associated with the superconducting qubit). We will take a linear relation

ω k = |k|,
where k is the momentum, setting the plasmon velocity to unity. Finally, we parametrize the coupling constant as g k = 2α|k|θ(ω c -ω k ). These assumptions accurately describe superconducting waveguides at characteristic energies that are well below the cutoff frequency ω c . As a result, the spectral density in the continuum limit reads

J(ω) = k>0 πg 2 k δ(ω-ω k ) = 2πα ω θ(ω c -ω),
where, in case of electric coupling, the dimensionless coupling strength α is proportional to the impedance of the waveguide. As is standard practice, it is useful to fold the problem onto a half-line, by defining even and odd modes,

a e k = 1 √ 2 (a k + a -k ) and a o k = 1 √ 2 (a k -a -k ) , (2)
so that the Hamiltonian (1) can be rewritten:

H = ∆ 2 σ x - σ z 2 k>0 g k (a e k + a e † k ) + k>0 ω k [a e † k a e k + a o † k a o k ]. (3)

B. Dynamical ansatz and quantum equations of motion

In the small impedance limit, α 1, it is customary to invoke the rotating wave approximation (RWA) [START_REF] Meystre | Elements of quantum optics[END_REF], where Hamiltonian (3) is truncated such that qubit levels dressed only by adjacent Fock states are included. However, this approximation breaks down already for α 0.1, and so for our purposes the model must be addressed in its full complexity. A particular defect of the RWA is the lack of many-body renormalizationthat is, the strong reduction of the bare tunneling energy ∆ to a smaller value ∆ R . This renormalization effect is, however, well-described by an alternative approach, where the dressing of the qubit levels occurs via coherent states [21,[26][27][28]30]. At the lowest degree of approximation, a single multimode coherent state is introduced for each qubit state, so that the time-dependent state vector takes the form of the following simple ansatz:

|Ψ(t) = p(t) |↑ e k [f k (t)a e † k -f * k (t)a e k ] |0 (4)
+q(t) |↓ e k [h k (t)a e † k -h * h (t)a e k ] |0 .
Here p(t) and q(t) are the time-dependent amplitudes of the dressed qubit states (and map the entire Bloch sphere), while f k (t) and h k (t) denote complex displacements of the associated bath oscillators dressing the qubit. The bath states assume thus a simplified form, where entanglement between the various modes is simply neglected. As a particular case, note that the approximate Silbey-Harris state that is obtained for the ground state of Hamiltonian (3) satisfies the relations (obtained by energy minimization)

p = -q = 1/ √ 2, f k = -h k ≡ f SH k = (1/2)g k /(ω k + ∆ R )
, where ∆ R is the renormalized qubit tunnel amplitude, an important parameter in what follows.

Only the even modes of the waveguide appear in the ansatz (4) as the odd modes are decoupled from the qubit. In Section III, the odd modes will be taken in their vacuum state and will not be considered in the dynamics. However, the transport conditions considered in Sec. V will require inclusion of the odd modes, which is trivially done since their evolution is given by the free part of Hamiltonian (3).

The quantum dynamics will be piloted by the real Langrangian density

L = Ψ(t)| i 2 - → ∂ t -i 2 ← - ∂ t -H|Ψ(t) , with
Euler-Lagrange type of equations of motion, as resulting from the Dirac-Frenkel time-dependent variational principle [START_REF] Kramer | Geometry of the Time-Dependent Variational Principle in Quantum Mechanics[END_REF]:

d dt ∂L ∂ ḟk = ∂L ∂f k , d dt ∂L ∂ ḣk = ∂L ∂h k , d dt ∂L ∂ ṗ = ∂L ∂p , d dt ∂L ∂ q = ∂L ∂q . (5)
This results in the set of dynamical equations:

i ḟk = ω k f k - g k 2 - q∆ 2p (f k -h k ) f |h , (6)
i ḣk = ω k h k + g k 2 - p∆ 2q (h k -f k ) h|f , (7)
i ṗ = ∆ 2 q f |h -p k g k 4 (f k + f * k ) + p k ω k |f k | 2 -i p 2 [ ḟk f * k -ḟ * k f k ], (8)
i q = ∆ 2 p h|f + q k g k 4 (h k + h * k ) + q k ω k |h k | 2 -i q 2 [ ḣk h * k -ḣ * k h k ], (9)
where

f |h = e k [f * k h h -|f k | 2 /2-|h k | 2 /2] .
In practice, Eqs. (6-7) are substituted into Eqs. (8)(9), so that an independent set of first-order non-linear differential equations is obtained, which can be efficiently solved by standard Runge-Kutta techniques, with a linear scaling of the computational cost in the number of bosonic modes.

Such dynamical equations have a long history from polaron physics [33][34][35] to dissipative quantum mechanics [31,32], and have been previously derived in the literature. Our purpose in this paper is to benchmark carefully the physical results that they lead to, in order to pinpoint the advantages and drawbacks in using them in the specific context of waveguide QED. We first consider in the next section the situation of quantum quenches, before turning to the investigation of scattering properties.

III. POPULATION DECAY AND COHERENCE BUILDUP A. Sudden quantum quench

We investigate here a standard protocol, where the qubit is initialized at time t = 0 in the state |↑ , while the bath is taken in the vacuum |0 . (Note that |↑ is not the excited state of the qubit but rather a quantum superposition of the ground and excited state.) The qubit then evolves at later times according to the full spin-boson Hamiltonian (3), and progressively relaxes to its many-body ground state while energy is dissipated into the bath. This theoretical problem has been considered by a great variety of methods, such as quantum Monte Carlo [START_REF] Egger | [END_REF]41], stochastic expansions [42][43][44], timedependent NRG [45,46], systematic variational dynamics [47], and analytical weak-coupling calculations [48][49][50][51], but mostly from the perspective of the qubit dynamics. Indeed, in addition to the relaxation properties of the qubit itself, we will examine here in depth the behavior of the states in the bath. This joint qubit and bath dynamics is approximated using the simple equations of motion Eqs. ( 6)-( 9), with the initial conditions p = 1, q = 0, f k = h k = 0. In fact, for numerical stability reasons, one must give a small non-zero value, typically q = 10 -3 at initial times.

