Rémi Robin
email: remi.robin@inria.fr

Nicolas Augier

Ugo Boscain

Mario Sigalotti

Ensemble qubit controllability with a single control via adiabatic and rotating wave approximations

Keywords: Averaging, control of quantum mechanical systems, spin dynamics, rotating wave approximation, adiabatic approximation AMS subject classification: 81Q93, 34C29, 81Q15

In the physics literature it is common to see the rotating wave approximation and the adiabatic approximation used "in cascade" to justify the use of chirped pulses for two-level quantum systems driven by one external field, in particular when the resonance frequency of the system is not known precisely.

Both approximations need relatively long time and are essentially based on averaging theory of dynamical systems. Unfortunately, the two approximations cannot be done independently since, in a sense, the two time scales interact. The purpose of this paper is to study how the cascade of the two approximations can be justified and how large becomes the final time as the fidelity goes to one, while preserving the robustness of the adiabatic strategy. Our first result, based on high-order averaging techniques, gives a precise quantification of the uncertainty interval of the resonance frequency for which the population inversion works. As a byproduct of this result, we prove that it is possible to control an ensemble of spin systems by a single real-valued control, providing a nontrivial extension of a celebrated result of ensemble controllability with two controls by Khaneja and Li.

Introduction

Consider a two-level system described by the Schrödinger equation

i dψ dt = E + α w(t) w(t) -E -α ψ. (1)
Here w : [0, T ] → R is a (sufficiently regular) function representing an external field, E > 0, and α ∈ [α 0 , α 1 ] is an unknown parameter representing the fact that the resonance frequency of the system 2(E + α) is not known precisely, but lies between 2(E + α 0 ) and 2(E + α 1 ). All along the paper we assume the condition [α 0 , α 1 ] ⊂ (-E, ∞), 0 ∈ (α 0 , α 1 ), guaranteeing that the eigenvalues of the matrix in equation ( 1) are never zero, independently of the value of α. The solution of (1) (that depends on α and w(•)) with initial condition ψ α w (0) = (0, 1) is the wave function ψ α w : [0, T ] → C 2 . One would like to find a function w(•) (the same for all values of α) such that, if at time zero the system is at the ground state (0, 1) (i.e., it is in the eigenstate corresponding to the eigenvalue -E -α), then at time T the system is close to a state of the form (e iθ , 0) for some θ ∈ R. In mathematical terms this can be rephrased as follows.

P: For every ε > 0, find a time T and an external field w : [0, T ] → R such that

|ψ α w (T ) -(e iθ , 0)| < ε,
for every α ∈ [α 0 , α 1 ] and for some θ ∈ R (possibly depending on ε, E, T, w, α).

In the mathematical literature it has been proved that problem P admits a solution when one replaces the real-valued function w by a complex-valued one, as in equation ( 2) below ( [START_REF] Beauchard | Controllability issues for continuousspectrum systems and ensemble controllability of Bloch equations[END_REF][START_REF] Li | Control of inhomogeneous quantum ensembles[END_REF][START_REF] Li | Ensemble control of Bloch equations[END_REF][START_REF] Maciel Neto | H 1 -control of an ensemble of half-spin systems replacing Rabi pulses by adiabatic following[END_REF]). As far as we are aware, the problem is open in the case of real-valued functions. The result proved in this paper (Theorem 3) solves problem P in a more general framework, in which there is an additional parameter dispersion on the coupling between the control and the system (that is, w(t) is replaced by δw(t) for δ in a compact interval of (0, +∞)). Solving P is a key ingredient to prove ensemble controllability of (1) with more general initial and final conditions. This celebrated problem has been solved in the case where w is replaced by a complex-valued control in [START_REF] Li | Ensemble control of Bloch equations[END_REF][START_REF] Li | Control of inhomogeneous quantum ensembles[END_REF] and [START_REF] Beauchard | Controllability issues for continuousspectrum systems and ensemble controllability of Bloch equations[END_REF].

The intuitive approach to tackle problem P, consists in the following two steps ( [START_REF] Mittleman | Introduction to the theory of laser-atom interactions[END_REF][START_REF] Shore | Coherent manipulations of atoms using laser light[END_REF][START_REF] Shore | Manipulating Quantum Structures Using Laser Pulses[END_REF][START_REF] Vitanov | Coherent manipulation of atoms and molecules by sequential laser pulses[END_REF]):

• use an external field oscillating at the resonance frequency 2E and having a small and slowly varying amplitude and a slowly varying phase, to simulate by rotating wave approximation (RWA, for short) a system driven by a complex-valued function (in a sense, this "duplicates" the number of available external fields);

• use an adiabatic strategy based on chirped pulses (i.e., pulses whose frequency is slowly increasing from a value below 2(E + α 0 ) to a value above 2(E + α 1 )) to drive the system from an eigenstate to the other one independently of the value of α. This second step substantially exploits the presence of a complex-valued external field and is called adiabatic approximation (AA, for short) [START_REF] Garwood | The return of the frequency sweep: Designing adiabatic pulses for contemporary NMR[END_REF][START_REF] Malinovsky | General theory of population transfer by adiabatic rapid passage with intense, chirped laser pulses[END_REF][START_REF] Shapiro | Quantum Control of Molecular Processes[END_REF][START_REF] Simon | Robust quantum dot exciton generation via adiabatic passage with frequency-swept optical pulses[END_REF][START_REF] Wu | Population inversion in a single InGaAs quantum dot using the method of adiabatic rapid passage[END_REF]. Alternative robust methods are developed, for example, in [START_REF] Jo | Robust two-level system control by a detuned and chirped laser pulse[END_REF][START_REF] Torosov | High-fidelity adiabatic passage by composite sequences of chirped pulses[END_REF].

However the RWA may affect the precision of the adiabatic strategy, as it has been remarked in [START_REF] Leghtas | Adiabatic passage and ensemble control of quantum systems[END_REF]. In order to detail in which sense the "cascade" of the two approximations introduced above may break down, let us give some quantitative estimate.

Rotating wave approximation

Consider a two-level system of the form

i dψ dt = E w(t) w * (t) -E ψ. (2)
Here we assume that the resonance frequency of the system is known precisely, hence we have no α. The symbol w * denotes the complex conjugate of w, which represents here a complex-valued external field. For every ε > 0, consider the external fields w ε (t) = 2εu(εt) cos(2Et + ∆(εt)),

w R ε (t) = εu(εt)e -i(2Et+∆(εt)) .

where u(•) and ∆(•) are two real-valued smooth functions defined on [0, T ], T > 0. We have the following.

Proposition 1. For ε > 0 let ψ wε and ψ w R ε be the solutions of (2) with initial condition ψ 0 ∈ C 2 corresponding to the external fields w ε and w R ε , respectively. Then

max t∈[0,T /ε] |ψ wε (t) - ψ w R ε (t)| converges to 0 as ε → 0.
The proof of this fact is well known. If one applies the unitary change of variables

ψ wε (t) = e -i(Et+∆(εt)/2) 0 0 e i(Et+∆(εt)/2) ψwε (t)
then ψwε (t) satisfies the Schrödinger equation

i d ψwε dt = ε -∆ (εt)/2 u(εt) u(εt)
∆ (εt)/2 + 0 e i(4Et+2∆(εt)) u(εt) e -i(4Et+2∆(εt)) u(εt) 0 ψwε .

Here ∆ indicates the derivative of the function ∆ : [0, T ] → R. Now, defining s = ε t, varying in the interval [0, T ], and ψwε (s) = ψwε (t/ε) we obtain

i d ψwε ds = -∆ (s)/2 u(s) u(s) ∆ (s)/2 + 0 e i(4Es/ε+2∆(s)) u(s) e -i(4Es/ε+2∆(s)) u(s) 0 =:B(s,ε) ψwε . (5)
The same change of variables on ψ w R ε gives rise to

i d ψw R ε ds = -∆ (s)/2 u(s) u(s) ∆ (s)/2 ψw R ε . (6)
Equations ( 5) and ( 6) differ only for the term B(s, ε). Since for every interval

[s 1 , s 2 ] ⊆ [0, T ] we have lim ε→0 s 2 s 1 B(s, ε) = 0
and B is uniformly bounded, we have that solutions of (5) converge uniformly in [0, T ] to solutions of (6) with the same initial condition. This is a classical averaging result that can be found, for instance, in [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]Chapter 8]. Coming back to the original variables one obtains that |ψ wε -ψ w R ε | converges uniformly to zero on the interval [0, T /ε]. This simple argument is very useful. We started with a system driven by one scalar function w and we obtain at the limit a system driven by a complex-valued control or, equivalently, system [START_REF] Augier | Effective adiabatic control of a decoupled hamiltonian obtained by rotating wave approximation[END_REF] where the controls are the two scalar functions u(t) and v(t) = ∆ (t)/2. A more detailed quantitative analysis permits to conclude that on [0, T /ε] we have

|ψ wε -ψ w R ε | = O(ε).
(See, for instance, [5, Appendix A] for a quantitative version of the averaging result mentioned above.) Higher order RWA can be obtained by considering higher-order averaging results.

In recent applications, it is sometimes necessary to use intense external fields. In these cases the RWA may become inaccurate, as pointed out in [START_REF] Ashhab | Two-level systems driven by large-amplitude fields[END_REF][START_REF] Cao | Dynamics and quantum zeno effect for a qubit in either a low-or high-frequency bath beyond the rotating-wave approximation[END_REF][START_REF] Scheuer | Precise qubit control beyond the rotating wave approximation[END_REF]. Thus it is crucial to have a precise quantification of the error.

Adiabatic approximation

We have seen in the previous section how to make the solutions of system (2) approximate those of system (2). We show here how such a system can be easily driven by adiabatic pulses.

Let us consider the case in which the energy of the system is not known precisely. We are then considering the system

i dψ dt = E + α w(t) w * (t) -E -α ψ, where α ∈ [α 1 , α 2 ]. (7)
Let us choose the pulse w in the form

w(t) = u(t)e -i(2Et+∆(t)) , (8)
where u(•) and ∆(•) are two real-valued smooth functions. This choice of control corresponds to (4) in which ε has been set equal to 1. Applying the change of variables

ψ(t) = e -i(Et+∆(t)/2) 0 0 e i(Et+∆(t)/2) Ψ(t),
we obtain

i dΨ ds = α -v(s) u(s) u(s) -α + v(s) Ψ. ( 9
)
where v(t) := ∆ (t)/2. Notice that the eigenvalues of the matrix in equation ( 9), seen as functions of the pair (u, v), coincide if and only if u = 0 and v = α, where a conical eigenvalue intersection occurs. Fix now v 0 < α 0 and v 1 > α 1 and consider a smooth path t → (u(t), v(t)) lying in the half-plane u > 0 except for the initial and final points, where u = 0 (see Figure 1).

Define

u ε (t) = u(εt), v ε (t) = v(εt).
Since the eigenvalues of the Hamiltonian in Equation ( 9) are the adiabatic theorem (see, e.g., [START_REF] Teufel | Adiabatic perturbation theory in quantum dynamics[END_REF]) ensures that, for ε > 0 small, the trajectory of ( 9) corresponding to (u ε , v ε ) and starting from (0, 1) stays close to the eigenvector associated with the negative eigenvalue. More precisely, we have the following estimate.

± (α -v(s)) 2 + u(s) 2 = 0,
Proposition 2. There exists C > 0 such that, for every α ∈ [α 0 , α 1 ] and every ε > 0, the solution Ψ of system (9) with initial condition (0, 1) and corresponding to the control

(u ε , v ε ) satisfies |Ψ(T /ε) -(e iθ , 0)| ≤ Cε for some θ ∈ R.
Going back to equation [START_REF] Beauchard | Controllability issues for continuousspectrum systems and ensemble controllability of Bloch equations[END_REF], the control corresponding to (u ε , v ε ) is

w ε (t) = u(εt)e i(Et+ ∆(εt) ε ) .
Such a law is called a (amplitude modulated) chirped pulse, since the range of frequency swept by the pulse is {2E + ∆ (s) | s ∈ [0, T ]}, which is independent of ε. For more details, see [START_REF] Augier | Adiabatic ensemble control of a continuum of quantum systems[END_REF].

