Adel Kara

Geneviève Fleury

Thermoelectric study of the time-dependent Resonant Level Model

We study the non-interacting time-dependent resonant level model mimicking a driven quantum dot connected through leads to two electronic reservoirs held at different temperatures and electrochemical potentials. Using a scattering approach, we provide analytical formulas of the time-dependent particle currents, heat currents, and input driving power under the wide-band limit approximation. We also derive Landauer formulas for the corresponding time-integrated quantities when the perturbation applied on the dot is of finite duration. Then, we focus on the case of a single square pulse, benchmark our analytical results against numerical ones that are valid beyond the wide-band limit, and perform numerical simulations for a smooth square pulse and a periodic square pulse train. Finally, we discuss whether the efficiency of the device in a stationary Seebeck configuration can be enhanced by driving the dot potential. We find numerically that the transient increase of the efficiency observed in some cases is eventually cancelled out at long times.

I. INTRODUCTION

Molecular junctions and quantum dots have emerged as promising experimental platforms for investigating energy transport on the nanoscale. They have been used in the last few years to investigate thermoelectric effects [START_REF] Dzurak | Thermoelectric signature of the excitation spectrum of a quantum dot[END_REF][START_REF] Scheibner | Thermopower of a kondo spin-correlated quantum dot[END_REF][START_REF] Rincón-García | Thermopower measurements in molecular junctions[END_REF][START_REF] Cui | Perspective: Thermal and thermoelectric transport in molecular junctions[END_REF][START_REF] Cui | Peltier cooling in molecular junctions[END_REF][START_REF] Prete | Thermoelectric conversion at 30 k in inas/inp nanowire quantum dots[END_REF][START_REF] Dutta | Direct probe of the seebeck coefficient in a kondo-correlated single-quantum-dot transistor[END_REF] and to implement e.g. heat engines [START_REF] Josefsson | A quantum-dot heat engine operating close to the thermodynamic efficiency limits[END_REF][START_REF] Josefsson | Optimal power and efficiency of single quantum dot heat engines: Theory and experiment[END_REF][START_REF] Jaliel | Experimental realization of a quantum dot energy harvester[END_REF][START_REF] Popp | Thermoelectricity of nearresonant tunnel junctions and their relation to carnot efficiency[END_REF] , heat valves [START_REF] Dutta | Single-quantum-dot heat valve[END_REF] , and thermal rectifiers [START_REF] Scheibner | Quantum dot as thermal rectifier[END_REF] in the stationary regime. The physics in those devices is often described by invoking the paradigmatic fermionic Resonant Level Model (RLM): a single localized electronic level with energy ε 0 (hereafter called the dot) is connected with coupling energies Γ α to two (or more) electronic baths α held at temperatures T α and electrochemical potentials µ α . While it is straightforward to calculate heat and charge currents for the non-interacting RLM in the stationary regime, the inclusion of electron-electron interactions in the dot and/or of time-dependent perturbations pushing the system out-of-equilibrium complicates considerably the model.

In particular, in spite of its apparent simplicity, the noninteracting time-dependent RLM is the subject of active research in the fields of high-frequency nanoelectronics and quantum thermodynamics. It has been extensively used in the theoretical literature to investigate dynamical charge [START_REF] Jauho | Time-dependent transport in interacting and noninteracting resonant-tunneling systems[END_REF][START_REF] Platero | Photon-assisted transport in semiconductor nanostructures[END_REF][START_REF] Gurvitz | Single-electron approach for time-dependent electron transport[END_REF][START_REF] Ridley | Current through a multilead nanojunction in response to an arbitrary time-dependent bias[END_REF][START_REF] Gurvitz | Zero-bias current induced by periodic drive of arbitrary shape[END_REF] and energy [START_REF] Esposito | Finite-time thermodynamics for a single-level quantum dot[END_REF][START_REF] Crépieux | Enhanced thermopower under a time-dependent gate voltage[END_REF][START_REF] Liu | Elastic transient energy transport and energy balance in a single-level quantum dot system[END_REF][START_REF] Esposito | Quantum thermodynamics: A nonequilibrium green's function approach[END_REF][START_REF] Ludovico | Periodic energy transport and entropy production in quantum electronics[END_REF][START_REF] Zhou | Boosting thermoelectric efficiency using time-dependent control[END_REF][START_REF] Daré | Time-dependent thermoelectric transport for nanoscale thermal machines[END_REF][START_REF] Yu | Full-counting statistics of transient energy current in mesoscopic systems[END_REF][START_REF] Entin-Wohlman | Heat currents in electronic junctions driven by telegraph noise[END_REF][START_REF] Lehmann | Time-dependent framework for energy and charge currents in nanoscale systems[END_REF][START_REF] Covito | Transient charge and energy flow in the wide-band limit[END_REF][START_REF] Haughian | Quantum thermodynamics of the resonant-level model with driven system-bath coupling[END_REF][START_REF] Bruch | Landauer-büttiker approach to strongly coupled quantum thermodynamics: Inside-outside duality of entropy evolution[END_REF][START_REF] Oz | Numerical approach to nonequilibrium quantum thermodynamics: Nonperturbative treatment of the driven resonant level model based on the driven liouville von-neumann formalism[END_REF][START_REF] Semenov | Transport and thermodynamics in quantum junctions: A scattering approach[END_REF][START_REF] Yu | Time-dependent thermoelectric transport in mesoscopic systems under a quantum quench[END_REF] transport with different techniques. Fundamental questions have been addressed with the aim of building a consistent theory of thermodynamics in the out-of-equilibrium quantum regime [START_REF] Ludovico | Periodic energy transport and entropy production in quantum electronics[END_REF][START_REF] Haughian | Quantum thermodynamics of the resonant-level model with driven system-bath coupling[END_REF][START_REF] Bruch | Landauer-büttiker approach to strongly coupled quantum thermodynamics: Inside-outside duality of entropy evolution[END_REF][START_REF] Oz | Numerical approach to nonequilibrium quantum thermodynamics: Nonperturbative treatment of the driven resonant level model based on the driven liouville von-neumann formalism[END_REF][START_REF] Semenov | Transport and thermodynamics in quantum junctions: A scattering approach[END_REF] . The possibility of performing thermodynamic tasks such as heat pumping [START_REF] Cuansing | Nonadiabatic particle and energy pump at strong system-reservoir coupling[END_REF] , cooling [START_REF] Daré | Time-dependent thermoelectric transport for nanoscale thermal machines[END_REF] or heatto-work conversion [START_REF] Crépieux | Enhanced thermopower under a time-dependent gate voltage[END_REF][START_REF] Zhou | Boosting thermoelectric efficiency using time-dependent control[END_REF][START_REF] Yu | Time-dependent thermoelectric transport in mesoscopic systems under a quantum quench[END_REF] has also been investigated. In particular, in Ref. [START_REF] Zhou | Boosting thermoelectric efficiency using time-dependent control[END_REF] , the authors reported on a boost of thermoelectric efficiency of a driven heat engine modeled by the time-dependent RLM. More advanced models [START_REF] Rey | Nonadiabatic electron heat pump[END_REF][START_REF] Juergens | Thermoelectric performance of a driven double quantum dot[END_REF][START_REF] Bhandari | Geometric properties of adiabatic quantum thermal machines[END_REF][START_REF] Harunari | Maximal power for heat engines: Role of asymmetric interaction times[END_REF][START_REF] Mayrhofer | Stochastic thermodynamic cycles of a mesoscopic thermoelectric engine[END_REF] have been considered as well, e.g. by including Coulomb interaction and/or by describing a two-level dot or two dots instead of one.