By symmetry, the two-level system shows no polarization along the z-axis in its ground state (this applies only in the delocalized phase α < 1, otherwise spontaneous polarization does occur). Due to the presence of the transverse field ∆, one expects precession of the spin and so damped oscillation of P (t) ≡ σ z (t) reaching zero in the long-time limit. In addition, it is known that the dynamics crosses over from an underdamped to an overdamped regime as dissipation reaches values around α = 0.5. Finally, one expects the oscillations to occur at a renormalized frequency ∆ R ∆(∆/ω c ) α/(1-α) with a damping rate Γ α∆ R .

One remarkable achievement of the single-coherentstate dynamics is that all these non-trivial features of the sudden quench dynamics are qualitatively obtained, as can be seen from the top panel in Fig. 1. Comparison to the existing literature shows however that the precise form of the population decay obtained from this single coherent-state approximation is not fully accurate. First, the exact renormalized qubit splitting differs from the single coherent state result by numerical factors that can be sizable, as was shown recently by extensive CSE calculations in the ground state [22,23]. Second, we see that for α = 0.5, the qubit is still slightly underdamped, while it is known [50] that the dynamics should be strictly overdamped, without any sign change in P (t).

However, a more problematic and qualitative issue arises when monitoring the qubit coherence σ x (t) , which was not reported in previous studies [31,32]. The lower panel in Fig. 1 shows that the qubit coherence builds as expected qualitatively, but that the long-time limit is in stark disagreement with the value found from the single-coherent-state Silbey-Harris approximation in the ground state [21,27,30]. Note that the single coherent state ansatz (4) captures very precisely the ground state for α < 0.2, so the discrepancy in the long time value of σ x (t) is unexpected. The origin of the problem lies in the states of the bath that carry energy away from 2). At short distances, the Silbey-Harris ground state is stabilized, while at large distances a wavepacket is propagating away from the qubit located at the origin. The lack of factorization, f (x) = g(x) for x > 220, is associated with spurious correlations between the emitted wavepacket and the two-level system, that are responsible for the improper relaxation seen in Fig. 1.

the qubit. By monitoring the bath displacements f k and h k in real space (as obtained by Fourier transform), one can see in Fig. 2 that for distances x 1/∆ R , an entanglement cloud [15] forms between the qubit and the waveguide, which maps perfectly onto the Silbey-Harris predictions for the ground state (dotted line). This confirms that the origin of the improper relaxation must lie in the propagating waves, that are seen in the plot at larger distances. This can be understood physically in the limit α 1 as follows. Let us define the bare qubit ground and excited eigenstates |g = (|↑ -|↓ )/ √ 2 and |e = (|↑ +|↓ )/ √ 2. Our initial state reads:

|Ψ(t = 0) = |↑ ⊗|0 = (1/ √ 2) |g ⊗|0 +(1/ √ 2) |e ⊗|0
. At weak dissipation, the configuration |g ⊗ |0 is close to the actual ground state, and does dot experience any time evolution. In contrast, the excited state |e ⊗ |0 will decay towards the low energy state of the qubit [START_REF] Meystre | Elements of quantum optics[END_REF] while emitting a single photon at the resonant frequency ω = ∆. Thus the complete wavefunction in the long time limit should read:

|Ψ(t = ∞) = 1 √ 2 |g ⊗ |0 + 1 √ 2 |g ⊗ a † ∆ |0 (10) = (|↑ -|↓ ) √ 2 ⊗ |0 + a † ∆ |0 √ 2 . ( 11
)
Clearly the qubit states and the emitted photons are unentangled in the long time limit, which is in contrast to the outcome of the single coherent state approximation. Indeed, from Fig. 2, one sees that the real space displacements f (x) and h(x) are close to each other, but not strictly equal. This generates some spurious correla- tions with the qubit states, hence the improper value for the long-time coherence.

B. Adiabatic switching

These moderate artifacts, which are found in previous out-of-equilibrium NRG calculations as well [52], are related to the sudden form of the quench. Indeed, if the qubit is driven adiabatically, the correct relaxation occurs within the single coherent-state scheme. To show this, we still subject the qubit to a time-dependent polarization field (t) along the z-axis, but now switch it off gradually. The resulting coherence σ x (t) is shown in Fig. 3 for three values of the switching time.

For a short switching time, the long-time value of the coherence is underestimated, as for the sudden quenches. However, for a more adiabatic pulse, good convergence towards the ground state value is recovered. At the same time, one can check in Fig. 3 that the emitted wavepacket is factorized with respect to the short distance cloud: indeed, the real space displacements f (x) and h(x) are now equal to each other at large distances, ensuring proper factorization.

These observations thus show the merits and drawbacks of the popular single-coherent-state dynamics, also known as the Davidov dynamical ansatz [31,32]. After a sudden quench, the polarization dynamics and final wavefunction near the origin are captured reasonably well, but the coherence and eventual disentanglement of the qubit and the traveling photon are not. For a sufficiently adiabatic quench, all quantities are correctly captured by the single-coherent-state dynamics.

IV. SCATTERING PROPERTIES: NUMERICAL RENORMALIZATION GROUP

In this second part of the paper, we wish to study a typical situation in quantum optics in the specific context of waveguide QED: we investigate the scattering properties of photons in the two-terminal setup depicted in Fig. 4, but in a domain of ultra-strong coupling. This question was investigated by many authors in a regime of weaker coupling, α 1, using for instance the RWA [53][54][55] or extensions to scattering of the Wigner-Weisskopf theory [56] (based on Fock state truncation). In order to tackle the case of large α, we first develop a numerically exact approach to the photon scattering properties in the weak intensity limit. This is done by using the numerical renormalization group method [57,58]. The results provide an important benchmark for the coherent state dynamics treated in the next section and, indeed, for any calculations of scattering in waveguide QED.