Combination of RWA and AA and statement of the population inversion result

What one would like to do is to consider the two approximations in cascade, in order to induce a transition from the state (0, 1) to (1, 0) (up to a phase) for an ensemble of systems parameterized by α ∈ [α 0 , α 1 ] using a real-valued external field. The cascade of the two approximations is expected to behave well in many experimental setups, such as in NMR, due to the separation of timescales between the RWA and the AA. However, for intense external fields or in presence of large parametric dispersions, the outcome of the cascade is more challenging to predict and quantify precisely. Let us denote by ε 1 the small parameter that in the RWA was called ε and by ε 2 what in the AA was called ε. A formal cascade of the two approaches yields a control law of the form

w ε 1 ,ε 2 (t) = 2ε 1 u(ε 1 ε 2 t) cos 2Et + ∆(ε 1 ε 2 t) ε 2 ,
where u(•) and v(•) are the same functions as those used in Proposition 2.

The hope is that the pulse w ε 1 ,ε 2 , for ε 1 and ε 2 small, induces approximately a transition from the state (0, 1) to a state of the form (e iθ , 0) in time T /(ε 1 ε 2 ). The two approximations are, however, competing: when one decreases ε 2 (better AA), one needs the RWA to be true for a longer time as the final time is of order 1/(ε 1 ε 2 ). On the other hand, decreasing ε 1 deteriorates the performances of the AA:

1. The error on the adiabatic theorem depends of the gap between the eigenvalues, which goes to zero as ε 1 → 0;

2. The range of frequencies swept by the pulse is {2E + ε 1 ∆ (s) | s ∈ [0, T ]}, that is, the allowed dispersion on the frequency is shrinking as ε 1 goes to zero.

As a consequence, this method can only work when α = 0. Under this restriction, and for suitable relations between ε 1 and ε 2 as they both go to zero, the cascade of the two approximations can be proved to work (see [START_REF] Augier | On the compatibility between the adiabatic and the rotating wave approximations in quantum control[END_REF][START_REF] Augier | Effective adiabatic control of a decoupled hamiltonian obtained by rotating wave approximation[END_REF]). Another possibility would be to fix ε 1 small and to hope that the limit as ε 2 → 0 makes the RWA work as well. Nevertheless, the k-th order RWA is usually valid up to a time of order 1 ε k 1 , whereas we would need the time to be of order 1 ε 1 ε 2 . In fact, without restriction on the allowed frequency, simulations show that convergence does not hold, as illustrated in Figure 2. An approach to tackle the issue of the shrinking interval of frequencies swept by the pulse is to divide ∆(ε 1 ε 2 t) by ε 1 ε 2 and not just by ε 2 . We claim that an external field of the type

w ε 1 ,ε 2 (t) = 2ε 1 δu(ε 1 ε 2 t) cos 2Et + ∆(ε 1 ε 2 t) ε 1 ε 2 , ( 10
)
where δ is a positive constant, can induce a transition for the robust population transfer problem, provided that the relative order between ε 1 and ε 2 satisfies some suitable constraint as both parameters go to zero and under some further assumptions on the range [α 0 , α 1 ]. This is detailed in the following theorem.

Theorem 3. Assume that v 0 < 0 < v 1 are such that 3(E + v 0 ) ≥ E + v 1 . Fix T > 0 and u, ∆ : [0, T ] → R smooth (e.g., u ∈ C 2 and ∆ ∈ C 3 ) such that 1.
(u(0), ∆ (0)) = (0, 2v 0 ) and (u(T ), ∆ (T )) = (0, 2v 1 );

2. ∀s ∈ (0, T ), u(s) > 0 and ∆ (s) ≥ 0.

Denote by ψ α ε 1 ,ε 2 the solution of (1) with initial condition ψ α ε 1 ,ε 2 (0) = (0, 1) and control w ε 1 ,ε 2 as in [START_REF] Jo | Robust two-level system control by a detuned and chirped laser pulse[END_REF]. Then, for every N 0 ∈ N, for every compact interval I ⊆ (v 0 , v 1 ), there exist C N 0 > 0 and η > 0 such that for every α ∈ I and every (ε

1 , ε 2 ) ∈ (0, η) 2 , |ψ α ε 1 ,ε 2 ( 1 ε 1 ε 2 ) -(e iθ , 0)| < C N 0 max(ε 2 /ε 1 , ε N 0 -1 1 /ε 2 )
for some θ ∈ R. Moreover, the constant C N 0 can be taken locally uniform with respect to the parameter δ > 0 appearing in [START_REF] Jo | Robust two-level system control by a detuned and chirped laser pulse[END_REF].

Roughly speaking, ε 2 /ε 1 is the AA error and ε N 0 -1 1 /ε 2 the RWA error. We define the fidelity of a pulse as the quantity inf

θ |ψ α ε 1 ,ε 2 ( 1 ε 1 ε 2 ) -(e iθ , 0)| (also denoted | ψ α ε 1 ,ε 2 ( 1 ε 1 ε 2 )|e 1 |).
It is a natural measure of the transition rate induced by a pulse. Thus, by playing on the integer N 0 and on the order of magnitude between ε 1 and ε 2 , we can express the fidelity attained by the strategy above in terms of the duration of the pulse.

Corollary 4. Taking ε 1 = ε 2/N 0 2 (N 0 ≥ 3) leads to an error of the order T 2/N 0 -1 1+2/N 0 , where T = 1/(ε 1 ε 2 ) is the duration of the pulse w ε 1 ,ε 2 .
Remark 5.

• As an example, one can apply Theorem 3 with

T = 1, δ = 1, ∆(s) = v 0 -v 1 π sin(πs) + (v 0 + v 1 )s and u(s) = 1 -cos(2πs), s ∈ [0, 1]. More explicitly, w ε 1 ,ε 2 (t) = 2ε 1 (1 -cos(2πε 1 ε 2 t)) cos 2Et + (v 0 -v 1 ) sin(πε 1 ε 2 t) πε 1 ε 2 + (v 0 + v 1 )t .
All the simulations in this paper use this pulse scheme and some compare to the complexvalued pulse

w R ε 1 ,ε 2 (t) = ε 1 (1 -cos(2πε 1 ε 2 t)) exp 2iEt + i (v 0 -v 1 ) sin(πε 1 ε 2 t) πε 1 ε 2 + i(v 0 + v 1 )t .
• By taking N 0 large, one can get, for each η > 0, a fidelity close to one at order T -1+η , to compare with the standard O(T -1 ) of the adiabatic theorem.

ε 1 = 1, ε 2 = 0.1, v 0 = -0.5, v 1 = 0.5. Assumption 4E + 3∆ (s) -2α < 0 is satisfied if and only if α < 0.
• The assumption 3(E + v 0 ) ≥ E + v 1 ensures non-overlapping of some characteristic frequencies (cf. Lemma 25). It could be replaced by the weaker one: 4E + 3∆ -2α > 0 for every α ∈ [α 0 , α 1 ] and everywhere in [0, T ]. Nevertheless, asking this condition to be valid for every compact subinterval [α 0 , α1 ] of (v 0 , v 1 ) is equivalent to the inequality

3(E + v 0 ) ≥ E + v 1 .
Numerical simulations suggest that the inequality 4E + 3∆ -2α > 0 is sharp in the following sense: if for a given α, 4E + 3∆ (s) -2α < 0 for some s ∈ [0, T ], an inequality as in Theorem 3 seem not to hold. As an illustration, in Figure 3 we observe that for α ≥ 0 (condition 4E + 3∆ -2α>0 not satisfied), the accuracy of the RWA is worse than for α < 0 (condition 4E + 3∆ -2α>0 satisfied).

Remark 6. Many questions concerning the combination of the RWA and AA remain open.

In particular we do not know if a version of Theorem 3 holds with ε 1 fixed, small enough, and ε 2 going to 0.

Concerning systems with higher number of levels (possibly infinite), we expect the techniques developed in this paper to work. Nevertheless, such an extension seems not trivial.

We postpone the proof of Theorem 3 to Section 3. This proof is technical and is sketched in Sections 3.1 and 3.2 (see also Remark 34).

Application to the ensemble control problem

We denote by σ x , σ y , σ z the Pauli matrices given by

σ x = 0 1 1 0 , σ y = 0 -i i 0 , σ z = 1 0 0 -1 , (11)
and by SU 2 the special unitary group of degree 2. We recall that its Lie algebra su 2 is generated by iσ x , iσ y , and iσ z .

There is a natural distance on SU 2 induced by the norm of endomorphism on C 2 , which we denote

• . Let v 0 < 0 < v 1 and 0 < δ m ≤ δ M . Let D = [v 0 , v 1 ] × [δ m , δ M ]
be the compact set of the dispersion parameters and endow

F := C 0 (D, SU 2 ) with the usual distance d F (f, g) := max d∈D f (d) -g(d) .
Li and Khaneja proved in [START_REF] Li | Ensemble control of Bloch equations[END_REF] the following ensemble operator controllability result.

Theorem 7 [START_REF] Li | Ensemble control of Bloch equations[END_REF]. For any control bound K > 0, any target distribution M F ∈ F, and any ε > 0, there exist some T > 0 and controls u, v ∈ L ∞ ([0, T ], [-K, K]) such that the solution of the equation

i d dt M (α, δ, t) = ((E + α)σ z + δu(t)σ x + δv(t)σ y )M (α, δ, t), M (α, δ, 0) = I 2 , ∀(α, δ) ∈ D ( 12
)
satisfies d F (M (•, •, T ), M F (•, •)) < ε.
Remark 8.

• The result was originally stated on SO 3 for the Bloch sphere, the extension to SU 2 stated in Theorem 12 is straightforward.

• This is a very strong ensemble controllability result, as it tackles the controllability of the semigroups.

We extend here this result to the problem of a qubit driven by a single real control, thus replacing Equation ( 12) by

i d dt M (α, δ, t) = ((E + α)σ z + δu(t)σ x )M (α, δ, t), M (α, δ, 0) = I 2 , ∀(α, δ) ∈ D. (13)
One of the key ingredients of the proof of Theorem 7 is the existence of an adiabatic pulse inducing a propagator U ∈ F such that max d∈D min θ∈[0,2π] U (d)(0, 1) T -(e iθ , 0) T is arbitrarily small. Theorem 3 ensures the following corollary.

Corollary 9. Suppose that 3(E + v 0 ) > E + v 1 .
Then, for any K > 0 and any ε > 0, there exist T > 0 and a control u

∈ L ∞ ([0, T ], [-K, K]) such that the solution of Equation (13) satisfies max (α,δ)∈D min θ∈[0,2π] M (α, δ, T )(0, 1) T -(e iθ , 0) T < ε.
Based on Corollary 9, we will prove the following result, which generalizes Theorem 7 under the extra assumption on the α-dispersion.

Theorem 10. Suppose that 3(E + v 0 ) > E + v 1 . Let > 0, M F ∈ F, and K > 0. Then there exist T > 0 and u ∈ L ∞ ([0, T ], [-K, K]) such that the solution of Equation (13) satisfies d F (M (•, •, T ), M F (•, •)) < ε.
The proof, sketched below, is an adaptation of the arguments used in [START_REF] Li | Ensemble control of Bloch equations[END_REF].

Let R = {M (•, •, T ) | T > 0, M is a solution of (13) for some u ∈ L ∞ ([0, T ], [-K, K])}.
It is clear that R and its closure R are semigroups of F. We have to prove that R = F. Lemma 11. For all t in R, (α, δ) → e -t(E+α)iσz is in R.

Proof. Using a null control in (13) during a time t ≥ 0, we get (α, δ) → e -t(E+α)iσz belongs to R.