In this paper, we study thermoelectric transport in the noninteracting RLM when the energy ε 0 of the dot is made timedependent. As in Refs. [START_REF] Covito | Transient charge and energy flow in the wide-band limit[END_REF][START_REF] Bruch | Landauer-büttiker approach to strongly coupled quantum thermodynamics: Inside-outside duality of entropy evolution[END_REF][START_REF] Kara Slimane | Simulating time-dependent thermoelectric transport in quantum systems[END_REF] , the dot is coupled from the remote past to two baths through ideal (non-interacting) leads. Dissipation only occurs in the baths and a static temperature and/or electrochemical potential bias can be applied between the two baths. We use a scattering approach [START_REF] Kara Slimane | Simulating time-dependent thermoelectric transport in quantum systems[END_REF][START_REF] Michelini | Asymmetry induces long-lasting energy current transients inside molecular loop circuits[END_REF] to describe time-dependent thermoelectric transport in our model. This approach is not restricted to weak dot-bath couplings and is valid for arbitrary driving beyond the adiabatic limit. It is only assumed that the driving is switched on from a given instant t = 0 (i.e. ε 0 (t < 0) = V 0 constant) so that the timedependent scattering state inside the dot and the leads can be calculated [START_REF] Gaury | Numerical simulations of time-resolved quantum electronics[END_REF] by evolving in time the stationary scattering state defined for t < 0. In Ref. [START_REF] Kara Slimane | Simulating time-dependent thermoelectric transport in quantum systems[END_REF] , we used this approach to develop a numerical technique [START_REF]The t-Kwant extension to time-dependent thermoelectric transport is[END_REF] -based on the Tkwant software [START_REF] Kloss | Tkwant: a software package for time-dependent quantum transport[END_REF][START_REF]The t-Kwant package is a free (open-source) software[END_REF] -for simulating time-dependent thermoelectricity in quantum systems. We used the driven RLM as a benchmark and performed numerical simulations of dynamical Peltier cooling in a two-dimensional quantum point contact. Our results pointed to a negative role of time-dependent perturbations for thermoelectric cooling but did not allow us to comprehend and disentangle the physical effects at stake. Here, we focus our interest on the time-dependent RLM and leverage our scattering approach to provide an analytical description of time-dependent thermoelectric transport within the so-called wide-band limit (WBL) approximation. Our analytics serve as guidelines for identifying interesting operating regimes that can be investigated subsequently numerically. In particular, inspired by Ref. [START_REF] Zhou | Boosting thermoelectric efficiency using time-dependent control[END_REF] , we ask ourselves whether or not the efficiency of the stationary device in a Seebeck configuration can be enhanced by driving the dot. We discuss the subtleties of defining a time-resolved efficiency and draw the empirical conclusion from the analysis of our numerous numerical data that the time-dependent driving does not bring an advantage in terms of net thermodynamic efficiency in the RLM. Even though the efficiency can be increased in the transient regime, the net efficiency at long times does not exceed the stationary efficiency in the investigated wide parameter range.

The outline of the paper is as follows. We introduce in Sec.II our time-dependent RLM and in Sec.III the basics of the time-dependent scattering theory of thermoelectric transport. This theory is used in Sec.IV within the WBL approximation to derive (semi-)analytical formulas describing timedependent thermoelectric transport in the RLM. Sec.V is devoted to the peculiar case where a single square pulse is applied on the dot. Potential applications for time-dependent thermoelectric energy harvesting are discussed in Sec.VI. We For times t ≤ 0, the model Hamiltonian is static but the system can be driven out of equilibrium by applying a temperature bias T L -T R and/or an electrochemical potential bias µ Lµ R across the left and right electronic reservoirs attached to the leads Additionally, for times t > 0, the onsite energy ε 0 in the dot can be varied in time with e.g. a back gate.

conclude in Sec.VII.

II. MODEL

We consider a one-dimensional (1D) discretized version of the time-dependent RLM sketched in Fig. 1. A central site 0 with time-dependent onsite energy ε 0 (t) is attached through nearest-neighbor hopping terms γ L and γ R to two left (L) and right (R) semi-infinite 1D chains playing the role of the leads. In the latter, the nearest-neighbor hopping term is denoted by γ and the onsite energies are taken equal to zero for simplicity. The tight-binding Hamiltonian of our system reads

H(t) = H 0 (t) + ∑ α=L,R H α + ∑ α=L,R H 0α (1)
where

H 0 (t) = ε 0 (t)c † 0 c 0 is the dot Hamiltonian, H L = -γ ∑ i≤-1 c † i-1 c i + h.c. and H R = -γ ∑ i≥1 c † i+1 c i + h.c
. are the Hamiltonians of the left and right leads respectively, while

H 0L = -γ L c † 0 c -1 + h.c. and H 0R = -γ R c † 0 c 1 + h.c
. are the tunneling Hamiltonians between the dot and the leads. Here, c i and c † i denote the annihilation and creation operators of an electron at site i. Moreover, each lead α (= L or R) is attached from the remote past to an electronic reservoir characterized by its (static) electrochemical potential µ α and temperature T α . Importantly, we assume that the dot onsite energy is constant for t ≤ 0 and equal to ε 0 (t ≤ 0) = V 0 while timedependent perturbations are switched on for t > 0. We use the notation ε 0

(t) = V 0 +V (t) with V (t ≤ 0) = 0.
Throughout the paper, we take e = h = k B = 1 where e is the electron charge, h the reduced Planck constant, and k B the Boltzmann constant.