A. Elastic and inelastic transport coefficients from the numerical renormalization group

We start by defining the real-time retarded equilibrium spin susceptibility:

χ(t) = - i 4 θ(t) GS|[σ z (t), σ z (0)]|GS (12)
with |GS the full many-body ground state (note that χ(t) is a purely real function). Inserting a complete eigenbasis of states {|a } with respective energies E a , one readily obtains:

χ(t) = - 1 2 θ(t) a | GS|σ z |a | 2 sin[(E a -E GS )t]. ( 13
)
This leads to the frequency-resolved spin susceptibility (decomposed into real and imaginary parts):

χ(ω) ≡ χ (ω) + iχ (ω) = +∞ -∞ dt e iωt χ(t). (14)
In particular, in terms of the Lehman spectral decomposition onto the complete eigenbasis, the imaginary part reads:

χ (ω) = π 4 a | GS|σ z |a | 2 [δ(ω + E a -E GS ) (15) -δ(ω -E a + E GS )] . (16)
One thus obtains from the above decomposition a sum rule that will be important in what follows:

+∞ -∞ dω χ (ω) Sign(ω) = π 2 . (17)
Now, from linear-response theory [12,13] and using an exact identity for the Green funciton of the bosonic modes,

G kk (ω) ≡ a † k (ω)a k (-ω) = δ kk ω -ω k + g k g k χ(ω) (ω -ω k )(ω -ω k ) (18)
which relates the scattering matrix to the qubit response, one obtains the reflection and transmission coefficients,

R(ω) = (2παω) 2 |χ(ω)| 2 (19) T (ω) = (2παω) 2 [χ (ω)] 2 + [1 -2παωχ (ω)] 2 , ( 20
)
as well as the total inelastic deficit,

γ(ω) ≡ 1-R(ω)-T (ω) = 4παωχ (ω)-2(2παω) 2 |χ(ω)| 2 .
(21) As they are derived from linear response theory, the reflection and transmission probabilities here are those of a single incoming photon.

These quantities are shown in Fig. 5 for increasing values of dissipation. The reflection/transmission coefficient shows as expected a peak/dip at the qubit absorption frequency ∆ R , which progressively renormalizes to smaller values as α increases. At the same time, one notes that the peak value in R(ω) is slightly lower than unity, with a deviation that increases with enhanced dissipation. This is in contrast to the approximate results of Ref. 12 but in agreement with exact calculations at the Toulouse point [13,57], which read:

χ Toulouse (ω) = 1 2πω 1 ω + iΓ log(1 -2iω). ( 22
)
As a consequence, the inelastic deficit becomes more and more important as α grows, due to stronger photon nonlinearities, as shown in the bottom panel. One notes in particular that the deficit γ(ω) peaks above ∆ R and shows very long tails, which are reminiscent of the inelastic contribution to scattering for fermionic Kondo impurities [59], although the physical quantities do not correspond strictly here. As previously noted by Goldstein et al. [13], the total inelastic losses are quite important at strong dissipation, and reach above 20% at α = 0.6. We note in addition a surprising result from the NRG: while the reflection deficit is sizable [for large impedance, typically up to 10% deviation from unitary scattering at the peak value of R(ω)], the transmission deficit is very small-the transmission dip goes to a tiny (yet non-zero) value. It is practically not possible to see this small background in the middle panel of Fig. 5, but one can verify from the exact Toulouse formula at α = 0.5 that the minimal transmission is of order 10 -3 , a very small number to which we cannot give a simple physical interpretation at this stage.

B. Analytical comparisons

We provide here some analytical insights, comparing previously derived theories to our numerical simulations. We also derive new phenomenological formulas that match the NRG results quite well, and that could be used in practice for easier comparisons to experiments. We do not include checks to the exact Toulouse limit [13], as it has been shown earlier [57] that the NRG reproduces the spin susceptibility quite accurately at α = 0.5.

First, we compare the NRG data to approximate results based on the rotating wave approximation used routinely in quantum optics [START_REF] Meystre | Elements of quantum optics[END_REF], which leads to the following t-matrix for a unidimensional waveguide:

t RWA (ω) = ω -∆ ω -∆ + iΓ . ( 23
)
In this approximation, the qubit level is not renormalized, but acquires a lifetime, with a rate Γ ∝ α∆. This leads to the reflection and transmission coefficients:

R RWA (ω) = |1 -t RWA (ω)| 2 = Γ 2 (ω -∆) 2 + Γ 2 (24) T RWA (ω) = |t RWA (ω)| 2 = (ω -∆) 2 (ω -∆) 2 + Γ 2 . ( 25
)
Clearly, scattering is unitary in the RWA, since R RWA (ω) = 1 -T RWA (ω). Due to the lack of inelastic effects, the RWA lineshape should become less accurate as dissipation increases. We see indeed sizable deviations when α > 0.1 in the upper panel of Fig. 6 by a comparison to NRG for the reflection coefficient. Note that we allow here as fitting parameters for the RWA lineshape the renormalized qubit frequency and linewidth, which improves the agreement somewhat artificially. This defect of the RWA has motivated improved perturbative calculations by K. Le Hur [12], based on an approximate form for the spin susceptibility that derives from analogy to the Non Interacting Blip Approximation (NIBA) [37]:

χ NIBA (ω) = ∆ R ∆ 2 R -ω 2 -i2παω∆ R . ( 26
)
From Eqs. ( 19)-( 20), this yields an approximate form of the reflection and transmission coefficients,

R NIBA (ω) = (2Γω) 2 (∆ 2 R -ω 2 ) 2 + (2Γω) 2 (27)
T NIBA (ω) = (∆ 2 R -ω 2 ) 2 (∆ 2 R -ω 2 ) 2 + (2Γω) 2 , ( 28
)
from which the RWA expression (24) is recovered in the limit Γ ∆ R . These formulas can be seen, however, to again satisfy R NIBA (ω) + T NIBA (ω) = 1 and thus miss inelastic losses, as pointed out in Ref. 13. The comparison to NRG is nevertheless much more satisfactory than with the RWA, as seen in the lower panel of Fig. 6 only discrepancy comes again from the lack of inelastic contributions within the NIBA expression.