Let us prove that the result also holds for t < 0. Set an arbitrary ε > 0. By Corollary 9, there exists

U ε ∈ R such that max d∈D min θ∈[0,2π] U ε (d)(0, 1) T -(e iθ , 0) T < ε.
Using Euler angle decomposition, there exist three functions a ε , b ε , c ε from D to [0, 2π] (not necessarily continuous) such that U ε (d) = e aε(d)iσz e bε(d)iσx e cε(d)iσz for every d ∈ D. In particular, max d∈D |b ε (d) -π| is of order ε, so that sup d∈D U ε (d) -Ũ ε (d) is also of order ε, where Ũ ε (d) := e aε(d)iσz e πiσx e cε(d)iσz . For all t > 0, we have that (α, δ) → e -t(E+α)iσz is in R, by using the control u ≡ 0. Using the relation e -πiσx e riσz e πiσx = e -riσz , r ∈ R, we deduce that Ũ ε (d)e -t(E+α)iσz Ũ ε (d) = e aε(d)iσz e -πiσx e cε(d)iσz e -t(E+α)iσz e a ε (d)iσz e πiσx e cε(d)iσz = e t(E+α)iσz , for every d = (α, δ) in D. This shows that (α, δ) → e t(E+α)iσz is at distance of order ε from an element of R, concluding the proof.

Lemma 12. Let u ∈ R. Then (α, δ) → e uδiσx is in R.
Proof. Consider first the case |u| ≤ K. Setting V n (α, δ) = e (-(E+α)iσz+uδiσx)/n , one can easily check that the sequence ((α, δ) → (V n (α, δ)e t(E+α)iσz/n ) n ) n∈N is in R and converges to (α, δ) → e uδiσx in F. This concludes the case |u| ≤ K. We deduce the general case using the fact that R is a semigroup.

Let g = {X ∈ C 0 (D, su 2 ) | ∀t ∈ R, e tX ∈ R}. (14)
Thus (α, δ) → δiσ x and (α, δ) → (E + α)iσ z belong to g. The space C 0 (D, su 2 ) has a natural addition, product, and Lie bracket. Moreover, it has the structure of Banach algebra using as norm the sup norm, denoted by | • | ∞ . Before concluding the proof of Theorem 10, let us to show that g is a Lie algebra by proving that it is stable by addition and Lie bracket.

Lemma 13. The set g defined in [START_REF] Maciel Neto | H 1 -control of an ensemble of half-spin systems replacing Rabi pulses by adiabatic following[END_REF] is stable under addition and Lie brackets:

[g, g] ⊂ g, g + g ⊂ g.
Proof. Pick X, Y ∈ g. Let us first prove that e t[X,Y ] ∈ R for every t ∈ R.

F (U ( s/n) n , e s[X,Y ] ) ≤ C(s/n) 3/2 (n -1) ≤ Cs 3/2 n -1/2 , s ∈ [0, 1].
As a consequence, e s[X,Y ] ∈ R for every s ∈ [0, 1]. Applying the same reasoning to -X instead of X, we get that e s[X,Y ] ∈ R for every s ∈ [-1, 1]. We conclude the proof of the stability under Lei bracket by using the semigroup structure of R.

Concerning the stability under addition, set V (s) = e sX e sY and notice that V (s) ∈ R for every s ∈ R. Noticing that ) , we deduce that e t(X+Y ) ∈ R for every t ∈ R.

V (t/n) n d F ---→ n→∞ e t(X+Y
Denote by ad X (Y ) = [X, Y ] the adjoint representation both in su 2 and in C 0 (D, su 2 ). We recall the Pauli matrices commutation laws

[iσ x , iσ y ] = -iσ z , [iσ y , iσ z ] = -iσ x , [iσ z , iσ x ] = -iσ y .
After some straightforward computations, one gets Finally, let ρ > 0 be such that (a 1 , a 2 , a 3 ) → e a 1 iσx+a 2 iσy+a 3 iσz is a diffeomorphism between a neighborhood of 0 in R 3 and the ball of radius ρ centered at I 2 in SU 2 . Then for every

ad 2l δiσx (ad 2k+1 (E+α)iσz (δiσ x )) = (-1) l+k (E + α) 2k+1 δ 2l+1 iσ y , ad 2l+1 δiσx (ad 2k+1 (E+α)iσz (δiσ x )) = (-1) l+k (E + α) 2k+1 δ 2l+2 iσ z , ad (E+α)iσz ad 2l δiσx (ad 2k+1 (E+α)iσz (δiσ x )) = (-1) l+k+1 (E + α) 2k+2 δ 2l+1 iσ x .
M F ∈ F such that d F (M F , I 2 ) < ρ there exist f 1 , f 2 , f 3 ∈ C 0 (D, R) such that M F (d) = e f 1 (d)iσx+f 2 (d)iσy+f 3 (d)iσz . Thus M F ∈ R.
Since R is a semigroup, we deduce that R is both open and closed in F, yielding that R = F. This concludes the proof of Theorem 10.

Proof of Theorem 3.1 A first change of variables

Let w ε 1 ,ε 2 be as in [START_REF] Jo | Robust two-level system control by a detuned and chirped laser pulse[END_REF]. In order to recast the equation

i d dt ψ = Hψ = ((E + α)σ z + w ε 1 ,ε 2 σ x )ψ
in the interaction frame, set

ψ I (t) = e i(E+α)σzt ψ(t), E 1 (t) = 2αt - ∆(ε 1 ε 2 t) ε 1 ε 2 , E 2 (t) = 4Et + 2αt + ∆(ε 1 ε 2 t) ε 1 ε 2 ,
and notice that

i d dt ψ I = H I ψ I ,
where

H I (t) = -(E + α)σ z + e i(E+α)σzt H(t)e -i(E+α)σzt = ε 1 u(ε 1 ε 2 t) 0 e iE 1 (t) + e iE 2 (t) e -iE 1 (t) + e -iE 2 (t)
0 .

We will assume without loss of generality that T = 1. For E ∈ R, define

A(E) = 0 e iE e -iE 0 , B(E) = 0 -ie iE ie -iE 0 . ( 15
)
In terms of these new notations, we can rewrite

H I (t) = ε 1 u(ε 1 ε 2 t)A(E 1 (t))+ε 1 u(ε 1 ε 2 t)A(E 2 (t)), t ∈ [0, 1 ε 1 ε 2 ].
In the usual first order RWA setting, one neglects the term containing the factor A(E 2 ), which is highly oscillating compared to the first one. A standard method to justify this, is to use a change of variables close to the identity (see, e.g., [START_REF] Rouchon | Quantum systems and control 1[END_REF] and [START_REF] Sanders | Averaging methods in nonlinear dynamical systems[END_REF]). Inspired by this, we introduce the notation

f 1 (t) = d dt E 1 (t), f 2 (t) = d dt E 2 (t), (16)
and we apply the unitary change of variables

ψI (t) = exp iε 1 u(ε 1 ε 2 t) f 2 (t) B(E 2 (t)) ψ I (t). ( 17
)
The dynamics of ψI are characterized by the Hamiltonian

HI (t) = i d dt cos ε 1 u(ε 1 ε 2 t) f 2 (t) I + i sin ε 1 u(ε 1 ε 2 t) f 2 (t) B(E 2 (t)) cos ε 1 u(ε 1 ε 2 t) f 2 (t) I -i sin ε 1 u(ε 1 ε 2 t) f 2 (t) B(E 2 (t)) + cos ε 1 u(ε 1 ε 2 t) f 2 (t) I + i sin ε 1 u(ε 1 ε 2 t) f 2 (t) B(E 2 (t)) H I cos ε 1 u(ε 1 ε 2 t) f 2 (t) I -i sin ε 1 u(ε 1 ε 2 t) f 2 (t) B(E 2 (t)) .
Notice that the first term can be rewritten as

-ε 1 u(ε 1 ε 2 t)A(E 2 (t)) + O(ε 2 1 ), so that HI (t) = ε 1 A(E 1 (t)) + O(ε 2 1 + ε 2 1 ε 2 )
, where the notation O(•) is defined as follows. Definition 14. Let R be a (ε 1 , ε 2 )-parameterized function in the following sense: for every Remark 15.

• We have |ψ I -ψI | = O(ε 1 ). Moreover, from the hypotheses of Theorem 3, we have u(0) = u(1) = 0, thus ψI (0) = ψ I (0) and ψI ( 1

ε 1 ε 2 ) = ψ I ( 1 ε 1 ε 2 ).
• Let ψ rwa be the solution of the Schrödinger equation with initial condition ψ I (0) and Hamiltonian ε 1 u(ε 1 ε 2 t)A(E 1 (t)). Then it turns out that |ψ rwa ( 1

ε 1 ε 2 )-ψI ( 1 ε 1 ε 2 )| = O(ε 1 /ε 2 ) (see Lemma 29).
To prove convergence as (ε 1 , ε 2 ) → 0 in a suitable asymptotic regime, it would thus be enough to show that the dynamics of ε 1 u(ε 1 ε 2 t)A(E 1 (t)) induce a transition between (0, 1) and (1, 0) up to a phase, in the regime ε 1 ε 2 . Nevertheless this is not the case (recall that 'standard' adiabatic theorem cannot be applied since ε 1 is not fixed) as illustrated in Figure 4.

Figure 4: Taking v 0 = -0.5, v 1 = 0.5, E = 0, and α = 0, we observe that the fidelity does not converge to 1 as (ε 1 , ε 2 ) → 0 in the regime ε 1 ε 2 . The plot corresponds to the choice

ε 1 = ε 2 2 .

Idea of the proof

We aim at providing correction terms to the Hamiltonian ε 1 u(ε 1 ε 2 t)A(E 1 (t)), in order to improve the order of the averaging approximation. For this we will repeat a procedure similar to the one introduced in Equation ( 17). At each step the expression of the obtained effective Hamiltonian is more complicated but provides a more accurate estimate of the final state. Then it will be possible to apply adiabatic theory to prove transition for the effective Hamiltonian. More precisely, we will prove the following theorem.

Theorem 16. Let α ∈ (v 0 , v 1 ) and assume that E + α > 0 and 4E -3∆ (s) > 2α for every s ∈ [0, 1]. Then, for every N 0 ∈ N there exists a Hamiltonian H RWA of the form

H RWA (t) = ε 1 h 1 (ε 1 ε 2 t)A(E 1 (t)) + ε 2 1 h 2 (ε 1 ε 2 t)B(E 2 (t)) + ε 2 1 h 3 (ε 1 ε 2 t)σ z , (18)
with h 1 , h 2 , h 3 polynomials in (ε 1 , ε 2 ) with coefficients in C ∞ ([0, 1], R), such that the solution ψ RWA of the Cauchy problem

i d dt ψ RWA = H RWA ψ RWA , ψ RWA (0) = ψ I (0), satisfies |ψ RWA ( 1 ε 1 ε 2 ) -ψ I ( 1 ε 1 ε 2 )| = O(ε 2 1 ε 2 + ε N 0 -1 1 /ε 2 )
. More precisely, there exist h j,p,q ∈ C ∞ ([0, 1], R), for j = 1, 2, 3, p = 0, . . . , N 0 , and q = 0, 1, such that

1. h 1 = u + N 0 p=1 1 q=0 ε p 1 ε q 2 h 1,p,q with h 1,p,0 (0) = h 1,p,0 (1) = 0, 2. h 2 = N 0 p=0 1 q=0 ε p 1 ε q 2 h 2,p,q with h 2,p,0 (0) = h 2,p,0 (1) = 0, 3. h 3 = N 0 p=0 1 q=0 ε p 1 ε q 2 h 3,p,q with h 3,p,0 (0) = h 3,p,0 (1) = 0.
After that, we will prove that H RWA induces a transition between eigenstates with an error of order O(ε 2 /ε 1 ), which will be enough to prove Theorem 3.

The rotating wave approximation

Definition 17. Define the algebra S of slow functions as the set of all (ε 1 , ε 2 )-parameterized functions f (in the sense of Definition 14) such that for every t ∈ [0,

1 ε 1 ε 2 ], f ε 1 ,ε 2 (t) = g(ε 1 ε 2 t) for some smooth g : [0, 1] → R independent of (ε 1 , ε 2 ). The quantity sup t∈[0, 1 ε 1 ε 2 ] |f ε 1 ,ε 2 (t)| is independent of (ε 1 , ε 2
) and provides a norm, endowing S with the structure of Banach algebra.