III. SCATTERING THEORY FOR TIME-DEPENDENT THERMOELECTRIC TRANSPORT

We use a wave-function approach [START_REF] Kara Slimane | Simulating time-dependent thermoelectric transport in quantum systems[END_REF][START_REF] Michelini | Asymmetry induces long-lasting energy current transients inside molecular loop circuits[END_REF][START_REF] Gaury | Numerical simulations of time-resolved quantum electronics[END_REF] to study timedependent quantum transport in our model. The central objects of the theory are the scattering states Ψ αE (t). Since the leads are time-independent, the incoming part of Ψ αE (t) is made of a single plane wave at energy E coming from lead α (α and E being thus used to label the scattering states). On the contrary the outgoing part of Ψ αE (t) writes in general as a superposition of outgoing plane waves at different energies E ′ , the dot onsite energy ε 0 (t) being time-dependent. For instance, the scattering state Ψ LE n (t) originating from the left lead and evaluated at site n reads in the leads

Ψ LE n<0 (t) = χ → n (t, E) + dE ′ 2π χ ← n (t, E ′ ) r(E ′ , E) (2a)
Ψ LE n>0 (t) = dE ′ 2π χ → n (t, E ′ ) d(E ′ , E) (2b)
where r(E ′ , E) is the probability amplitude for an electron with an energy E coming from the left lead to be reflected with an energy E ′ , d(E ′ , E) is its probability amplitude to be transmitted to the right lead with an energy E ′ , while χ → n and χ ← n are plane waves propagating in the leads from left to right and right to left respectively i.e.

χ → n (t, E) = 1 √ |v(E)| e -iEt+ik(E)n (3a) χ ← n (t, E) = 1 √ |v(E)| e -iEt-ik(E)n (3b)
v(E) and k(E) being the plane wave velocity and momentum satisfying

|v(E)| = 4γ 2 -E 2 , E = -2γ cos k(E), k(E) > 0.
Similar formula can be written for the scattering state Ψ RE n̸ =0 (t) originating from the right lead, with reflection and transmission amplitudes r ′ (E ′ , E) and d ′ (E ′ , E). In particular, in the static case, transport is elastic and

d(E ′ , E) = 2πδ (E -E ′ )d 0 (E), d ′ (E ′ , E) = 2πδ (E - E ′ )d ′ 0 (E), r(E ′ , E) = 2πδ (E -E ′ )r 0 (E), r ′ (E ′ , E) = 2πδ (E - E ′ )r ′ 0 (E)
where δ is the Dirac distribution and d 0 (E), d ′ 0 (E), r 0 (E), and r ′ 0 (E) are notations for the scattering amplitudes of the time-independent problem. Within this framework, the particle current I N L (t) = I N 0,-1 (t) flowing from the left lead to the dot -evaluated between the sites -1 and 0 -and the particle current I N R (t) = I N 0,1 (t) flowing from the right lead to the dot -evaluated between the sites 1 and 0 -are given by

I N α (t) = 2γ α ∑ β dE 2π f β (E)Im[(Ψ β E ±1 (t)) * Ψ β E 0 (t)] (4)
with a sign

+ [-] in the subscript ± for α = R [L]. Here f β (E) = 1/{1 + exp[(E -µ β )/T β ]} is the Fermi-Dirac distri-
bution of the lead β and the sum over β is implicitly done over both leads L and R. Note that particle number conservation implies

I N L (t) + I N R (t) = dρ 0 dt ( 5
)
where

ρ 0 (t) = ∑ β dE 2π f β (E)|Ψ β E 0 | 2 is the particle density in the dot.
The corresponding heat currents I H α (t) defined by 47

I H α (t) = - d dt ⟨H α + 1 2 H 0α ⟩ -µ α I N α (t) (6)
read in terms of the scattering states 41

I H α (t) = ε 0 (t) 2 -µ α I N α (t) + ∑ β dE 2π f β (E) Im[(Ψ β E ∓1 ) * γ L γ R Ψ β E ±1 -(Ψ β E ±2 ) * γγ α Ψ β E 0 ] (7)
where the upper [lower] sign has to be chosen in ± and ∓ for α = R [L]. Finally the input power P(t) corresponding to the driving of the dot energy level writes

P(t) = -V (t) dρ 0 dt . (8)
Hereafter, we use one exact numerical technique and one approximate semi-analytical technique to calculate

I N L (t), I N R (t), I H L (t), I H R (t)
, and P(t). For numerical simulations, we use the Tkwant software [START_REF] Kloss | Tkwant: a software package for time-dependent quantum transport[END_REF][START_REF]The t-Kwant package is a free (open-source) software[END_REF] together with its extension package [START_REF] Kara Slimane | Simulating time-dependent thermoelectric transport in quantum systems[END_REF][START_REF]The t-Kwant extension to time-dependent thermoelectric transport is[END_REF] for thermoelectric transport. In brief, Tkwant computes the scattering states Ψ β E (t) by solving the time-dependent Schrödinger equation for the open system with a Runge-Kutta solver. This requires a subtle treatment of the leads [START_REF] Weston | Linear-scaling source-sink algorithm for simulating time-resolved quantum transport and superconductivity[END_REF] . Then, the integral over the energy E in Eqs.( 4) and ( 7) is done in momentum space with a (Gauss-Kronrod) adaptative scheme. This allows us to compute I N α (t), I H α (t), and P(t) exactly for a given (arbitrarily high) level of precision and an arbitrary form of V (t). While this numerical approach can help us to explore dynamical thermoelectric transport in the RLM, it is insufficient to provide a physical understanding of the mechanisms at stake. Therefore, to gain more physical insight, we also construct a semi-analytical approach outlined in the next section.

IV. SEMI-ANALYTICAL TREATMENT IN THE WIDE-BAND LIMIT A. Time-resolved formulas

It is possible to simplify the formulas above under the wideband limit (WBL) approximation. The WBL is reached in our model (1) by increasing [START_REF] Covito | Transient charge and energy flow in the wide-band limit[END_REF] the bandwidth 4γ of the 1D leads, keeping fixed the coupling energies

Γ L ≡ 2γ 2 L /γ and Γ R ≡ 2γ 2 R /γ. In practice, this is achieved by doing the substitu- tions γ → λ γ, γ L → √ λ γ L , and γ R → √ λ γ R , λ being a scaling parameter. When λ → ∞, the self-energy Σ α (E) of each lead α converges to a pure imaginary energy-independent value, Σ α (E) → -iΓ α /2,
and the WBL is reached.