The NIBA expression can, in fact, be corrected phenomenologically by noticing a simple problem in the formula for the spin susceptibility (26). Indeed, the sum rule ( 17) is increasingly violated as α grows:

+∞ -∞ dω χ NIBA (ω)Sign(ω) = π + 2 atan 2-(2πα) 2 2πα √ 4-(2πα) 2 2 4 -(2πα) 2 ≡ π 2 N α . (29)
One recovers N α → 1 for α → 0, but in general the deviation from the sum rule can be large, and, as we will see, accounts for the missing inelastic contribution. We thus define an improved phenomenological formula:

χ phen. (ω) = 1 N α ∆ R ∆ 2 R -ω 2 -i2παω∆ R , (30)
which satisfies by construction the exact sum rule. This leads to the following expressions for the reflection, transmission, and inelastic responses:

R phen. = 1 N 2 α (2Γω) 2 (∆ 2 R -ω 2 ) 2 + (2Γω) 2 (31)
T phen. = (∆ 2 R -ω 2 ) 2 + (1 -N -1 α ) 2 (2Γω) 2 (∆ 2 R -ω 2 ) 2 + (2Γω) 2 (32)
γ phen. = N α -1 N 2 α 2(2Γω) 2 (∆ 2 R -ω 2 ) 2 + (2Γω) 2 . (33)
The comparison to the NRG data in the top panel of Fig. 7 gives now excellent agreement at α ≤ 0.2 for the reflection coefficient, and reproduces well the magnitude of the inelastic deficit, although the shift of the peak position and the long tails in γ(ω) are not correctly accounted for. We also see from the bottom panel of Fig. 7 that large deviations occur in γ(ω) for larger α values; this is expected [60] because a slower decay of the form χ (ω) ∼ 1/(ω log 2 ω) is known to occur in the Kondo regime for ω ∆ R . As a result, the inelastic contribution is even more underestimated in this regime.

V. SCATTERING PROPERTIES: COHERENT STATE DYNAMICS A. Coherent state formalism for a two-terminal waveguide

The problem we want to study is the scattering of a coherent state wavepacket off the Silbey-Harris ground state of the spin-boson model. The strategy is to define the proper initial condition for the wavepacket in leftright space (the full waveguide), then transform it to an even-odd basis, and propagate the resulting state. Since only even modes couple to the qubit, the equations of motion ( 6)-( 9) can be used, while the odd modes are freely propagating. After the scattering process has occured, a transformation back to the left-right basis provides the final wavefunction, which allows one to find the transmitted and reflected intensity.

Mathematically, we denote the initial scattering wavepacket by z x or z k in position space or momentum space, respectively (this excludes the entanglement cloud associated to the Silbey-Harris ground state). These are related by the Fourier transform convention,

z x = ∞ -∞ dk √ 2π e +ikx z k . (34)
The even and odd parts of the wavepaket are then defined strictly for k > 0 as

z e k = 1 √ 2 (z k + z -k ) and z o k = 1 √ 2 (z k -z -k ) , (35)
which can be inverted to yield

z k = 1 √ 2 (z e k + z o k ) and z -k = 1 √ 2 (z e k -z o k ) . ( 36
)
Note that the first expression in Eq. ( 36) gives the rightgoing wave (k > 0), while the second one is the left-going wave (-k < 0). We choose to use a Gaussian wavepacket defined as

z k = √ n 1 2πσ 2 1 4 e -(k-k 0 ) 2 4σ 2
e -i(k-k0)x0 e -ik0x0/2 , (37) corresponding to a signal initially centered around x 0 , with mean wavenumber k 0 , spatial extent 1/σ, and a total intensity corresponding to n photons on average. The corresponding real space wavepacket is then

z x = √ n 2σ 2 π 1 4
e -(x-x0) 2 σ 2 e +ik0(x-x0) e +ik0x0/2 .

(38) Note that these amplitudes are both normalized so that

∞ -∞ dx|z x | 2 = ∞ -∞ dk|z k | 2 =
n. Now let us define the actual photon content of the wavepacket, using here coherent states, which are better suited for our simulations than scattering Fock states. The creation operator for a photon in the wavepacket state is defined as

a † z = +∞ -∞ dk z k a † k = +∞ -∞ dx z x a † x . (39)
A coherent state with an average of n photons in this wavepacket is thus:

e -n/2 e a † z |0 = e a † z -az |0 . (40)
One can construct even and odd creation and destruction operators for bosons in a given mode k > 0 by analogy to the transformations made previously on the wavepacket:

a e k = 1 √ 2 (a k + a -k ) and a o k = 1 √ 2 (a k -a -k ) . ( 41
)
Then the creation operator for the wavepacket state is

a † z = +∞ 0 dk z k a † k + z -k a † -k (42) = +∞ 0 dk z k a e † k + a o † k √ 2 + z -k a e † k -a o † k √ 2 (43) = +∞ 0 dk z e k a e † k + z o k a o † k = a e † z + a o † z . (44)
Notice that the even and odd sector operators commute,