Remark 18.

• The functions f 1 and f 2 defined in (16) are slow.

• S is isometric to the Banach algebra C ∞ ([0, 1], R).

• Given f ∈ S, its t-derivative ḟ defined by ḟε 1 ,ε

2 (t) = d dt f ε 1 ,ε 2 (t) is such that 1 ε 1 ε 2 ḟ ∈ S.
For every j ∈ Z, let us introduce the notations

Λ j = (j + 1)E 1 -jE 2 , φj = jE 1 -jE 2 , λ j = (j + 1)f 1 -jf 2 , φ j = jf 1 -jf 2 . ( 19
)
Definition 19. Define the set

G = {±Z(Λ p ), ± cos(Φ p )σ z , ± sin(Φ p )σ z | Z ∈ {A, B}, p ∈ Z}.
We say that an element of G is oscillating if its associated integer p is different from 0.

Lemma 20. G has the following stability properties:

1. ∀p ∈ Z, ∀X ∈ G, cos(Φ p )X and sin(Φ p )X are in span R G;

2. ∀X, Y ∈ G, i[X, Y ] ∈ span R G; 3. ∀X, Y ∈ G, XY X ∈ span R G.
Proof. The first point is a consequence of the fact that {Φ p | p ∈ Z} is a group for the addition. Thus, for every p, q ∈ Z,

2 cos(Λ p ) cos(Φ q )σ z = cos(Φ p + Φ q )σ z + cos(Φ p -Φ q )σ z ∈ span R G.
Moreover, 2 cos(Φ p )A(Λ q ) = A(Λ q+p ) + A(Λ q-p ) ∈ span R G. The remaining cases can be checked similarly.

For the second point, for every E, E , E ∈ R,

i[A(E), A(E )] = -2 sin(E -E )σ z , i[A(E), cos(E )σ z )] = 2 cos(E )B(E) = B(E + E ) + B(E -E ), i[cos(E )σ z , cos(E )σ z ] = 0.
Using the fact that, for every p ∈ Z, A(Λ p -π/2) = B(Λ p ), we obtain that i[A(Λ p ), G] ∈ span R G. Similar results can easily be obtained for B(Λ p ), cos(Φ p )σ z , and sin(Φ p )σ z .

The last point relies on the relations

A(E)A(E )A(E) = A(2E -E ), cos(E )A(E)σ z A(E) = -cos(E )σ z , 2 cos 2 (E )σ z A(E)σ z = A(E) + 1 2 (A(E + 2E ) + A(E -2E )), 2 cos 2 (E ) cos(E )σ 3 z = cos(E ) + 1 2 (cos(2E + E ) + cos(2E -E )) σ z .
Definition 21. Define the vector space E as the set of entire series in (ε 1 , ε 2 ) with coefficients in the set span S G, i.e., j,k≥0

ε j 1 ε k 2 g∈H j,k s g g | H j,k ⊂ G finite, s g ∈ S, j,k≥0 ε j 1 ε k 2 g∈H j,k |s g | < ∞ for (ε 1 , ε 2 ) small enough .

The elimination procedure

In order to generalize [START_REF] Rouchon | Quantum systems and control 1[END_REF], we introduce the operation of elimination of an oscillating term of a coefficient of E.

(E = Λ p if Z ∈ {A, B} or E = Φ p if Z(E) ∈ {cos(E)σ z , sin(E)σ z }). Suppose that f = Ė (which is necessarily slow) is nowhere vanishing. Fix j ≥ 1, k ≥ 0, s ∈ S and let c = ε j 1 ε k 2 s.
The operation of elimination of cZ(E) from H is defined as

El(c, Z(E))(H) = i d dt exp i(c/f )Pr(Z)(E) exp -i(c/f )Pr(Z)(E) + exp i(c/f )Pr(Z)(E) H exp -i(c/f )Pr(Z)(E) , (20)
El(c, Z(E))(ψ) = exp ic/f Pr(Z)(E) ψ.

In fact, the elimination procedure is the generalization of the change of variables in Equation [START_REF] Rouchon | Quantum systems and control 1[END_REF]. It transforms the Hamiltonian dynamics i d dt ψ = Hψ into the dynamics i d dt η = El(c, Z(E))(H)η, where η = El(c, Z(E))ψ. The term elimination is motivated by the following lemma, stating that the procedure described above generates in the transformed Hamiltonian only terms of degree higher than ε j 1 ε k 2 .

Lemma 24. Take H, Z(E), j, k, c as in Definition 23. Then El(c,

Z(E))(H) ∈ E. Besides, if H = O(ε 1 ) then El(c, Z(E))(H + cZ(E)) = H + O(ε j+1 1 ε k 2 ).
Proof. First recall that for each matrix M such that M 2 = I and each c ∈ R, exp(icM ) = cos(c)I + i sin(c)M . As A(E) 2 = B(E) 2 = σ 2 z = I, we can give an explicit expression for El(c, Z(E))(H).

Let us start from the case Z(E) = A(Λ p ), for which we have

El(c, A(Λ p ))(H) = J 1 + J 2 + J 3 , (21)
where

J 1 = i d dt (c/f ) -sin(c/f )I + i cos(c/f )B(Λ p ) cos(c/f )I -i sin(c/f )B(Λ p ) = - d dt (c/f )B(Λ p ), J 2 = -sin(c/f ) d dt B(Λ p ) cos(c/f )I -i sin(c/f )B(Λ p ) = -f sin(c/f )A(Λ p ) cos(c/f )I -i sin(c/f )B(Λ p ) , J 3 = cos(c/f )I + i sin(c/f )B(Λ p ) H cos(c/f )I -i sin(c/f )B(Λ p ) .
The term J 1 is obviously an element of E. Besides, cos(c/f ) and sin(c/f ) are entire series in ε 1 , ε 2 with coefficients in S. Thus,

J 2 = -f sin(c/f ) cos(c/f )A(Λ p ) -f sin 2 (c/f )σ z
is also an element of E. The last term to be considered is

J 3 = cos 2 (c/f )H + cos(c/f ) sin(c/f )i[B(Λ p ), H] + sin 2 (c/f )B(Λ p )HB(Λ p ).
Thanks to Lemma 20, J 3 is then the sum of elements of E.

Let us now assume that H = O(ε 1 ) and focus on the order of each term (in the case

Z(E) = A(Λ p )). We notice that J 1 = O(ε j+1 1 ε k+1 2 ) as d dt (s/f ) = O(ε 1 ε 2 ) and J 2 = -cA(Λ p )+O(ε j+1 1 ε k 2 ). Finally, J 3 = H + (c/f )i[B(Λ p ), H] + O(ε j+1 1 ε k 2 ). As H = O(ε 1 ), we get (c/f )i[B(Λ p ), H] = O(ε j+1 1 ε k 2 ). Thus El(c, Z(E))(H + cZ(E)) = -cZ(E) + H + cZ(E) + O(ε j+1 1 ε k 2 ).
The same computations as above work for the case Z(E) = B(Λ p ).

In the case Z(E) = cos(Φ p )σ z we have

El(c, cos(Φ p )σ z )(H) = J 1 + J 2 + J 3 , (22)
where

J 1 = - d dt (c/f ) sin(Φ p )σ z , J 2 = -c cos(Φ p )σ z , J 3 = (cos(c/f sin(Φ p ))I + i sin(c/f sin(Φ p ))σ z )H(cos(c/f sin(Φ p ))I -i sin(c/f sin(Φ p ))σ z ).
Note that sin(c/f sin(Φ p )) and cos(c/f sin(Φ p )) can be developed as entire series in ε 1 , ε 2 with coefficients in S cos(Φ q ) and S sin(Φ q ) for q ∈ Z. Lemma 20 ensures that El(c, cos(Φ p )σ z )(H) is an element of E. The computations of the order of the terms when H = O(ε 1 ) are similar to those made above, and one can apply the same reasoning to El(c, sin(Φ p )σ z )(H).

A key assumption of Lemma 24 above is that f is nowhere vanishing. The following result ensures that this is the case for all frequencies of the oscillating terms in G.

Lemma 25. Let j ∈ Z be nonzero. Then the functions λ j and φ j , defined as in [START_REF] Scheuer | Precise qubit control beyond the rotating wave approximation[END_REF], are nowhere vanishing in [0, 1

ε 1 ε 2 ]. Proof. Let us first prove that 2f 1 (t) < f 2 (t), ∀t ∈ 0, 1 ε 1 ε 2 , (23)
where we recall that f 1 , f 2 are defined in [START_REF] Mittleman | Introduction to the theory of laser-atom interactions[END_REF]. Indeed,

2f 1 (t) -f 2 (t) = 2α -4E -2∆ (ε 1 ε 2 t) -∆ (ε 1 ε 2 t) < 2v 1 -4E -6v 0 ,
where we used the inequality α < v 1 and the fact that, according to the hypotheses of Theorem 3, ∆ is increasing from 2v 0 to 2v 1 . The inequality 2v 1 -4E -6v 0 = 2(E +v 1 )-6(E +v 0 ) ≤ 0, corresponding to the assumption 3(E + v 0 ) ≥ E + v 1 of Theorem 3, concludes the proof of [START_REF] Simon | Robust quantum dot exciton generation via adiabatic passage with frequency-swept optical pulses[END_REF]. Moreover,

f 2 (t) = 4E + 2α + ∆ (t) ≥ 4(E + v 0 ) ≥ 4(E + v 1 ) 3 > 0, ∀t ∈ 0, 1 ε 1 ε 2 .
In particular,

f 1 -f 2 = -4E -2∆ ≤ -4(E + v 0 ) < 0.
This implies that φ j never vanishes for j = 0. Finally, for j > 0,

λ j = (j + 1)f 1 -jf 2 = (j -1)(f 1 -f 2 ) + 2f 1 -f 2 < 0, and, similarly, λ j = (j + 1)(f 1 -f 2 ) + f 2 > 0 for j < 0.

Algorithm description

We can now introduce an algorithm to simplify the Hamiltonian H I . The cleaning operation cl p(p 0 , q 0 ), with p 0 ≤ p, consists in eliminating from H I all oscillating terms of degree ε p 1 ε q 2 for p ≤ p q < q 0 and p ≤ p 0 q = q 0 in lexicographic order on (p, q). The algorithm is constructed by induction, as follows:

4. H r,N 0 is an entire series in ε 1 with coefficients in span S G, 5. H r is an entire series in ε 1 , ε 2 with coefficients in span S G.

Proof. Points 1 and 3 follow from Lemma 27, while points 2, 4, and 5 follow from Lemma 24.

Noticing that, in particular, cl

N 0 (N 0 , 1) = ε 1 H N 0 + ε 2 1 ε 2 H N 0 + O(ε 3 1 ε 2 2 + ε N 0 1 ), we introduce the truncation H RWA = ε 1 H N 0 + ε 2 1 ε 2 H N 0 of cl N 0 (N 0 , 1) and we denote by ψ RWA the solution of i d dt ψ RWA = H RWA ψ RWA , ψ RWA (0) = ψ N 0 (0), (24)
where ψ N 0 = cl N 0 (N 0 , 1). Notice that, even if we are using the same notation ψ RWA , we are considering here a RWA of higher-order than the one discussed in Remark 15.