Introducing the Fourier transforms r(t, E) and d(t, E) of the reflection and transmission amplitudes r(E ′ , E) and d(E ′ , E), e.g.

r(t, E) = dE ′ 2π e -iE ′ t r(E ′ , E) , (9)
we find in the WBL (see Appendix A)

I N α (t) = dE 2π I N α (t, E) (10)
where

I N L (t, E) = f L (E)[1 -|r(t, E)| 2 ] -f R (E)|d(t, E)| 2 (11)
I N R (t, E) = f R (E)[1 -|r ′ (t, E)| 2 ]-f L (E)|d(t, E)| 2 (12)
and

I H α (t) = dE 2π (E -µ α ) I N α (t, E) + Γ α Γ ᾱ f α (E) + f ᾱ (E) E|d(t, E)| 2 + Im [d * ∂ t d] - Γ α Γ ᾱ f α (E) Re [∂ t A(t, E)] (13)
where

A(t, E) = -ie iEt d(t, E) and ᾱ = R [L] if α = L [R].
Note that the alternative [START_REF] Crépieux | Enhanced thermopower under a time-dependent gate voltage[END_REF][START_REF] Zhou | Boosting thermoelectric efficiency using time-dependent control[END_REF] heat current ĨH α (t) defined by ĨH 6) obeys Eq.( 13) without the last term in its right hand side. Obviously, dt

α (t) = -d dt ⟨H α ⟩ -µ α I N α (t) instead of Eq.(
I H α (t) = dt ĨH α (t) when V (t)
is a pulse of finite duration. Finally, the input power reads

P(t) = -V (t) dE 2π f L (E) Γ R + f R (E) Γ L ∂ t |d(t, E)| 2 . ( 14
)
We point out that our equations for P(t) and I N α (t) on one hand, and I H α (t) and ĨH α (t) on the other hand, are an equivalent reformulation in the time-dependent scattering approach of the formulas derived in Ref. [START_REF] Jauho | Time-dependent transport in interacting and noninteracting resonant-tunneling systems[END_REF] and Refs. [START_REF] Crépieux | Enhanced thermopower under a time-dependent gate voltage[END_REF][START_REF] Zhou | Boosting thermoelectric efficiency using time-dependent control[END_REF][START_REF] Daré | Time-dependent thermoelectric transport for nanoscale thermal machines[END_REF][START_REF] Kara Slimane | Simulating time-dependent thermoelectric transport in quantum systems[END_REF] respectively, within the non-equilibrium Green's function formalism. The bond between the two approaches is provided by the generalized Fisher-Lee formula for time-dependent transport, [START_REF] Gaury | Numerical simulations of time-resolved quantum electronics[END_REF] linking the scattering amplitudes to the retarded Green's function.

B. Time-integrated formulas

The formula given above are valid under the WBL approximation, with no assumption on the shape of the pulse V (t) in the dot. We now assume that the duration of the pulse V (t) is finite (i.e. V (t) → 0 when t → ∞) and introduce the following time-integrated quantities induced by the time-dependent drive

∆N α = ∞ 0 dt [I N α (t) -I N α (t = 0)] ( 15
)
∆Q α = ∞ 0 dt [I H α (t) -I H α (t = 0)] (16)
W ext = ∞ 0 dt P(t) (17)
where α = L or R. ∆N α and ∆Q α correspond respectively to the dynamically injected number of particles and heat flowing from lead α, while W ext is the driving work. We also define

T dyn (E) = ∞ 0 dt |d(t, E)| 2 -|d 0 (E)| 2 . ( 18
)
In Appendix A, we show

T dyn (E) = ∞ 0 dt |r 0 (E)| 2 -|r(t, E)| 2 (19) = ∞ 0 dt |r 0 (E)| 2 -|r ′ (t, E)| 2 (20)
with dE T dyn (E) = 0, and

∆N L = -∆N R = dE 2π T dyn (E)[ f L (E) -f R (E)] (21)
which can be seen as a generalized Landauer formula as previously noticed in Refs. [START_REF] Gaury | Numerical simulations of time-resolved quantum electronics[END_REF][START_REF] Gurvitz | Generalized landauer formula for time-dependent potentials and noise-induced zero-bias dc current[END_REF] . Moreover, we have

∆Q α = dE 2π (E -µ α )T dyn (E)[ f α (E) -f ᾱ (E)] - Γ α Γ α + Γ ᾱ W ext ( 22
)
where ᾱ

= R [L] if α = L [R].
Note that in Ref. [START_REF] Entin-Wohlman | Heat currents in electronic junctions driven by telegraph noise[END_REF] , a similar expression was derived for the heat current in the peculiar case of a dot driven by a random telegraph noise V (t) and after averaging over random processes. On the contrary, Eq.( 22) is valid for an arbitrary V (t) of finite support. Finally, as it will be convenient in the following, we introduce the notation

w(E) = -∞ 0 dt V (t)∂ t |d(t, E)| 2 .
We have dE w(E) = 0 and in virtue of Eq.( 14)

W ext = dE 2π f L (E) Γ R + f R (E) Γ L w(E) . ( 23
)
Let us add a few comments. First, we check that

∆Q L + ∆Q R + (µ L -µ R )∆N L +W ext = 0 ( 24
)
as expected from the first law of thermodynamics. Second, if the two reservoirs are at equilibrium (i.e.

f L = f R ), then ∆N L = ∆N R = 0 and ∆Q α reduces to ∆Q α = Γ α /(Γ α + Γ ᾱ )W ext .
Thus, the input driving energy W ext is dissipated as heat in the two reservoirs [START_REF] Moskalets | Scattering Matrix Approach to Non-Stationary Quantum Transport[END_REF] . The dissipation to the left and to the right reservoirs is asymmetric when the left-right symmetry is broken, here with Γ L ̸ = Γ R . On the contrary, if the two reservoirs are out-of-equilibrium (i.e. f L ̸ = f R ), the application of the pulse results in an additional transfer of particles (∆N L = -∆N R ̸ = 0) which is accompanied by a transfer of heat corresponding to the first term in the right-hand side of Eq.( 22) (taking the form of a generalized Landauer formula). Since the external input energy W ext intervenes with a negative sign for both sides L and R in Eq.( 22), it can e.g reduce the amount of heat leaving the hot reservoir and increase heat going to the cold reservoir (as we will see in Fig. 6). Finally, to point out the role of an asymmetric coupling Γ L ̸ = Γ R between the dot and the leads, it is convenient to introduce the parameters a = (Γ R -Γ L )/(Γ R + Γ L ), with |a| ≤ 1, and

Γ = (Γ L + Γ R )/2. With these notations, T dyn (E, a) = (1 -a 2 )T dyn (E, a = 0) (25)
and

W ext (a) =W ext (a = 0) - a Γ dE 2π [ f L (E)-f R (E)]w(E, a = 0) . ( 26
)
Thus T dyn is only re-normalized when the lead-dot coupling is made asymmetric while Eq.( 26) follows from energy conservation (i.e. from Eq.( 24)). This leaves apparently no room

(a) (b) (c) (d) (e) (f)
FIG. 2. T dyn (left) and w (right) for a square pulse (as given by Eqs.( 29) and ( 30)), as a function of E -V 0 and ∆V , for three values of ∆t (0.1Γ -1 (top panels), 1Γ -1 (middle panels), and 10Γ -1 (bottom panels)). Taking Γ as the energy unit yields colormaps that are independent of the value of Γ.

for interesting effects of left/right asymmetry and our numerical exploration -though empirical and non-exhaustive -supports this statement. Therefore, except in Eqs.( 28), (29) (30) that we will keep general, we will assume in the rest of the manuscript and in all figures that left and right couplings are equal i.e. Γ L = Γ R = Γ.