[a e † k , a o k ] = [a e † z , a o z ] = 0. ( 45
)
Thus, the coherent state wavepacket can finally be written as:

e -n/2 e a † z |0 = e a o † z e -n/2 e a e † z |0 = e a o † z e a e † z -a e z |0 . (46)
The final step in the initialization of the state vector is to combine the scattering wavepacket coherent state z k with the Silbey-Harris ground state with displacement

f SH k = (1/2)g k /(ω k + ∆ R ).
Since the even and odd sectors are completely independent, the Silbey-Harris state affects only the even sector. For the spin-up projection of the wavefunction, we have thus,

e a e † z -a e z |Ψ ↑ = e k>0 (z e k a e † k -z e * k a e k ) (47)
× e k>0 (f SH k a e † k -f SH * k a e k ) |0 ,
where for compactness we have switched to sums over k instead of integrals. The two exponentials can be combined, keeping in mind that there may be a phase from the commutator in the standard relation e A e B = e A+B e 1 2 [A,B] , which is valid since the commutator here is just a c-number. Thus the initial state in the ↑ projection is

e a e † z -a e z |Ψ ↑ = e 1 2 k>0 (z e k f SH * k -z e * k f SH k ) (48) × e k>0 [(f SH k +z e k )a e † k -(f SH k +z e k ) * a e k ] |0 .
For the spin-down projection, one simply replaces f SH k by -f SH k without changing the sign of z e k , so that our total initial wavefunction reads:

|Ψ(t = 0) = 1 √ 2 |↑ e 1 2 k>0 (z e k f SH * k -z e * k f SH k ) (49) × e k>0 [(f SH k +z e k )a e † k -(f SH k +z e k ) * a e k ] |0 - 1 √ 2 |↓ e 1 2 k>0 (-z e k f SH * k +z e * k f SH k ) × e k>0 [(-f SH k +z e k )a e † k -(-f SH k +z e k ) * a e k ] |0 .
One can thus use this state as initial condition for the equations of motion ( 6)-( 9), with

f k (0) = f SH k + z e k , h k (0) = -f SH k + z e
k , and p(0) = -q(0) = 1/ √ 2. Note that the odd sector, which decouples from the qubit, is trivially evolving in time according to i żo k = ω k z o k . After this time-evolution is performed, the output must be analyzed. Because of the constrained ansatz (4), In the quantum optics regime α = 0.05, the reflection coefficient obtained from the variational dynamics (symbols) matches the RWA lineshape (dashed line), which was however corrected with the proper renormalized qubit splitting. Parameters here are ∆/ωc = 0.1 and the variational dynamics is done at weak power (n = 0.2) in order to be in the linear response regime. The wavepacket width is taken very small σ = 0.0005 to allow good spectral resolution.

the output is again a single coherent state, but one has to recombine the even sector output with the odd sector trivial evolution. After a long enough time T so that the outgoing wavepacket is disentangled from the Silbey-Harris cloud, one obtains the total displacement

f k (T ) = f SH k + z e k,out
, separating into the ground state contribution and an outgoing scattering part. Combining with z o k (T ) = e -iω k t z o k (0) using Eq. ( 36), this provides the final outgoing coherent state displacement z k,out in the physical momentum basis. One can then define transmission and reflection coefficients as the ratio of the outgoing (in each possible terminal) to the ingoing power:

T = k>0 k |z k,out | 2 k>0 k |z k,in | 2 and R = k<0 |k| |z k,out | 2 k>0 k |z k,in | 2 . (50)

B. Comparison of NRG and coherent state dynamics results for the transport coefficients

In this last section, we present transport calculations using the single coherent state dynamics and benchmark them in the linear response regime against the controlled results from NRG, including a comparison to RWA as well. Let us first consider the quantum optics regime α 1 where the RWA can be trusted, as shown in Fig. 8 for α = 0.05 and an incident coherent state wavepacket whose intensity is small. We see excellent agreement between the two methods; thus, the coherent state approach performs very well in the quantum optics regime [note that an effective qubit splitting was added by hand in the RWA curve].

Increasing dissipation up to α = 0.2, we assess the va- lidity of both the coherent state dynamics and the RWA by comparing to the numerically exact NRG results in Fig. 9. While the RWA results are rather inaccurate, the coherent state dynamics is in excellent agreement with the NRG, showing its utility well beyond the usual quantum optics regime.

Being a non-equilibrium method, the coherent state dynamics can access regimes beyond linear response, which the present formulation of the NRG is not able to achieve. We show for instance in Fig. 10 how the transport coefficients evolve as input power is turned on, increasing the average number of photons in the incoming coherent state wavepacket up to n = 5. Having checked our numerical integration carefully, we attribute the large level of noise in these results to the restricted form of the ansatz (4) and not to numerical inaccuracies. This is consistent with our general observation that artifacts typically arise for strong temporal perturbations in the single coherent state dynamics, which indeed increase at larger power.

The general trend of the curves are however clear [12]: due to saturation effects of the two-level system by the large flux of photons, the reflection is reduced at increasing power. Clear deviations from the linear response result occur for an average photon number n 5. This value can be understood from the fact that the qubit excitation rate nσ = 5 × 0.005 = 0.025 becomes then comparable to the qubit decay rate Γ α∆ R = 0.2 × 0.15 0.03. Saturation effects are indeed expected to be governed by the renormalized qubit frequency at strong coupling.

VI. CONCLUSION

We have investigated here the relaxation and transport properties of a single qubit side-coupled to a large impedance transmission line, using complementary many-body methods. The numerical renormalization group (NRG) allowed us to accurately compute the linear-response transport coefficients, and showed good agreement for a simpler approach using the variational time-evolution of a wavefunction based on a single coherent state. This coherent state dynamics was benchmarked also regarding relaxation properties, and despite some success in predicting the onset of overdamped dynamics, showed some physical artifacts such as improper relaxation to the correct steady state. We plan to overcome these difficulties in future works by systematically extending the dynamical ansatz along the lines of Ref. [22], allowing us also to tackle the transport problem in a more controlled way.