Lemma 29. We have the following estimates:

1. |ψ N 0 ( 1 ε 1 ε 2 ) -ψ I ( 1 ε 1 ε 2 )| = O(ε 2 1 ε 2 ); 2. |ψ N 0 ( 1 ε 1 ε 2 ) -ψ RWA ( 1 ε 1 ε 2 )| = O(ε 2 1 ε 2 + ε N 0 -1 1 /ε 2 ).
Proof. By Lemma 27, all the changes of variable used for obtaining cl N 0 (N 0 , 0) from

H I are of the form El(c, Z(E))(ψ) with c = ε p 1 s, s ∈ S 0 . Thus ψ I (0) = cl(N 0 , 0)(0) and ψ I ( 1 ε 1 ε 2 ) = cl N 0 (N 0 , 0)( 1 ε 1 ε 2 )
. Such changes of variable preserve the state at the initial and final time. After that we applied finitely many changes of variable of the form ψ → exp(iε p 1 ε q 2 sZ(E))ψ with p ≥ 2 and q = 1. Thus

sup t∈[0, 1 ε 1 ε 2 ] | cl N 0 (N 0 , 0)(t) -ψ N 0 (t)| = O(ε 2 1 ε 2 ), (25)
which concludes the proof of the first estimate. Notice that

d dt |ψ N 0 -ψ RWA | 2 = 2 Re i ψ N 0 -ψ RWA |cl N 0 (N 0 , 1)ψ N 0 -H RWA ψ RWA = 2 Re(i ψ N 0 -ψ RWA |H RWA (ψ N 0 -ψ RWA ) + i ψ N 0 -ψ RWA |(cl N 0 (N 0 , 1) -H RWA )ψ N 0 ) ≤|ψ N 0 -ψ RWA |O(ε 3 1 ε 2 2 + ε N 0 1 ).
Thus,

2 d dt |ψ N 0 -ψ RWA | ≤ O(ε 3 1 ε 2 2 + ε N 0 1 ),
and we conclude by integrating over [0, 1 ε 1 ε 2 ]. This concludes the proof of Theorem 16.

Two scales adiabatic approximation

The goal of this part is to prove the following lemma: Lemma 30. There exists δ > 0 such that the solution ψ RWA of [START_REF] Teufel | Adiabatic perturbation theory in quantum dynamics[END_REF]

satisfies |ψ RWA ( 1 ε 1 ε 2 ) - (e iθ , 0)| ≤ M ε 2 /ε 1 for some θ ∈ R (possibly depending on ε 1 , ε 2 , α) for (ε 1 , ε 2 ) ∈ (0, δ) 2 .
With a slight abuse of notation, let us say in this section that a (ε 1 , ε 2 )-parametric function f is a O(g(ε 1 , ε 2 )) (respectively, a Ω(g(ε 1 , ε 2 ))) if there exist M, δ > 0 such that [START_REF] Vitanov | Coherent manipulation of atoms and molecules by sequential laser pulses[END_REF] Recall that

∀ε 1 , ε 2 ∈ (0, δ) 2 , ∀s ∈ [0, 1], |f ε 1 ,ε 2 (s)| ≤ M g(ε 1 , ε 2 ) (respectively, |f ε 1 ,ε 2 (s)| ≥ M g(ε 1 , ε 2 )).
H RWA (t) = ε 1 h 1 (ε 1 ε 2 t)A(E 1 (t)) + ε 2 1 h 2 (ε 1 ε 2 t)B(E 2 (t)) + ε 2 1 h 3 (ε 1 ε 2 t)σ z ,
with h 1 , h 2 , and h 3 given by Theorem 16. We introduce the unitary change of variables ψ slow (t) = U (t)ψ RWA (t) with

U (t) = e i(αt- ∆( 1 2 t) 2 1 2 ) 0 0 e -i(αt- ∆( 1 2 t) 2 1 2
)

.

The notation ψ slow is motivated by the fact that the Hamiltonian corresponding to its evolution is slow in the sense that it only depends on the slow variable s = ε 1 ε 2 t, also known as macroscopic or reduced time. More precisely, i d dt ψ slow (t) = H slow (ε 1 ε 2 t)ψ slow (t), where

H slow (s) = ε 1 h 1 (s)σ x + ε 2 1 h 2 (s)σ y + α - ∆ (s) 2 + ε 2 1 h 3 (s) σ z . (27)
We cannot directly apply a 'standard adiabatic theorem' to describe the evolution of ψ slow because the adiabatic path depends on (ε 1 , ε 2 ).

The eigenvalues of H slow (s) are

±ω ε 1 ,ε 2 (s) = ± (ε 1 h 1 (s)) 2 + (ε 2 1 h 2 (s)) 2 + (α -∆ (s)/2 + ε 2 1 h 3 (s)) 2 , s ∈ [0, 1].
Using a Taylor series development, we have ω ε 1 ,ε 2 = Ω(ε 1 ). Thus, for (ε 1 , ε 2 ) small enough, ω ε 1 ,ε 2 does not vanish. As a consequence, we can introduce the spectral projector

P ε 1 ,ε 2 (s) of H slow (s) on the negative eigenvalue. Consider γ ε 1 ,ε 2 : [0, 1] → S 2 such that H slow (s) = ω ε 1 ,ε 2 (s)γ ε 1 ,ε 2 (s) • σ where a • σ = a 1 σ x + a 2 σ y + a 3 σ z .
We want to approximate P ε 1 ,ε 2 and its derivatives by the spectral projector on the negative eigenvalue for the simplified Hamiltonian Hslow = ε 1 uσ x + (α -∆ /2)σ z and its derivatives.

Lemma 31. Let -ω ε 1 (s) be the negative eigenvalue of the Hamiltonian Hslow (s) = ε 1 u(s)σ x + (α -∆ (s)/2)σ z and Pε 1 (s) be the spectral projector on -ω ε 1 (s), s ∈ [0, 1]. Then

• |P ε 1 ,ε 2 -Pε 1 | = O(ε 1 ), • | d ds P ε 1 ,ε 2 -d ds Pε 1 | = O(1), • | d 2 ds 2 P ε 1 ,ε 2 -d 2 ds 2 Pε 1 | = O(1/ε 1 ).
Proof. First, remark that for every nonnegative integer ν

d ν ds ν H slow (s) = d ν ds ν Hslow (s) + d ν ds ν R(ε 1 , ε 2 , s), (28)
where

d ν ds ν R(ε 1 , ε 2 , s) = O(ε 2 1
). For H ∈ isu 2 \ {0}, define the orthogonal projector P (H) as the projector on the negative eigenvalue of H. The map P is C ∞ and positively homogeneous of degree 0 on isu 2 \ {0}.

For every r > 0, let B r be the Euclidean ball of center 0 and radius r in M 2 (C). Denote by K the compact set isu 2 ∩ ∂B 1 . The differential dP is positively homogeneous of degree -1,

since ∀H, h ∈ isu 2 , dP H (h) = dP H |H| h |H| .
As a consequence, for

H ∈ su 2 \ B r , |dP H | ≤ sup L∈K |dP L | r
. Thus, there exists a universal constant C > 0 such that P is

C r -Lipschitz continuous on isu 2 \ B r . Moreover, consider r(ε 1 , ε 2 ) := inf s∈[0,1] ω ε 1 ,ε 2 (s)/2 = Ω(ε 1 ). As H slow -Hslow = O(ε 2
1 ), for ε 1 , ε 2 small enough we can assume that the segment [H slow (s), Hslow (s)] ∩ B r(ε 1 ,ε 2 ) is the empty set for every s ∈ [0, 1]. Then, applying P to the equality (28) for ν = 0, we obtain

|P ε 1 ,ε 2 (s) -Pε 1 (s)| ≤ C r(ε 1 , ε 2 ) |R(ε 1 , ε 2 , s)| ≤ M ε 1 , ∀s ∈ [0, 1].
For the second point, we have The third point is obtained by the same kind of argument.

Remark 32. The Hamiltonian Hslow (s) = ε 1 u(s)σ x + (α -∆ (s)/2)σ z is given by the first order RWA. The fact that ε 1 appears in front of the pulse is obviously of utter importance for the estimation of the RWA error but also means that the 'adiabatic path' is shrinking to the conical eigenvalue intersection. In fact, it is worse than just the shrinking of the spectral gap, as the derivative of the spectral projector is blowing up near the conical intersection (see Figures 5 and6).

Define γε 1 (s), s ∈ [0, 1], by the relation Hslow (s) = ωε 1 (s)γ ε 1 (s) • σ and denote by ( θ, φ) the spherical coordinates of γε 1 . Hence X = sin( θ/2)e 1 -e i φ cos( θ/2)e 2 is an eigenvector of Hslow (s) associated with the negative eigenvalue and Pε 1 = sin 2 ( θ/2) -e -i φ sin( θ/2) cos( θ/2) -e i φ sin( θ/2) cos( θ/2) cos 2 ( θ/2) . Lemma 33. Under the assumptions of Theorem 3, we have:

1. |P ε 1 ,ε 2 (0) -P e 1 | = O(ε 2 1 ε 2 ) and |P ε 1 ,ε 2 (1) -P e 2 | = O(ε 2 1 ε 2 )
, where P e i is the orthogonal projector on Ce i ; 2.

1 0 |( d ds P ε 1 ,ε 2 )(s)| 2 ds = O(1/ε 1 ); 3. 1 0 |( d 2 ds 2 P ε 1 ,ε 2 )(s)|ds = O(1/ε 1 ); 4. 1 0 | 1 ωε 1 ,ε 2 (s) 2 d ds P ε 1 ,ε 2 (s)|| d ds H slow (s)|ds = O(1/ε 2 1
). Proof. Point 1 is a simple consequence of points 1, 2 and 3 in Theorem 16.

Concerning the other three points, thanks to Lemma 31 we are left to prove the corresponding estimates for Pε 1 ,ε 2 and Hslow . We recall that θ(s) = arccos (α -∆ (s)/2)/ω ε 1 (s) . We can bound the transverse velocity of γε 1 (s) by its total velocity ωε 1 (s)

d ds θ(s) ≤ (ε 1 d ds u(s)) 2 + (∆ (s)/2) 2 = O(1), thus d ds θ = O(1/ε 1 ). Using formula (29), it is clear that | d ds Pε 1 (s)| | d ds θ| and | d 2 ds 2 Pε 1 (s)| | d ds θ| 2 +| d 2 ds 2 θ|
, where stands for inequality up to an universal multiplicative constant. As ∆ ≥ 0, θ is increasing and

1 0 d ds θ(s) 2 ds ≤ sup s∈[0,1] d ds θ(s) 1 0 d ds θ(s) ds ≤ π sup s∈[0,1] d ds θ(s) = O(1/ε 1 ).
Moreover, bounding the transverse acceleration of γε 1 (s) by its total acceleration, we have ds , for some θ ∈ R. Finally, Lemma 30 together with Theorem 16 conclude the proof of Theorem 3 for a given α and δ. To get uniformity on the range of α, notice that the algorithm does not depend on α elsewhere than in the expression of E 1 and E 2 (see [START_REF] Mittleman | Introduction to the theory of laser-atom interactions[END_REF]). For the adiabatic part, if we restrict α to a compact interval I ⊂ (v 0 , v 1 ), the estimates of Lemma 33 can be taken uniform with respect to α. The uniformity with respect to δ is straightforward.

2 d ds ωε 1 (s) d ds θ(s) -ωε 1 (s) d 2 ds 2 θ(s) ≤ (ε 1 d 2 ds 2 u(s)) 2 + (∆ (s)/2) 2 = O(1
Remark 34. Now that we have detailed the whole proof, we want to stress some of its key points.

1. The changes of variables applied iteratively in order to eliminate the oscillating terms of the Hamiltonian induce a very small error (of order ε 2 ε 2 1 ) on the initial and the final state (Lemma 29), whereas the error is of order ε 1 if one look at the entire trajectory.

2. The frequencies which appear during the algorithm are of very special type (pf 1 -pf 2 and (p + 1)f 1 -pf 2 for p integer) allowing us to perform as many changes of variables as we need and to give a simple condition implying that all such frequencies are nonzero.

3. Each change of variables yields a more complicated Hamiltonian. Fortunately, when we study the adiabatic dynamics of such an Hamiltonian, we can neglect all the terms except for those appearing in the first order RWA.

4. The first order RWA induces a population transfer in the limit ε 2 ε 1 .

Numerical simulations

We present in this section some numerical simulations illustrating the results stated in Theorem 3. In all simulations we use the chirp scheme presented in Remark 5 with E = 1, v 0 = -0.5, and v 1 = 0.5. Figure 7 shows the behavior of the distance from the target state as a function of ε 1 , ε 2 represented in log scale. The AA error appears clearly, reflecting the fact that one needs ε 2 ε 1 in order to have a fidelity close to 1. The figure also shows that the strategy has better performances than those anticipated theoretically in Theorem 3.