V. SINGLE SQUARE PULSE

In this section, we assume that a single square pulse of amplitude ∆V and duration ∆t is applied on the dot, i.e.

V (t) = ∆V Θ(t)Θ(∆t -t) ( 27
)
where Θ is the Heaviside function (with Θ(0) = 1). We push forward the semi-analytical approach described above and validate the results by comparing them to the ones obtained numerically with Tkwant. We study the convergence of the numerical results to the WBL and observe how the results are modified when the discontinuous jumps of the square pulse are made smooth.

A. Semi-analytics in the wide-band limit When V (t) is a square pulse, the transmission amplitude d(t, E) takes a simple form in the WBL. It can be derived from Eq.(A4a) in Appendix A using e.g. the residue theorem after having calculated the time integral defining K. We find (for arbitrary Γ L , Γ R )

d(t, E) = e -iEt d 0 (E) if t ≤ 0, d(t, E) = e -iEt d 1 (E) + e -(Γ+iV 1 )t d 01 (E) if 0 ≤ t ≤ ∆t, d(t, E) = e -iEt d 0 (E) + [e -(Γ+iV 1 )∆t -e -i∆tE ]× e -[Γ+iV 0 ](t-∆t) d 01 (E) if ∆t ≤ t . (28)
Here we have introduced the notations

V 1 = V 0 + ∆V , d 1 (E) = d 0 (E -∆V ), and d 01 (E) = d 0 (E) -d 1 (E)
, and as before Γ = (Γ L + Γ R )/2. T dyn (E) and w(E) introduced in Sec.IV B can be calculated analytically as well. Using Eqs.(A2a) and ( 28), it can be shown that

T dyn (E) = D 10 ∆t + e -Γ∆t cos(β )-1 2Γ Γ L Γ R D 2 10 - ∆V 2 Γ D 0 D 1 Γ L Γ R -2e -Γ∆t sin(β )D 10 E -V 0 Γ L Γ R D 0 + V 1 -E Γ L Γ R D 1 ( 29
)
w(E) = -D 10 ∆V + ∆V 2 Γ L Γ R D 0 D 1 2e -Γ∆t [(E -V 0 ) cos(β ) -Γ sin(β )] -∆Ve -2Γ∆t (30)
where

D 0 (E) = |d 0 (E)| 2 , D 1 (E) = |d 1 (E)| 2 , D 10 (E) = D 1 (E) -D 0 (E), and
β (E) = ∆t(V 1 -E).
The two quantities are plotted in Fig. 2 as a function of E -V 0 and ∆V for three values of ∆t. They can be positive or negative and satisfy the sum rules dE T dyn (E) = dE w(E) = 0 as noticed in Sec.IV B. For large ∆t ≫ Γ -1 (bottom panels in Fig. 2), T dyn and w reduce formally to their adiabatic limits, i.e. T dyn ≈ D 10 ∆t and w ≈ -D 10 ∆V (though the variation of the dot energy level at t = 0 and ∆t occurs instantaneously). When ∆t becomes comparable to Γ -1 or smaller, the colormaps of T dyn and w get modified and in particular, oscillations appear. Besides, a thorough graphical analysis of the function w(E) allows us to conclude that dE f α (E)w(E) ≥ 0 (knowing that dE w(E) = 0), and hence that W ext ≥ 0 for a square pulse in virtue of Eq. (23). Thus, the source driving the dot supplies energy to the junction, which eventually heats up the reservoirs.

B. Analytics vs numerics

By inserting Eq.( 28) into Eqs.( 10)-( 14), we can compute semi-analytically I N α (t), I H α (t), and P(t), the integrals over the energy being done numerically. The results are plotted in Fig. 3 for a given configuration and are compared to the exact Tkwant results for two values of λ . Note that the analytical results in the WBL are independent of Γ when energies and times are expressed in units of Γ and Γ -1 respectively, while the Tkwant results do not depend on λ , γ, and Γ separately but only on the ratio λ γ/Γ. In the limit of large λ (= 200) corresponding to the WBL, the two sets of curves overlap perfectly, with the exception of the heat currents near t = 0 and t = ∆t. Indeed in the WBL, the heat currents display singularities [START_REF] Covito | Transient charge and energy flow in the wide-band limit[END_REF][START_REF] Kara Slimane | Simulating time-dependent thermoelectric transport in quantum systems[END_REF] at those two times where V (t) jumps instantaneously from 0 to ∆V and conversely. This is highlighted in Fig. 4(a) where data are zoomed in. However, it can be shown analytically that those two singularities are regularized after integration over time i.e. t m -∞ dt [I H α (t) -I H α (t = 0)] is well defined. As illustrated in Fig. 4(b), the λ -dependence of the Tkwant curves I H α (t) near the singularities becomes irrelevant in the limit of large λ once the heat currents are integrated over time and very good agreement is found between analytics and numerics in the WBL after integration.

In Fig. 4(c), we also study with Tkwant how the timeresolved heat current I H L (t) is modified when V (t) is not varied abruptly but continuously between 0 and ∆V , as

V (t) = ∆V 2 erf t ξ /2 -erf t -∆t ξ /2 (31)
erf being the error function and ξ (≲ ∆t) the characteristic time controlling the smoothness of the square-like pulse. Data are plotted for λ = 200 (i.e near the WBL) and different values of ξ . We find that I H L (t) is (almost) independent of ξ , as long as ξ ≲ 0.01Γ -1 . Small ξ -dependence is only visible in that range near the cusps at t = 0 and t = ∆t and their amplitude turns out to decrease with ξ . As shown in Fig. 4(d), after integration over time, I H L (t)dt is strictly independent of ξ for ξ ≲ 0.01Γ -1 . This is also true for the other quantities I H R (t), I N L (t), I N R (t), and P(t) (data not shown). Let us briefly discuss what this implies for experiments. Realistic square pulses that can be applied in experiments takes the form of Eq.(31) while the theoretical limit ξ → 0 yielding the square pulse (27) cannot be engineered. Taking 8 Γ ≈ 10 GHz for a realistic quantum dot operating at an average temperature T = 10Γ ≈ 0.5 K (to be consistent with parameter values taken in Fig. 4), we find that the square pulse model is relevant for discussing such experiments if ξ ≲ 1ps which is within reach (yet challenging) experimentally [START_REF] Bäuerle | Coherent control of single electrons: a review of current progress[END_REF] .