5 FIG. 1 .

 51 FIG. 1. (Color online) Population decay σz(t) (upper panel)and coherence buildup σx(t) (lower panel) of the two-level system for several values of dissipation α = 0.1, 0.3, 0.5 at ∆/ωc = 0.1. The dashed lines on the vertical axis of the lower panel denote the values for the ground state coherence that are expected from the static single-coherent state approximation. The discrepancy with the long time limit of σx originates in spurious correlations between the bath and the two-level system.

FIG. 2 .

 2 FIG. 2. (Color online)Real space displacements f (x) and h(x) associated with the ↑ and ↓ components of the wavefunction, respectively, shown for a long time after the instantaneous quench process discussed in the text (parameters are ∆ = 0.1 and α = 0.2). At short distances, the Silbey-Harris ground state is stabilized, while at large distances a wavepacket is propagating away from the qubit located at the origin. The lack of factorization, f (x) = g(x) for x > 220, is associated with spurious correlations between the emitted wavepacket and the two-level system, that are responsible for the improper relaxation seen in Fig.1.

FIG. 3 .

 3 FIG. 3. (Color online)Top panel: Coherence σx(t) as a function of time for three different switching processes (from sudden to adiabatic), that are given by the temporal profile of the local field shown in the rightmost middle panel. The saturation to the equilibrium value is obtained for an adiabatic switching, while the sudden quench shows improper relaxation. The leftmost middle panels shows the associated population decay σz(t) . Bottom panel: real space displacement f (x) and h(x), associated to the ↑ and ↓ components of the wavefunction respectively, shown for a long time after the adiabatic quench. Now factorization is correctly recovered, in agreement with the proper relaxation value of σx(t) at long times.

FIG. 4 .

 4 FIG. 4. (Color online)Waveguide setup considered here, where a qubit is side-coupled in a two-terminal geometry to a photonic waveguide. Transport properties are described by scattering processes depicted by a set of ingoing and outgoing wavepackets.

1 FIG. 5 .

 15 FIG. 5. (Color online) Reflection coefficient (top panel), transmission coefficient (middle panel) and total inelastic losses (bottom panel) from the NRG calculations at three values of dissipation α = 0.2, 0.4, 0.6 for ∆ = 0.1. The curves with largest α show a resonant peak at the smallest value of the renormalized qubit splitting ∆R.

FIG. 6 .FIG. 7 .

 67 FIG.6. (Color online) Reflection coefficient comparing the RWA Lorentzian lineshape Eq. (24) (dashed lines, upper panel) and the NIBA lineshape Eq. (27) of Ref. 12 (dashed lines, lower panel) to the exact NRG calculations (full lines) at values of dissipation α = 0.2, 0.6 for ∆ = 0.1. An effective renormalized qubit splitting ∆R and a renormalized linewidth Γ were used as fitting parameters within the RWA and NIBA formulas, for better comparison of the actual lineshapes.

2 RWAFIG. 8 .

 28 FIG. 8. (Color online)In the quantum optics regime α = 0.05, the reflection coefficient obtained from the variational dynamics (symbols) matches the RWA lineshape (dashed line), which was however corrected with the proper renormalized qubit splitting. Parameters here are ∆/ωc = 0.1 and the variational dynamics is done at weak power (n = 0.2) in order to be in the linear response regime. The wavepacket width is taken very small σ = 0.0005 to allow good spectral resolution.

FIG. 9 .FIG. 10 .

 910 FIG. 9. (Color online)In the ultrastrong coupling regime α = 0.2, the reflection coefficient obtained from the variational dynamics (symbols) compares favorably to the exact NRG lineshape (solid line), while the RWA (dashed line) presents clear deviations. The same parameters as in Fig.8were used.

ACKNOWLEDGMENTS

We thank Zach Blunden-Codd, Alex Chin, Ahsan Nazir, Nicolas Roch, Marco Schiró and Izak Snyman for stimulating discussions and acknowledge funding from the Fondation Nanosciences de Grenoble under RTRA contract CORTRANO. The work at Duke was supported by the U.S. DOE, Division of Materials Sciences and Engineering, under Grant No. DE-SC0005237.


Marie-Pierre Chopin

Art education: a bulwark against the post-democratic trends of modern states? Critical approach

to help students to develop their critical mind skills, to distinguish knowledge from opinions or beliefs, to be able to argue and respect thinking of

Introduction

The reflection for this communication started from a fairly simple fact: the appearance for ten years, in France, of a new objective for art education at school the development of "the critical mind" of pupils or students, future citizens. This critical mind would have become a necessity of so-called post-democratic societies (Crouch, 2004). Art could constitute a bulwark against a set of excesses or threats that the children will encounter in their life (proliferation of fake news, various forms of politico-media manipulation, etc.).

The terms "critical mind" explicitly appear in 2015 in the reference text on the "path to art education", in which the function of art is « to develop the faculty of judging and critical mind», « to express an aesthetic emotion and a critical judgment », « to exercise a critical look at his practice » and « to learn critical thinking skills ». This critical aim of art education is consistent with the recent evolution of educative policies in France. The teachers' reference text on professional skills requires them to "help students to develop their critical mind skills". Likewise, the text about the "Common Knowledge and Skills Base" sets the primary objective to open the knowledge and the "judgment and critical thinking/mind" of pupils.

First, we will show the historical context of the emergence of this '' new '' finality for art education. Then, we will question the effects of this critical finality, starting from an artistic and cultural education project we analyzed.

-About the critical dimension of art education: an history of cultural policies in France

As we said, the notion of critical mind appears recently. But Nonetheless, the critical dimension of art education (that is, the idea that art allows to develop "agency" and ability of people to think critically), is quite old. That's we propose to show going back to the history of art education policies.

It's possible to identify three phases in the evolution of these policies: the first, in the late 60s and the 70s, is marked by the "contained" influence of popular education movements; the second, in the 80' and early 90', is rather a phase of professionalization and relative depoliticization of art education; the third, begins in the late 90s and may appear as a phase of re-politicization, but far from the collective and activist dimensions of the 60s-70s.