Figure 8 shows the fidelity as a function of α, while ε 1 and ε 2 (and hence T ) are fixed.

Figure 9 shows the fidelity as a function of the reduced time for three values of α, while ε 1 and ε 2 (and hence T ) are fixed. We clearly see that the RWA produces large oscillations (of magnitude of order ε 1 ), which become much smaller at the endpoints, as described in Remark 34, point 3.

Finally, Figure 10 illustrates the conflict between the AA and RWA. At T = 0.05 fixed, for smaller ε 1 we observe that the RWA is more accurate as the thick line (1st order RWA) is closer to the highly oscillating one (the trajectory ψ 0 ε 1 ,ε 2 ). Nevertheless as ε 1 decreases, the ratio ε 2 /ε 1 increases and the AA becomes less accurate.

Figure 1 :

 1 Figure 1: An adiabatic path as the one applied in Proposition 2.

Figure 2 :

 2 Figure 2: Comparison of the real-valued and complex-valued chirp scheme of the first point of Remark 5 with E = 0.75, α = 0.25, ε 1 = 1, v 0 = -0.5, v 1 = 0.5. Notice that the assumptions of Theorem 3 are not satisfied.

Figure 3 :

 3 Figure 3: E = 0.75, α = 0.25, ε 1 = 1, ε 2 = 0.1, v 0 = -0.5, v 1 = 0.5. Assumption 4E + 3∆ (s) -2α < 0 is satisfied if and only if α < 0.

  Thus for any n, m ∈ N, and any sequence (b k,l ) k,l of real numbers, we havem k=0 n l=0 b k,l δ 2k+2 (E + α) 2l+1 iσ x ∈ g, l δ 2k+1 (E + α) 2l+1 iσ y ∈ g, l δ 2k+1 (E + α) 2l+2 iσ z ∈ g.By the Stone-Weierstrass theorem, for any continuous function f ∈ C(D, R) we can approximate f (d) (E+α)δ 2 uniformly on D by polynomials of the form m k=0 n l=0 b k,l δ 2k (E + α) 2l . This proves that f (d)iσ x ∈ g. With a similar argument, we get f (d)iσ ∈ g for = x, y, z.

Definition 22 .

 22 Define the operation Pr : G → G by the relations Pr(±A(Λ p )) = ±B(Λ p ), Pr(±B(Λ p )) = ∓A(Λ p ), Pr(± cos(Φ p )σ z ) = ± sin(Φ p )σ z , and Pr(± sin(Φ p )σ z ) = ∓ cos(Φ p )σ z . Definition 23. Let H ∈ E and Z(E) be an oscillating term of G

)Figure 5 :Figure 6 :

 56 Figure 5: Eigendirection corresponding to the negative eigenvalue of Hslow as a function of (u, ∆ ) ∈ R 2 , for ε 1 = 0.01 and α = 0.

Figure 7 :

 7 Figure 7: Log of the distance fromψ 0 ε 1 ,ε 2 ( 1 ε 1 ε 2 )to the orbit of (1, 0).

Figure 8 :

 8 Figure 8: Population transfer as a function of α for E = 1, ε 1 = 0.5 and ε 2 = 0.1.

Figure 9 :

 9 Figure 9: ε 1 = 0.5, ε 2 = 0.1 and α = 0. In thick line are the trajectories corresponding to the equivalent 1st order RWA system.

Figure 10 :

 10 Figure 10: ε 1 ε 2 = 0.05, α = 0. In thick line are the trajectories corresponding to the equivalent 1st order RWA system and in dotted line the theoretical AA trajectories.

  As dP is positively homogeneous of degree -1,d 2 P is positively homogeneous of degree -2. Thus H → dP H is C r 2 -Lipschitz continuous on isu 2 \ B r with C = sup L∈K |d 2 P L |. Thus, for ε 1 , ε 2 small enough,

		dP H slow	d ds	H slow -dP Hslow	d ds	H slow ≤	C r(ε 1 , ε 2 ) 2 H slow -Hslow ,
	and	dP Hslow	d ds	H slow -dP Hslow	d ds	Hslow ≤	C r(ε 1 , ε 2 )	d ds	H slow -	d ds	Hslow .
	Thus we get
				dP H slow	d ds	H slow = dP Hslow	d ds	Hslow + O(1).

d ds P ε 1 ,ε 2 (s) = dP H slow (s) d ds H slow (s) .

  ). ,ε 2 (s)|| d ds H slow (s)|ds can be upper bounded, up to a multiplicative constant, by To conclude the proof of Lemma 30, we deduce from [24, Corollary 2.3] the adiabatic estimateψ α ε 1 ,ε 2 1 ε 1 ε 2 -(e iθ , 0) ≤ ε 1 ε 2 | d ds P ε 1 ,ε 2 (1)| ω ε 1 ,ε 2 (1) + | d ds P ε 1 ,ε 2 (0)| ω ε 1 ,ε 2 (0) 2| d ds P ε 1 ,ε 2 (s)| 2 ω ε 1 ,ε 2 (s) + | d 2 ds 2 P ε 1 ,ε 2 (s)| ω ε 1 ,ε 2 (s) + | d ds P ε 1 ,ε 2 (s)|| d ds H slow (s)| 2ω ε 1 ,ε 2 (s) 2

	As d ds ωε 1 (s) ≤ (ε 1	d ds u(s)) 2 + (∆ (s)/2) 2 = O(1), we have
		0	1	d ds	ωε 1 (s)	d ds	θ(s) = O(1),
	leading to
		0	1	d 2 ds 2 ωε 1 (s) ds = O(1/ε 1 ).
	Concerning point 4, notice the integral ds P ε 1 1 1 0 | 1 ωε 1 ,ε 2 (s) 2 d 0 1 ωε 1 (s) 2 d ds ωε 1 (s) d ds θ(s) + 1 ωε 1 (s)	d ds	θ(s)	2	ds,
	which is of order 1/ε 2 1 .
		1
	+
		0

,

> 0, R ε 1 ,ε 2 is a real-valued function defined on the interval [0, 1 ε 1 ε 2 ]. We say that R = O(g(ε 1 , ε 2 )) with g : R 2 + → R + if there exist δ, C > 0 such that for every (ε 1 , ε 2 ) ∈ (0, δ) 2 and t ∈ [0, 1 ε 1 ε 2 ], we have |R ε 1 ,ε 2 (t)| ≤ Cg(ε 1 , ε 2 ).

Acknowledgements

The authors want to thank Steffen Glaser and Dominique Sugny for their very useful comments.

This work was supported by the ANR project SRGI ANR-15-CE40-0018, and by the ANR project Quaco ANR-17-CE40-0007-01.

• cl p (0, 0) = H I ;

• for 0 ≤ p < p, cl p (p + 1, q) is obtained from cl p (p , q) by eliminating one by one all its oscillating terms of degree (p + 1, q), using Lemma 24;

• cl p (0, q + 1) = cl p (p, q). Notice that, by construction, there is no term of degree (0, q + 1) in cl p (p, q).

Associated with the transformed Hamiltonian cl p 0 (p, q), we define cl p 0 (p, q) the variable obtained iteratively from ψ I by applying, at every use of Lemma 24, the corresponding transformation El.

Remark 26. According to Lemma 24, each elimination procedure produces only terms of higher degree, thus the algorithm yielding cl p 0 (p, q) ends after a finite number of steps.

When we apply the algorithm, we first deal with monomials of the type

The following lemma provides a useful property concerning their corresponding coefficients.

where H 1 is an entire series in ε 1 with coefficient in span S G and H 2 ∈ E (H 1 collects all the monomials of the type ε n 1 ε 0 2 ). Then the coefficients of H 1 are in span S 0 G.

Proof. Let us first consider the case q = 0. Then H = H 1 + ε 2 H 2 and we want to eliminate an element cZ(E) with c = ε p+1 1 s and s ∈ S 0 using Formula [START_REF] Shapiro | Quantum Control of Molecular Processes[END_REF]. Notice that

only consist of monomials of the type ε n 1 ε m 2 with m ≥ 1. On the other hand, the terms J 1 and J 2 in Equations ( 21) and ( 22) (and the corresponding ones for Z(E) = B(E) and Z(E) = sin(E)σ z ) are clearly in span S 0 G. Besides, the coefficients

also stay in span S 0 G, as S 0 is a subalgebra. In the case q = 0, the elimination of a term of degree (p, q) with q ≥ 1 does not impact the monomials of the type ε n 1 ε 0 2 , according to Lemma 24. Let G 0 be the set of non-oscillating elements of G.

Lemma 28. Assume that (23) holds. Then we have

1. H N 0 is a polynomial of degree N 0 -1 in ε 1 with coefficients in span S 0 G 0 , 2. H N 0 is a polynomial of degree N 0 -2 in ε 1 with coefficients in span S G 0 , 3. H r,N 0 is an entire series in ε 1 with coefficients in span S 0 G,


Dag H Olsen
email: dag.h.olsen@uia.no

Tom Roar Eikebrokk
email: tom.eikebrokk@uia.no

Kristian Aspø

Elaine Sajets

Co-creation for digitalization: A study of co-creation in Norwegian business clusters

Keywords: Co-creation, Clusters, Digitalization

There is a growing emphasis on digitalization in research and business practice. The rapid progress in digital technologies compel firms to innovate and transform their businesses. One way to improve the capacity to innovate and transform is to cooperate with others. However, there is a general lack of research on how co-creation among businesses can facilitate digitalization. This qualitative study explores how co-creation among businesses can stimulate and facilitate digitalization.

We have investigated co-creation activities involving companies in business clusters. This paper reports from a study of three business clusters in Norway. We conducted an inductive qualitative study comprising semi-structured interviews as the primary empirical data source. 12 interviews were carried out with informants from three clusters. We found that the co-creation arenas and activities in the business clusters stimulate and facilitate digitalization among the cluster companies. We also addressed the most significant drivers and barriers to cocreation to get a deeper understanding of the co-creation phenomenon.

Introduction

Digital innovation has caused disruptive changes to the economy [START_REF] Yoo | The tables have turned: How can the information systems field contribute to technology and innovation management research[END_REF], and there is rapid change in many industries due to digitalization and digital transformation [START_REF] Hartl | IT projects in digital transformation: a socio-technical journey towards technochange[END_REF]. The fast progress in digital technologies generates a pressure on firms to innovate and transform their businesses. Organizations need to understand how to implement digital technologies and innovative business concepts [START_REF] Stief | Transform to Succeed: An Empirical Analysis of Digital Transformation in Firms[END_REF]. Digitalization is very challenging for most organizations who struggle with understanding the opportunities and consequences to their business [START_REF] Bharadwaj | Digital business strategy: toward a next generation of insights[END_REF]. The task is especially challenging for small firms due to their general lack of resources [START_REF] Zach | ERP system implementation in SMEs: exploring the influences of the SME context[END_REF]. One way to develop the capability to innovate and transform is to cooperate with others [START_REF] Eikebrokk | Co-creation of IT-value in a cluster of small enterprises[END_REF]. Businesses are increasingly seeking multiple partners to be more successful in applying digital solutions and achieving digitalization [START_REF] Grover | Cocreating IT value: New capabilities and metrics for multifirm environments[END_REF]. However, instituting and effectively managing a co-creation strategy is challenging, and potentially leading to tensions between companies [START_REF] Gnyawali | Co-opetition between giants: Collaboration with competitors for technological innovation[END_REF].

Our aim has been to explore how firms can co-create with other businesses to better cope with the challenges and opportunities of digitalization. We have therefore looked for arenas where firms co-create, and we conjectured that business clusters would be appropriate arenas. Business clusters have some common elements of cooperation and common activities for the benefit of the cluster companies. We have therefore addressed how co-creation among businesses can contribute to digitalization among the cluster members. We thus raised the following research question: How does co-creation contribute to digitalization in business clusters? To answer this main research question, we raised three more specific research questions: What are the drivers for co-creation in the business clusters? what are the barriers for co-creation in the business clusters? and how does co-creation influence digitalization in the business clusters? By addressing the drivers for co-creation, we would improve our understanding of issues and activities that contributed to the co-creation of digitalization. By addressing the barriers for co-creation, we would get a better understanding of key issues that hampered the co-creation efforts. This would give us a deeper understanding of the how the co-creation activities addressed these issues in order to stimulate co-creation in the clusters.