Finally, we have also validated analytical formulas ( 21)-( 23), (29), and (30) for ∆N α , ∆Q α , and W ext with the help of Tkwant simulations. This is illustrated in Fig. 5 where

∆N L FIG. 3. I N L (a), I N R (b), I H L (c), I H R (d)
, and P (inset in (b)), as a function of time t, when a single square pulse is applied in the dot. In all panels, numerical data computed with Tkwant for two values of λ (20 (gray lines) and 200 (black lines)) are compared to the semianalytical results (red dashed lines) valid in the WBL λ → ∞ and given by Eqs.( 10)- (14). Each line is built up from evenly spaced points with a time step δt = 0.002Γ -1 . With the exception of the heat current plots around singularities at t = 0 and t = ∆t (see Fig. 4(a) for a zoom), the red and black lines are superimposed. Parameters:

Γ L = Γ R = Γ, T L = 12Γ, T R = 8Γ, µ L = -3Γ, µ R = 3Γ, V 0 = 25Γ, ∆V = 10Γ, Γ∆t = 0.1, and γ = Γ.
and ∆Q L are plotted as a function of the square pulse amplitude ∆V , other parameters being fixed. We observe that the convergence of Tkwant data to the analytical curves valid in the WBL depend on ∆V . Indeed, the square driving makes transport inelastic within an energy range centered around µ L ≈ µ R of width ∼ ∆V . When ∆V increases, some outgoing plane waves are excited at energies E ′ closer and closer to the boundaries ±2γλ of the lead conduction band and λ must be increased as well to reach the WBL.

VI. HEAT ENGINE EFFICIENCY

In this section, we focus on standard Seebeck configurations with e.g. T L > T R and µ L < µ R , and investigate whether driving the dot energy level with V (t) in the time-dependent RLM is beneficial or detrimental to the heat engine efficiency. We restrict our study to the WBL for which semi-analytical formula have been derived above. We consider in Fig. 6 a set of parameters T L > T R , µ L < µ R , and V 0 such that the system in the WBL is in a Seebeck configuration for t < 0, with a high heat-to-work (stationary) thermoelectric efficiency η st = ∆µI N L (t < 0)/I H L (t < 0) ≈ 0.58 ≈ 0.81η C , where ∆µ ≡ µ Rµ L and η C = 1 -T R /T L is the Carnot efficiency. Then for t > 0, we switch on the time-dependent perturbation V (t) and consider first that V (t) is a single square pulse as shown in Fig. 6(a). Here the amplitude ∆V of the square pulse is chosen small enough so that in an adiabatic picture, the sta- tionary RLM with a dot energy level ε 0 = V 0 + ∆V remains in a Seebeck configuration. Moreover its heat-to-work efficiency remains unchanged (±0.001) with respect to the stationary configuration with ε 0 = V 0 at t < 0. However, we will see hereafter that the exact transient response of the RLM (beyond the adiabatic limit) is significantly impacted by the time-dependent driving. In panels (b), (c), and (d) of Fig. 6, we plot with full lines I N α (t), I H α (t), and P(t) computed with Tkwant taking λ = 200. Data are in very good agreement with the ones (not shown) obtained from Eqs.( 10)-( 14) and (28). Additionally, we plot with dashed lines the corresponding quantities I N,ad α (t), I H,ad α (t), and P ad (t) in the adiabatic (quasi-static) approximation. Obviously, P ad (t) = 0 while the adiabatic particle and heat currents are given by the stationary Landauer-Büttiker formula

I N,ad α (t) = dE 2π D(t, E)[ f α (E) -f ᾱ (E)] (32)
I H,ad α (t) = dE 2π (E -µ α )D(t, E)[ f α (E) -f ᾱ (E)] (33)
( ᾱ = R [L] if α = L [R]
) with a transmission probability D(t, E) depending parametrically on time: D(t, E) = D 1 (E) for times FIG. 5. ∆N L (left) and ∆Q L (right) as a function of the square pulse amplitude ∆V , others parameters being the same as in Figs. 3 and4. The black lines show the analytical predictions in the WBL given by Eqs.( 21)-( 23), (29), and (30) while the symbols correspond to Tkwant data for λ = 15 (red squares), 20 (orange triangles), 50 (blue dots), and 100 (green circles). Dotted lines are guides to the eye. 0 ≤ t ≤ ∆t and D(t, E) = D 0 (E) elsewhere, D 0 (E) and D 1 (E) being defined in Sec.V A. The Tkwant curves are qualitatively different from the adiabatic ones. When V (t) is increased from 0 to ∆V > 0, electrons of higher energy are expelled from the dot to both baths which leads to a decrease of I N L (t) and I N R (t) (and conversely when V (t) goes back to 0). The same effect is observed for the heat currents I H L (t) and I H R (t). This peculiar behavior due to displacement particle and energy currents renders difficult a proper definition of an efficiency, as we will see in the following. Besides, 0 ≤ P(t) ≪ I H L (t), |I H R (t)| for all times. We now tackle the question of the thermodynamic efficiency of this driven heat engine. Defining such an efficiency in the time-dependent regime is particularly subtle. Here we use and comment on different possible definitions. We introduce first an instantaneous (time-resolved) efficiency η R t (t) = ∆µ(-I N R (t))/(I H L (t) + P(t)) plotted in blue in Fig. 6(e). Its denominator is the sum of the resources used to generate electric power, i.e. the input power P(t) ≥ 0 and the heat current I H L (t) > 0 leaving the hot bath. Importantly, for this parameter set, I H L (t) > 0 and I H R (t) < 0 for all times. Its numerator is the electric power generated by electrons flowing out towards the (right) bath of higher electrochemical potential µ R > µ L . For completeness, we also plot in red in Fig. 6(e) the instantaneous efficiency η L t (t) = ∆µI N L (t)/(I H L (t) + P(t)). While η L t (t) is only slightly affected by the driving, η R t (t) increases drastically in the transient regime, above η C and even above 1. Actually, we argue that such instantaneous (timeresolved) efficiencies are ill-defined for two main reasons. First, one may dispute the choice of the numerator in η L t (t) or η R t (t). In a stationary Seebeck configuration, the generated electric power is attributed to electrons leaving the hot bath (T L > T R ) and climbing an electrochemical potential bias ∆µ = µ Rµ L > 0. In the (non-interacting) time-dependent case, this picture is modified due to the presence of the displacement current (i.e. temporary particle storage in the dot, see Eq.( 5)) and one should include the electromagnetic environment of the device to define properly the output power