1960'-1970': a politization under control, the neutralization of popular education movements

Since the end of the 19th century, popular education movements have considered art and culture as a means of promoting collective emancipation. The terms "critical thinking" or "critical mind" were not used, but the idea is close: the goal of these activists was to provide citizens with "cultural" tools to think critically about the world. This is how many groups of intellectuals and artists developed political and cultural projects, outside the bureaucratic field, such as the popular universities (in connection with the intellectual field) or the Théâtre du peuple (in connection with the artistic field). This political function of art and culture is still found in all popular education movements, from the 1930s to the 1960s, in France.

One of the first steps in the institutionalization of modern art education policies is the famous Amiens congress of 1968. A somewhat "romanticized" story tells that this symposium is at the origin of the miraculous meeting of militants of popular education movements, national education actors, and politicians, gathered around a common observation and a shared desire to step outside the classical framework of education.

It's a bit true, but it does not exactly reflect the historical reality.

In fact, this conference is mainly organized by an association of scientists and especially politicians, gathered around the idea that science can enlighten governments. Popular education is much mobilized to "legitimize" the discourses produced, but the popular education activists themselves are largely absent from the organizers and participants of the conference. Or, in any case, they are "invisible". This is entirely consistent with what happens in the following years, a certain "depoliticization", or at least a form of "politicization under control", of art education, which is no longer the strongly militant discourse of popular education.

Nevertheless, in this period, art education is addressed to pupils and students who can be "educated", as citizens, through art and culture.

In the early 70s, the first projects set up in schools, jointly by the Ministry of Culture and the Ministry of Education, were designed to form a critical attitude toward the mass media, especially television, present in almost every home. This quotation from a report of the congress of Amiens shows this well: a kind of oscillation between the question of the individual and society:

"To speak of cultural training is to place as a primary goal the formation of a being: knowing how to inquire, reason, communicate; responsible, free and de-alienated but at the same time communitarian; balanced, knowing to discover his art of living by an adjustment constantly recreated to the changing environment".

1980 '-1990': an assumed professionalization and depolitization of art education

In the 1980s, Socialist Party came to power, and the Ministry of culture made reforms to speed up the professionalization of the artistic field, including in the field of art education. This is the result of a certain bureaucratization of the field of production of cultural policies: the first generation is more "militant" (with the belief that science and culture can help to move towards a better world); this one is more technocratic (with the main concern of making the state works well, even if ambitions have to be revised downwards).

During this period, "cultural mediation" grew, and professionals gradually replaced activists of cultural action. These two groups do not have the same story, their members do not have the same professional trajectories, and therefore they do not see their "mission" in the same way.

This has two consequences for art education:  On one hand a certain depoliticization of the aim of art education, even stronger than before: now, art education aims to transmit knowledge about art. The individual and collective emancipation, is pushed in the background and word "critical" only appears to talk about the criticism of artworks;  On the other hand (and this goes together): there is a redefinition of the "publics" of art education: they are always pupils (from primary school to high school), but now they are more specific groups (the poor, the inhabitants of popular districts,...). It is the same in the other fields from the 1980s, with the increasing of targeted policies.

End of 1990 '-2000': a paradoxical repolitization

Gradually, since the end of the 1990s and especially in the 2000, the term "critical mind" appears in the official texts. For Minister of culture Catherine Trautmann (member of the Socialist Party), art and culture "perform an indispensable function of critical questioning of our time" (1999 speech). Fifteen years later, the Minister of National Education Najat Vallau Belkacem (also member of the Socialist Party) says that art and cultural education allows "the development of the faculty of judging and critical thinking" (2015 speech).

NB : this periodization is probably a bit broad, and she could be refined by showing how the focus is more and more on the individual, all along the 2000' (with the affirmation of the notion of skills (from the reforms of education policies in 2005, under a right wing government. with the ideas of "foundation of competence", "common knowledge base", "personal skills booklet", etc.)

In a certain way, we can say that it is a kind of "repolitization" of art education. And in any case, it is an assignment of a social and political function, after about two decades rather focused on properly artistic issues. But this politization is very different from the approach promoted by activists of popular education. It is not a question of giving individuals critical tools to work for collective emancipation; the "new" goal remains at the individual level.

In this context of post-democracy (with fake news and all forms of media manipulation), the objective is to give the citizens "individual skills" to decrypt the media messages ... but at no time, the subject is the collective emancipation. This matches perfectly with the neoliberal turn of public action in France, and with the development of new public management, which is a very individualistic approach of the state action, focused on competition and elitism. It is also very consistent with the development of cognitive psychology and neuroscience approaches in the field of education, which promotes a very individual approach to transmission, including in the fields of art and culture. These transformations are also prevalent in educational policies, with the rise of the notion of "skills", which is used to "evaluate" the work of students, individually.

-How teachers and pupils deal with these recommendations: the case of an art education project in France

This historical approach will now be linked with outcomes of an ethnographic research conducted in 6 classes of primary school in France (about 130 pupils) in 2018 and 20191 . We analyzed an art education project (a dance project) called (in french) « Esprit de corps_critique ». This name links two ideas: on one hand the fact that the body can be the place for a mind development (threw the dance practice); and on the other hand, the fact that this mind should be critical. The project is thus based on the idea that threw the body (and the dance movement), pupils could develop their critical mind.

« Esprit de corps_critique » project (EDCC) At the end of the project (dance workshops, writing workshops, and discovery of the show), the project managers organize a debate with the pupils of all classes. During this " Débat d'idées" (Debate of ideas), pupils have to talk about the show and more generally about dance, sharing their point of view in an argumentative way.