The paper is organized as follows: The next sections present related work on cocreation. Then we present the research method, followed by the results and a discussion of potential implications for further research. Finally, we present the conclusion.

Background

The management literature has for many years acknowledged the significance of interdependence between enterprises, resulting in social relationships and networks [START_REF] Czakon | Network myopia: An empirical study of network perception[END_REF][START_REF] Grönroos | Critical service logic: making sense of value creation and cocreation[END_REF]. Such connections have provided the participants with the capacity to create opportunities for competitive advantage through new sources of information. A rising stream of research has conceptually described these collaborations as co-creation. This research can present substantial input to innovation processes [START_REF] Nambisan | Designing virtual customer environments for new product development: Toward a theory[END_REF]. These networks can enhance competitive power by forming shared sources of value creation through co-creation [START_REF] Prahalad | Co-creation experiences: The next practice in value creation[END_REF]. This collaboration is especially valuable when the market is dynamic, and the enterprises have limited resources for innovation. Firms collaborating in such networks contribute knowledge and resources in co-creating interpretations and responses. This co-creation relates to an array of corporate issues such as supply chain utilization, service innovations and of information technology implementation [START_REF] Kohlbacher | International marketing in the network economy: a knowledge-based approach[END_REF]. There is a dearth of research on the nature of co-creation in various contexts and how it can be managed [START_REF] Felzensztein | Coopetition in regional clusters: Keep calm and expect unexpected changes[END_REF][START_REF] Frow | Managing co-creation design: A strategic approach to innovation[END_REF]. There is a need for more research on issues such as how do companies that are not suppliers or customers to each other, collaborate horizontally in business networks, and how does co-creation as a dynamic process contributes to the value-creation of the enterprises and the whole ecosystem [START_REF] Nambisan | Designing virtual customer environments for new product development: Toward a theory[END_REF].

The literature has identified various factors that influences collaboration in a cluster. Lack of resources, both human and financial, can lead to hesitation among member companies when it comes to participating in innovation [START_REF] Eikebrokk | Co-creation of IT-value in a cluster of small enterprises[END_REF]. Trust has also been found to have a major impact on collaboration. In a case study by Sarker, Sarker, Sahaym and Bjørn-Andersen [START_REF] Sarker | Exploring value cocreation in relationships between an ERP vendor and its partners: a revelatory case study[END_REF], the authors found that trust, goodwill and commitment are important drivers for member companies to commit resources for generating new products and services. Other findings are issues such as lack of incentives, leadership and a clear common vision. Such issues are important barriers that must be dealt with to create a fertile co-operation environment [START_REF] Eikebrokk | Co-creation of IT-value in a cluster of small enterprises[END_REF]. There are many potential benefits from co-creating, and substantial benefits can come from sharing knowledge and data [START_REF] Eikebrokk | Co-creation of IT-value in a cluster of small enterprises[END_REF].

Knowledge sharing is seen as the heart of collaboration and can act as a driving force for change and better understanding of customer preferences [START_REF] Zainuddin | Configurability, maturity, and value co-creation in SaaS: an exploratory case study[END_REF]. Nevertheless, it is a difficult balancing act for many companies to share their knowledge without giving up competitive advantage [START_REF] Yoong | Knowledge sharing and business clusters[END_REF]. Furthermore, we find that increased competitiveness, increased usability of collaboration tools (if done correctly) and greater focus on customer data [START_REF] Eikebrokk | Co-creation of IT-value in a cluster of small enterprises[END_REF][START_REF] Giesbrecht | Service encounter thinklets: how to empower service agents to put value co-creation into practice[END_REF] are advantages that can be achieved by co-creation. In SMEs, we see that IT-facilitated collaboration between companies can increase, and it can make members more proactive. It can be a challenging task to get such a collaboration going, but it can lead to improved performance [START_REF] Chang | Shaping Proactivity For Firm Performance: Evaluating The Role Of It-Enabled Collaboration In Small And Medium Enterprises[END_REF]. In companies with few resources and employees, IT-facilitated collaboration systems can be helpful if they cooperate [START_REF] Eikebrokk | Co-creation of IT-value in a cluster of small enterprises[END_REF][START_REF] Chang | Shaping Proactivity For Firm Performance: Evaluating The Role Of It-Enabled Collaboration In Small And Medium Enterprises[END_REF], and customer loyalty and retention may increase upon establishment of such co-creation services [START_REF] Zainuddin | Configurability, maturity, and value co-creation in SaaS: an exploratory case study[END_REF]. We argue that resources are key to realize value form co-creation [START_REF] Schryen | Revisiting IS business value research: what we already know, what we still need to know, and how we can get there[END_REF]. We therefore adopted the resource-based view (RBV) of the firm [START_REF] Barney | Firm resources and sustained competitive advantage[END_REF][START_REF] Mata | Information technology and sustained competitive advantage: A resource-based analysis[END_REF] as an analytical lens.

Digital transformation can be defined as a major change in how business is conducted due to digital technology [START_REF] Hartl | IT projects in digital transformation: a socio-technical journey towards technochange[END_REF][START_REF] Bilgeri | How digital transformation affects large manufacturing companies' organization[END_REF][START_REF] Osmundsen | Digital Transformation: Drivers, Success Factors and Implications[END_REF]. The term digitalization usually has a more limited interpretation; it is about leveraging digital technology to modify socio-technical structures [START_REF] Osmundsen | Digital Transformation: Drivers, Success Factors and Implications[END_REF]. We have adopted the term digitalization, to also include minor changes to the business models. There are many definitions of digital innovation [START_REF] Osmundsen | Digital Transformation: Drivers, Success Factors and Implications[END_REF]. What is generally common elements is that it is perceived as something novel, and that it is based on digital technology [START_REF] Fichman | Digital innovation as a fundamental and powerful concept in the information systems curriculum[END_REF][START_REF] Yoo | Research commentary-the new organizing logic of digital innovation: an agenda for information systems research[END_REF]. Osmundsen et al. [START_REF] Osmundsen | Digital Transformation: Drivers, Success Factors and Implications[END_REF] build on these definitions, and emphasizes that digital innovation is a process (to innovate) as well as an outcome, and is about "combining digital technology in new ways or with physical components that enables socio-technical changes and creates new value for adopters."

Business clusters that we know today started back in the 1950s with the production of ceramic tiles [START_REF] Porter | Clusters and the new economics of competition[END_REF]. It was in a small Italian town named Sassuolo in the Emilia-Romagna area [START_REF] Porter | Clusters and the new economics of competition[END_REF]. It wasn't until the early 1990s that clusters and cluster projects really came into focus through the research by Michael E. Porter [START_REF] Porter | Location, competition, and economic development: Local clusters in a global economy[END_REF]. This created an interest in clusters and their potential for innovation, knowledge sharing, collaboration, networking and growth [START_REF] Malmberg | True clusters: a severe case of conceptual headache[END_REF]. Clusters can be defined as "[…] geographic concentrations of interconnected companies, specialized suppliers, service providers, firms in related industries, and associated institutions (e.g., universities, standards agencies, trade associations) in a particular field that compete but thus cooperate " [START_REF] Porter | Location, competition, and economic development: Local clusters in a global economy[END_REF].

We next present the research approach and the three business clusters we investigated.

The interviews were mostly conducted face-to-face at the companies' sites. A few interviews were conducted over telephone. The interviews lasted from 22 minutes to over one hour, and they were taped and fully transcribed. The interviews were largely dialogue-based. Secondary material in the form of document studies was also utilized [START_REF] Myers | The qualitative interview in IS research: Examining the craft[END_REF], and included information and documents available through the clusters' web pages. Due to requests for anonymity, no personal data were registered in the transcripts. The transcripts were then moved into the NVivo analysis tool. The empirical material was systematized and reduced [START_REF] Miles | Qualitative data analysis: an expanded sourcebook[END_REF], and we performed a content analysis. Business cluster A consists of approximately 100 member companies, mainly in the energy and maritime sector. They are suppliers of technology, products and services. The cluster focuses on building competence and conducting research and training with national and international partners. The goal of the cluster is to strengthen competitiveness, improve the development of new products and services, and to apply knowledge and technology to new markets in a sustainable way. The cluster arranges and facilitates meetings, conferences and courses, in Norway as well as abroad. The cluster also runs several types of projects for cluster members.

Cluster B consists of more than 100 member companies and over 40 municipalities and other public actors. The cluster focuses on innovation, research and business development within health and welfare technology. The cluster focuses on innovation, research, development and export of products and services. Through collaboration, knowledge sharing and business development, they aim to strengthen the member companies. The cluster also aims to develop the market for health and welfare technologyboth in the private and the public sector. There cluster arranges many events; it is everything from member meetings, innovation forums, conferences and courses for specific projects. They also have one innovation lab where member companies can test out various technologies.

Cluster C consists of just under 100 organizations from finance, insurance, academia and technology. The cluster focuses on financial technology, including innovation, growth and value creation. The main goal of the cluster is to make finance easy, and they work to increase export of Norwegian financial technology. The cluster organizes and facilitates several types of conferences, workshops and seminars -both in and outside Norway. In addition, there are many projects run by the cluster, where the idea for some of the projects has come from brainstorming events in the cluster.

Results

The interviews generated several drivers and barriers. The drivers and barriers emerged from the interview transcripts. Table 2 shows the drivers and barriers noted by the informants. An "x" means that the informants agree on this issue, and an "o" means that they do not fully agree on this issue. We found that the main drivers are knowledge sharing, networking and having a proficient cluster management. The main barriers are the competition between cluster companies and the costs of cluster activities for the small member businesses. The informants from the three clusters had similar conceptions of the term digitalization. A common element among informants from cluster A was that digitalization is about employing technology to make "things simpler", for example making information more accessible. Informant A2 noted that the industry is quite conservative, making it difficult to get support for disruptive changes. Two of the informants in cluster A viewed digitalization as primarily the application of existing technologies. The focus was thus on improving rather than replacing. Informant A2 provided the following examples: automated monitoring of windmills and automated monitoring of maintenance requirements. This would be through sensors to monitor and report. Informant A1 noted that "the technology in itself is not important; it is what you can do with it for the business." Informant A1 further commented that a recent report found that enterprises with 10 to 250 employees struggle the most with digitalization -"they are big enough to make it complicated, but do not have the resources needed."

All informants in cluster B had an almost equal view of digitalization, what it implies and what opportunities this may entail in the flow of information. It was perceived to be about how to use data or digital tools to improve processes, automate processes and get different systems for talking together. Informant B2 noted that their perception of digitization depends on the context. Within the enterprise, digitization focuses on the collection and use of data to improve various health indicators among patients. Beyond this, they pointed out that they are working to make municipalities understand "how the digital data makes it much easier for them to understand things". Informant B4 corroborated this view, and he emphasized that digitalization meant "[…] how can we improve efficiency and work smarter". It was further observed that digitization was also important for management in terms of change management. Informant B1 stated that "Digital tools or digital technology give you other opportunities to do things in a completely different way." Informant B1 brought up the term disruptive as a key word for "new ways to operate your [company] that new technology enables," and pointed out that digitalization has top priority in the health sector.

Informants from cluster C emphasized process automation and application of technology as key digitalization matters. Informant C1 noted that digitalization is about "how technology can be used for a good user experience [...] That's the key." Informant C4 from the cluster management also viewed customer experiences as the main focus. However, informant C4 emphasized that they need a precise definition. Informant C4 explained that "You have quite a few options then, with computing power and sensors and digital tools, to solve tasks that people had to solve [before]. That's probably the big picture digitization for me." Informant C2 explained that "Within our industry, digitalization is about creating, often moving certain tasks that have been manual over to smartphones, so that consumers can access it in better way. So, yes, it is really there to facilitate everyday life and optimize and automate processes to a large extent."