0 2 V [Γ] (a) (g)
0.02 0.03 In panels (b) to (f), each quantity is also evaluated within the quasistatic approximation in the WBL (gray dashed lines). Parameters:

I N L , -I N R [Γ] (b) I N L -I N R (h) 1.0 1.5 2.0 I H L , -I H R [Γ 2 ] (c)
Γ L = Γ R = Γ, T L = 87Γ, T R = 25Γ, µ L = -26Γ, µ R = 26Γ, V 0 = 55Γ, ∆V = 2Γ, ∆t = 0.08/Γ, γ = Γ,
and λ = 200. (Right) Same as left panels when the square pulse V (t) is repeated periodically for t > 0, with a period τ = 0.1/Γ, as shown in (g). The inset in (l) shows the convergence of η net to the steady state efficiency (black dotted line).

of the driven heat engine. Second, the definition of the timeresolved heat current I H L (t) intervening in the denominator of η L t (t) or η R t (t) is also controverted. In e.g. Refs. [START_REF] Crépieux | Enhanced thermopower under a time-dependent gate voltage[END_REF][START_REF] Zhou | Boosting thermoelectric efficiency using time-dependent control[END_REF] , the last term in the right hand side of Eq.( 13) due to the leaddot coupling contribution is not included [START_REF] While | defined by Eq.(6) is given by Eq[END_REF] . Besides, I H L (t) is defined at the dot-lead interface while heat is eventually dissipated later on and on a different timescale in the bath attached to it [START_REF] Oz | Numerical approach to nonequilibrium quantum thermodynamics: Nonperturbative treatment of the driven resonant level model based on the driven liouville von-neumann formalism[END_REF] . Moreover, one may argue that only the thermoelectric contribution to I H L (t) (i.e. the first term in the right hand side of Eq.( 13)) and not the whole I H L (t) should be considered as a resource for the engine since the driving -whose contribution is counted by adding P(t) in the denominator of η L t (t) or η R t (t) -also heats up the baths and hence contributes to I H L (t). Before ending this discussion, we put forward an alternative definition of the efficiency by noticing that the relevant quantity for application purposes is not the instantaneous efficiency but the net efficiency of the engine at the end of the experiment. Thus we introduce

N α (t) = t 0 du I N α (u) (34)
Q α (t) = t 0 du I H α (u) (35)
W ext (t) = t 0 du P(u) (36)
which correspond to the net number of particles (N α (t)) and the net heat (Q α (t)) flowing out from lead α, as well as the net driving work (W ext (t)), measured from time t = 0 at which V (t) is switched on to the running time t. Then we define the two corresponding net efficiencies η L net (t) = L L (t) + W ext (t)) and η R net (t) = ∆µ(-N R (t))/(Q L (t) +W ext (t)) plotted respectively in red and in blue in Fig. 6(f). Note that W ext (t) ≪ Q L (t) so that its contribution in the denominator of η α net (t) and in Q L (t) (given at long times by Eq.( 22)) is negligible. We find that η L net (t) is always (slightly) smaller than the stationary efficiency η st . On the contrary, η R net (t) increases drastically in the transient regime. The behavior of these two efficiencies can be qualitatively interpreted: when the dot level undergoes the Heaviside jump upwards, the now higher-energy electrons residing in it temporarily flow outwards to both leads. This has the transient effect of (i) increasing the net number of electrons (climbing the chemical potential step) that flow to the cold lead ; (ii) decreasing the net particle current leaving the hot lead ; (iii) increasing the net heat current going to the cold lead ; (iv) decreasing the net heat leaving the hot lead. These explain both the behavior of η R net (t) and η L net (t) during the upwards jump, a similar analysis can be made for the downward jump that follows. Finally, both η L net (t) and η R net (t) converge to η st at long times.

Since η R net (t) can be increased at short times by driving the dot with a single square pulse, let us now investigate the same device when V (t) is cycled as

V (t) = ∆V ∑ n∈ N Θ(t -nτ)Θ(nτ + ∆t -t) . (37)
The period τ of the pulse train is chosen so as η R net (t = τ) in Fig. 6(f) is close to its maximum. Our results are summarized in the right panels of Fig. 6. We see in particular in panel (l) that after the transient increase of η R net (t) at short times, the convergence time of η R net (t) to the steady-state efficiency

η ss = ∆µN τ,∞ L /(Q τ,∞ L + W τ,∞ ext
) is much larger than the one of η R net (t) to η st in the case of a single square pulse (see panel (f)). However, we find η ss ≈ η st (with a discrepancy of about 0.002). Thus the transient increase of the efficiency which may appear advantageous for thermoelectric applications is eventually cancelled out in the long time limit.

Roughly speaking, what is won at the beginning is eventually given back.

We end up this section with a remark concerning the validity range of our results. We focused the discussion above on the analysis of Fig. 6 valid for a given set of parameters. Yet actually, we also performed a systematic study in the case of a single square pulse by deriving analytical expressions of N α (t), Q α (t), and W ext (t) in the WBL. Though such expressions are lengthy (not shown), their numerical evaluation is almost immediate whereas using Tkwant with a large λ and integrating subsequently over time is much more expensive in computation time. This approach allowed us to investigate about 200000 configurations in the parameter space (T L > T R , µ L < µ R ,V 0 , ∆V, ∆t), the choice of Γ value playing no role after proper scaling as noticed in Sec.V B. We restricted ourselves to parameter sets leading to Seebeck configurations in the stationary cases ε 0 = V 0 and ε 0 = V 0 + ∆V , and explored randomly the parameter space by keeping roughly equidistant points with respect to the corresponding stationary efficiencies. We used for that purpose the Adaptive Python package [START_REF] Nijholt | Adaptive: parallel active learning of mathematical functions[END_REF] . We monitored η R net (t) and found many configurations yielding behaviors qualitatively similar to the one shown in Fig. 6(f), while other configurations showed a transient decrease of η R net (t) or oscillating decaying behaviors around η ss . The parameter set in Fig. 6 was chosen randomly among the most promising ones i.e. the ones giving the largest transient net efficiency and the largest increase with respect to η ss , keeping η R net (t) > η ss for all times. Then, only this configuration was subjected to a cyclic square driving. Moreover, the value of the period τ was chosen so as to increase significantly the convergence time of η R net to η ss ≈ η st . Other simulations were also run for a few other values of τ and showed similar behaviours i.e. a convergence of the net efficiency to a constant value very close to η st . Hence, as discussed above, we found no advantage on the net efficiency of driving the dot as the transient increase of η R net (t) cannot be leveraged in practical heat engines that need to operate for long times. Our conclusion is nevertheless purely empirical and still lacks a rigorous proof.