Two outcomes about the critical aim of the project EDCC

The corpus of the research is based on 90 hours of observations, 8 interviews, and a set of quantitative data not used for this presentation2 ): For the designers of the EDCC project, critical mind is a generic skill, allowing pupil to understand his relationship to the world. In fact, this project allowed pupils to live a particular experience and it is difficult to say if this could develop generic skills, dispositions to think in a critical way. In fact, an a priori didactical analysis (Brousseau, 1998), show that the production of a discourse about a show is, finally, linked to a competence of critical apprehension of an artwork (here a dance show). We've to note that this competence is precisely the aim claimed in art education policies in the 1980s (the 2nd period identified above). In the case of dance, the body experience coupled with the discourse of journalists (not the one of teachers, this is an important point) could allow the achievement of this specific objective. The statistic survey, always in progress, should show if all pupils, regardless of school level, socio-cultural position or out-of-school cultural practices, take advantage from the project in the same way.

To summarize this first outcome, in the EDCC project, the aim potentially achieved is the development of a critical approach to artwork. However, threw the idea of "critical mind", it was more directly a critical approach to the world which was aimed by the designers.

If, of course, a critical approach to artwork could generate a critical approach to the world, in the long run, this distinction between what was officially aimed and what was probably achieved, is important. It invites us, not to think that teachers, artists or journalists committed in the project failed in their mission, but rather to question the relevance of such broadly objectives for art education, at least on two aspects: first, is it possible to develop this kind of disposition in such short times?; and more generally, we could imagine that the assignation of this kind of objective could have some counterproductive effects. The second outcome seems to confirm this hypothesis.

Outcome 2: Art pushed in the background?

The second outcome is based on the "a posteriori" didactic analysis of the project. It concerns the effects of the fact of bringing the development of critical mind to the fore.

Although the teachers immediately became fully committed to the project at the start of the year, with great enthusiasm for this critical dimension, the observations show that they were more prudent as it progresses, including into the debate organized at the end of the project. One way to understand Isabelle's words is to say that the announced official objective (to develop the critical mind) has occupied such a place in the project that teachers feel as a pressure the obligation to account for the supposed effects of the project: to be able to debate about artwork and more than this. The interviews with the teachers show their disarray (and that of their students), faïnelly, to see this cognitive and linguistic aim supplant artistic aim: knowing and practicing dance.

Elsa (teacher, primary school) : To be honest, when I talked to the pupils about this restitution, they looked at me with great big eyes, like this! [We laugh]. I explained that there were four classes, for a debate about the critical mind. I felt they were confused…

Everything happens as if the aim assigned by the art education policies (to develop the critical mind of the pupils), led Elsa (but more widely all the teachers of the panel) to focalize so much on this dimension that art were pushed in the background. "Critical mind" became, in this project, a kind of new object of teaching.

Conclusion

To conclude, this communication showed that the critical dimension of art education has been reconfigured threw the fifty last years. For 10 years, the idea that our society became post democratic served as a justification for a new relationship between art, education and critical exercise. But, by imposing the very specific notion of critical mind (reinforcing the idea of an individual cognitive ability), it could paradoxically contribute to making art a secondary subject of artistic education.

  Extracts from the decree of 1 July 2015 on art and cultural education path (Official journal on 7 July 2015) (National Education, France) « to develop the faculty of judging and critical mind» « to express an aesthetic emotion and a critical judgment » « to exercise a critical look at his practice » « to learn critical thinking skills »

  EDCC project was designed by a team of the National Center of Choreographic Development (NCCD), an important label for cultural organization in France. The NCCD is an intermediary between cultural policies and people engaged in art education projects (teachers, pupils, artists). Lauranne, the project leader, also top manager of the NCCD, explains how the EDCC project occurs: Lauranne : The project starts during the french presidential election on 2017 (...) Are artists able to influence the development of the critical mind of citizens? This question was important because today each citizen get used to lies and fake news… information is given in a sort of dangerous instantaneity. It started in this situation. All the professionals of cultural organizations are proud to say that they are able to influence the development of the critical mind… But, is this real? During 5 weeks, pupils and their teacher work with:  An artist (choreograph and/or dancer of a company). 8 hours are allowed to dance workshop, based on the show of the company. Each class see this show during the project;  A journalist, dance critic. 6 hours are allowed to a writing workshop in order to teach the basics of the analysis of a dance piece. At the end of the project, each class prepares a collective critic of the show they have seen. Students discover the practice of dance criticism.

Table 1 -Corpus of the ethnographic research on the project EDCC

 1

	Observations (90h)	48h of art practice workshop
		36h of writing workshop
		6h of show on stage
	Interviews (8)	2 designers of the project
		6 teachers
	Other datas	 Pre-tests and post-tests with the
		pupils
		 Grilles d'évaluation des niveaux
		scolaires et des effets subjectifs du
		projet
		 Survey questionnaire on families
		 Survey questionnaire on teachers
	Outcome 1: What pupils can learn in this kind of project

  In 2018, 4 classes were committed in the project (we observed only two): but only one class accepted to participate to the debate. All the teachers were skeptical about the interest of it: We had to organize a debate, a discussion between all the pupils. And, all the teachers, we all have the same point of view. Me, I sent a mail to my colleagues, only to them, not to the NCCD (she laughs), and like me, they didn't know what to do ! The leaders of the project want to have a restitution of what we did, but I don't think that putting all the classes together is a good idea. Because it was the same project, but with different people. I don't know what we can really share… And the kids, they'll get upset, agitated...

Isabelle (teacher, primary school):

En

2019, les observations ont été réalisées avec une jeune chercheure étudiante en master : Loîcia Collot.

Ce volet de la recherche, réalisé avec Loîcia Collot, repose sur la passation : 1. de pré-tests et post-tests auprès des élèves pour construire des indicateurs possibles de l'esprit critique et produire des comparaisons de mesures ; 2. des questionnaires aux familles pour caractériser les positions socio-culturelles des élèves, les pratiques d'éducation familiales et leur pratiques culturelles ;

Des grilles d'évaluation et des questionnaires aux enseignants pour caractériser les positions scolaires des élèves et les pratiques éducatives des professeurs.