The interviews showed that the three clusters have several arenas for co-creation. The informants from cluster A emphasized arenas such as workshops, conferences, courses and co-creation projects. The project manager from the cluster administration estimated that they currently had approximately 30 cluster projects, either ongoing or about to start. Two of the informants from this cluster emphasized a test laboratory that they sponsor, where a university is the main owner. This is a lab where the cluster holds workshops with member companies, where they are exposed to emerging technologies, such as AR, VR and 3D printing. This has been seen a way to extend the member companies' scope outside the oil and gas industry. Informant A3 noted that "Then one realizes that many of these companies needed such a test lab. Then it was no longer a one-sided focus on drilling and oil exploration, it was a [quite] broader [focus]." Several informants underlined that the oil and gas industry is very conservative, and that it was very beneficial that the member companies were exposed to opportunities and challenges related to new technology. Informant A1 noted that "We also have workshops on strategy and business development […]. How will the new digital technology influence [the cluster companies]? What do they need to do today to [be] where they want to be 5 to 10 years [from now]?" We see from these examples that this cluster has many co-creation activities that encourage and support digitalization efforts among cluster companies. Table 3 shows which co-creation arenas was noted by the informants.

The informants from cluster B would use terms like building relationships and networks, and raising competencies and innovation, when describing the cluster activities. The cluster has focus on creating programs that would be beneficial for the members, efforts to raise competencies among members, co-creation projects and fundraising. This cluster focuses on health technology, and it has therefore a strong relationship to the public sector, since this marked is dominated by the public healthcare system. Informant B1 asserted that: "[…] since it is about health [technology] in the public sector, we need to [focus on] developing the market too. It is quite immature. […] We work quite a lot with developing the Norwegian health industry." Informant B2 corroborated this, and he added that "[…] having the local municipalities as members meant quite a lot, since then we had a sparring partner we could learn from." We therefore conclude that the co-creation activities in this cluster clearly promotes digitalization among cluster companies. The informants from cluster C emphasized activities stimulating co-creation such as networking and knowledge sharing. Informant C1 noted that the cluster facilitates travels to industry conferences and gathers members for joint travels: "they dine together, have breakfast together, they attend [conference sessions] together, so [they] become good friends. [Networks] are made there". This contributes to stronger ties between people in the cluster, and it creates a stronger cluster identity and a network between people in the cluster companies. All informants emphasized the importance of networking activities for gaining deeper understanding of the challenges and opportunities related to digitalization. Informant C2 noted that they have found a business partner (in the cluster) that is proficient on machine learning, through the cluster networking activities. Informant 3 pointed out the importance of being visible in the cluster, and that the network they get access to, is very interesting, and described that this is an "investment in the future". We therefore find that significant co-creation activities were aimed at improving digital competencies and skills among cluster companies, and thus to the digitalization of these companies.

Discussion

We have explored how co-creation contributes to digitalization in three Norwegian business clusters. We found that co-creation is an important vehicle for digitalization among the individual member companies. By addressing the drivers and barriers for co-creation, we were able to achieve a better understanding of co-creation as a phenomenon, how co-creation manifests itself and the important factors related to achieving co-creation in practice. We will elaborate on the various factors below. We conjectured that three drivers were particularly significant for achieving co-creation in the clusters. First, networking activities were perceived as a major driver for co-creation. This is consistent with the literature on business clusters (e.g. [START_REF] Sölvell | The cluster initiative greenbook[END_REF]), which views network building as one of the most common cluster goals. All informants in the three cases emphasized this as one of the main drivers for membership. Some companies joined the cluster to get access to new customers or partners, while others saw the cluster as an opportunity to develop new competences and capabilities. Regardless of the corporate intention, the clusters arrange various activities that facilitate networking among the members. We conjecture that such activities were essential to create arenas for co-creation. These activities could be reserved for cluster members, or they could include public institutions outside the cluster as well as members of other clusters. This is also consistent with Porter's [START_REF] Porter | Location, competition, and economic development: Local clusters in a global economy[END_REF] finding that involving and activating cluster members of all sizes is important for collaboration and community.

Second, related to this, we found that knowledge sharing was also a major driver for co-creation. The companies see clusters as a place to go to meet other companies in an informal setting. Informal arrangements like this have proven to lead to friendship which is important when it comes to building trust [START_REF] Bergh | Entrepreneurs learning together: The importance of building trust for learning and exploiting business opportunities[END_REF]. When the members of a cluster increase their trust in each other it can lead to more knowledge sharing. Networking activities create relationships and build trust between companies. This can create an arena for exchanging knowledge and experience among the cluster members. This is consistent with the literature, which points out that knowledge exchange is central in collaboration [START_REF] Eikebrokk | Co-creation of IT-value in a cluster of small enterprises[END_REF], and a driving force for change [START_REF] Zainuddin | Configurability, maturity, and value co-creation in SaaS: an exploratory case study[END_REF]. In our literature review, we uncovered two activities related to knowledge sharing that is consistent with our findings [START_REF] Waits | The added value of the industry cluster approach to economic analysis, strategy development, and service delivery[END_REF]:

• Co-inform: Common to all three clusters is the newsletter that the cluster management sends out to the cluster members by email. The newsletters convey activities and opportunities in the cluster, as well as providing information on ongoing and new projects. They also have websites where, among other things, they present the cluster members.

• Co-learn: Both Cluster A and Cluster B offer their members access to a test lab where they can access technology that they might not otherwise be able to afford. Such labs are important for companies, regardless of available resources, to test new technological trends and get access to useful knowledge. In this way they may get the ability to identify appropriate technology. Research has demonstrated that testing out new technological trends is important in order to further develop your business processes [START_REF] Sölvell | The cluster initiative greenbook[END_REF], and that digital technology can have a direct impact on how businesses innovate [START_REF] Yoo | The tables have turned: How can the information systems field contribute to technology and innovation management research[END_REF].

Third, we also found that cluster management was also an important driver. Many of the respondents emphasized that it was easy to get access to the cluster management. This made it easy take initiatives, for example to make request for training or to propose topics for cluster seminars. We therefore conjecture that a strong cluster management is important for initiating and supporting co-creation activities.

We identified two significant barriers. First, competition among the members is a significant barrier. We saw that informants generally are positive to sharing competence, but also that a majority of them are hesitant to share strategies or concepts where time to market is an issue. The literature also emphasizes the perils of sharing knowledge with competitors, and it may relinquish competitive advantage [START_REF] Yoong | Knowledge sharing and business clusters[END_REF].

Second, most informants did not perceive membership costs as a significant problem, especially informants from large enterprises. However, the informants that believed that costs was an issues, felt that there should not be a membership fees for small businesses. Also, several informants from small enterprises noted that the costs of participating in cluster activities was sometimes prohibiting small member companies from participating. This is consistent with the literature, which has demonstrated that limited human and economic resources can be a significant challenge for co-creation [START_REF] Eikebrokk | Co-creation of IT-value in a cluster of small enterprises[END_REF]. Several informants noted that the projects could last a long time, and this would reduce the interest among the participants. The longer the time between the workshops, the higher the likelihood would be of members withdrawing from a project [START_REF] Osmundsen | Digital Transformation: Drivers, Success Factors and Implications[END_REF].

The activities in clusters are usually based on input from its members. Most informants noted that it was easy to get access to the cluster management, and that it was easy to propose topics for activities or projects in the clusters. It was generally perceived that it was a short way from proposing a new topic to set up a project through a brainstorming session. This can lead to digital innovation through the co-creation of new products or services [START_REF] Frow | Managing co-creation design: A strategic approach to innovation[END_REF][START_REF] Osmundsen | Digital Transformation: Drivers, Success Factors and Implications[END_REF]. This supports the argument that clusters drive innovation. Sölvell et al. [START_REF] Sölvell | The cluster initiative greenbook[END_REF] argue that there are three critical arguments for why innovation tend to be connected with clusters: 1) the need for incremental reduction of technical and economic uncertainty, 2) the need for repeated and continuous interaction between related firms and specialized institutions (including research and education), and 3) The need for face-to-face contact in the exchange and creation of new knowledge. Our findings demonstrated that the three clusters in this study facilitate resolution of all those needs.

We saw that the clusters offered the members arenas where co-creation processes could take place. The clusters would arrange brainstorming events where members from various topical areas could discuss ideas and solutions related to perceived market needs, facilitate co-creation projects, and aid in the application for external funding. Such processes were vital to stimulate co-creation of digital innovation and digitalization in all three clusters.

Based on the findings and the discussion above, we argue that co-creation can be an important avenue for digitalization, digital innovation or digital transformation of companies. We contend that this is particularly true for the companies that have limited human and financial resources, such as small and medium sized enterprises. Such companies have limited resources to implement digital technologies and innovative business concepts. We therefore see co-creation arenas, such as business clusters, as beneficial for the digitalization of such companies.

Conclusion

This study has explored how co-creation can contribute to digitalization among companies in business clusters. We found that the co-creation arenas and activities within the business clusters stimulate and facilitate digitalization among the cluster companies.

The study also addressed the most significant drivers and barriers to co-creation to get a deeper understanding of the co-creation phenomenon. Co-creation arenas, such as business clusters, can be very valuable for companies that do not have substantial resources to devote to innovation activities, such as digitalization and digital transformation. This research has several limitations. It was performed in three clusters with a small number of informants and, therefore, has limited generalizability. Further research should explore this pertinent issue in other contexts, using the present research as a basis for subsequent quantitative studies to provide generalizable results. Such results can contribute to a better understanding about how organizations can co-create with other organizations in order to innovate and transform, and thus contributing to the theory on co-creation. This will be particularly beneficial for resource poor organizations, such as small and medium sized enterprises.

Table 1 .

 1 Overview over interviewees.

	Cluster Informant	Position	Company	Role in cluster	Interview	Duration
				size		date
	A	A1	Project leader	N/A	Cluster mgmt	02.15.2019 01:02
	A	A2	Senior engineer	SME	Member company 03.12.2019 00:58
	A	A3	CTO	Large Member company 03.21.2019 01:08
	A	A4	CEO	SME	Member company 05.09.2019 00:54
	B	B1	CEO	N/A	Cluster mgmt	02.26.2019 00:44
	B	B2	CEO	SME	Member company 04.01.2019 00:53
	B	B3	Project leader	SME	Member company 04.11.2019 00:34
	B	B4	Project leader	Large Member company 04.23.2019 00:35
	C	C1	Member	N/A	Cluster mgmt	04.11.2019 00:58
	C	C2	CEO	SME	Member company 04.02.2019 00:35
	C	C3	CEO	SME	Member company 04.02.2019 00:33
	C	C4	Member	N/A	Cluster board	04.02.2019 00:22

Table 2 .

 2 Drivers and barriers for digitalization.

		Cluster A	Cluster B	Cluster C
	Informants: A1 A2 A3 A4 B1 B2 B3 B4 C1 C2 C3 C4
	Drivers
	Knowledge sharing	x x x x x x x x x x x x
	Networking	x x x x x x x x x x x x
	Creating visibility		x	x x	x	x
	Access to resources	x		x	x x
	Insight into new markets x x		x
	Cluster management	x x	x	x x	x x
	Barriers
	Time	x x	x	x x x x
	Costs	x	o x o o o x o	x
	Competition	x	x	x x x x x
	Little activity	x		x	x x	x

Table 3 .

 3 Co-creation arenas noted.

		Cluster A	Cluster B		Cluster C
		A1 A2 A3 A4 B1 B2 B3 B4 C1 C2 C3 C4
	Hackatons	x			x	x
	Conferences	x x x	x x x x x x x
	Courses	x x		x x	x x x	x
	Digitalization Lab.	x	x	x x x x
	Seminars	x			x	x
	Brainstorming sessions	x x		x x	x x	x

We conducted an inductive qualitative study comprising semi-structured interviews as the primary empirical data source. 12 interviews were carried out with informants from three clusters, see table1for an overview of the informants.