VII. CONCLUSION

We have studied thermoelectric transport in the noninteracting time-dependent RLM. We have first used our scattering approach to construct a compact analytical framework describing thermoelectric transport within the WBL approximation but beyond the adiabatic and weak coupling limits. Our time-resolved expressions of particle and heat currents as well as input power are the counterparts in the scattering approach of formulas previously derived in the literature [START_REF] Jauho | Time-dependent transport in interacting and noninteracting resonant-tunneling systems[END_REF][START_REF] Crépieux | Enhanced thermopower under a time-dependent gate voltage[END_REF][START_REF] Zhou | Boosting thermoelectric efficiency using time-dependent control[END_REF][START_REF] Daré | Time-dependent thermoelectric transport for nanoscale thermal machines[END_REF][START_REF] Kara Slimane | Simulating time-dependent thermoelectric transport in quantum systems[END_REF] using the non-equilibrium Green's function technique. Our time-integrated formulas also reproduce formally expressions given in previous works [START_REF] Entin-Wohlman | Heat currents in electronic junctions driven by telegraph noise[END_REF][START_REF] Gaury | Numerical simulations of time-resolved quantum electronics[END_REF][START_REF] Gurvitz | Generalized landauer formula for time-dependent potentials and noise-induced zero-bias dc current[END_REF] though the energy-related quantities were calculated in Ref. [START_REF] Entin-Wohlman | Heat currents in electronic junctions driven by telegraph noise[END_REF] for a telegraph noise while Eqs.( 22) and ( 23) are valid for arbitrary pulses (of finite duration). Moreover, we have pushed forward our analytical approach in the peculiar case where the dot is driven with a single square pulse and performed numerical simulations to study the convergence towards the WBL and deviations appearing when the square pulse is smoothed. Our analytics allowed us to spot interesting regimes that we studied numerically in a second step so as to address the cases where the pulse is repeated periodically. Considering a device in a stationary Seebeck configuration, we have studied the possibility of enhancing the efficiency of the heat engine by time-dependent driving. We have observed that the net thermodynamic efficiency of the device, defined over a cycle of a periodic drive or over the duration of an experiment (i.e. after switching on and before switching off the driving), is not enhanced with respect to the stationary efficiency. The advantage brought by the driving that may appear in the transient regime at short times is lost in the long run. It is noteworthy that, at a rough qualitative level, similar conclusions were drawn in Ref. [START_REF] Snyder | Supercooling of peltier cooler using a current pulse[END_REF] reporting on classical experiments of Peltier cooling driven by a current pulse.

Our work contributes to the growing literature using the RLM as a test bed to investigate transport and thermodynamics in out-of-equilibrium nanodevices. It brings new insights into the field of time-dependent (non-adiabatic) quantum thermoelectricity but also suffers from severe limitations of the employed model. Indeed, the presence of a finite displacement current in our non-interacting RLM plays a crucial role in the behavior of the thermodynamic efficiencies in Sec.VI and this picture will be drastically modified after inclusion of electron-electron interactions. Moreover, it is obviously delicate to define the efficiency of a driven heat engine without including the electrostatic environment of the junction (i.e. the load). Both ingredients could be included in future numerical simulations, either with a self-consistent mean-field approach or with more advanced techniques.

FIG. 1 .

 1 FIG.1. Sketch of the time-dependent Resonant Level Model studied in this paper. A central site playing the role of the dot is attached to two left (L) and right (R) leads modeled by semi-infinite 1D chains. For times t ≤ 0, the model Hamiltonian is static but the system can be driven out of equilibrium by applying a temperature bias T L -T R and/or an electrochemical potential bias µ Lµ R across the left and right electronic reservoirs attached to the leads Additionally, for times t > 0, the onsite energy ε 0 in the dot can be varied in time with e.g. a back gate.

FIG. 4 .

 4 FIG. 4. (a) Zoom of Fig.3(c) with additional Tkwant data for λ = 100 (green line) and λ = 150 (blue line). The amplitude of the cusps at t = 0 and t = ∆t increases with λ . (b) Integral t m 0 dt[I H L (t) -I H L (0)]/Γ of data shown in (a) as a function of λ , with t m = 1.75Γ -1 . Empty [full] circles correspond to the numerical (Simpson's) integration of Tkwant data with a time step δt = 0.002Γ -1 [δt = 0.0001Γ -1 ]. The dashed line indicates the value of the integral calculated in the WBL with Simpson's rule using points of the dashed line in (a) separated by δt = 0.002Γ -1 . The full line indicates this value when the integral over time is done analytically. (c) I H L (t) calculated with Tkwant for a smoothed square pulse, with different values of the characteristic length ξ (Γξ = 0.0001 (black line), 0.01 (red line), 0.04 (green line), 0.1 (blue line), and 0.2 (purple line)) and for Γ∆t = 0.5 and λ = 200. Other parameters are the same as in Fig.3. (d) Integral t m 0 dt[I H L (t) -I H L (0)]/Γ of data shown in (c) as a function of ξ (circles) with t m = 2.5Γ -1 (for t ≥ 2.5Γ -1 , I H L (t) is (almost) independent of ξ ). The dashed line serves as a guide to the eye.

FIG. 6 .

 6 FIG. 6. Transient increase of the heat engine efficiency. (Left) Particle currents I N L,R (b), heat currents I H L,R (c), input power P (d), instantaneous efficiency η t (e), and net efficiency η net (f) as a function of time t when the single square pulse V (t) shown in (a) is applied. The different curves in (e) and (f) correspond to different definitions of the efficiencies (η L t/net (in red), η R t/net (in blue), see main text). The horizontal black line indicates the Carnot efficiency value η C .In panels (b) to (f), each quantity is also evaluated within the quasistatic approximation in the WBL (gray dashed lines). Parameters:Γ L = Γ R = Γ, T L = 87Γ, T R = 25Γ, µ L = -26Γ, µ R = 26Γ, V 0 = 55Γ, ∆V = 2Γ, ∆t = 0.08/Γ, γ = Γ,and λ = 200. (Right) Same as left panels when the square pulse V (t) is repeated periodically for t > 0, with a period τ = 0.1/Γ, as shown in (g). The inset in (l) shows the convergence of η net to the steady state efficiency (black dotted line).

ACKNOWLEDGMENTS

We acknowledge the financial support of the Cross-Disciplinary Program on Numerical Simulation of CEA, the French Alternative Energies and Atomic Energy Commission. We thank Christophe Goupil for bringing Ref. 54 to our attention.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Appendix A: Derivation of generic formulas in the wide-band limit

We outline here the derivation of Eqs.( 10)-( 14) and ( 19)- (22) obtained in the WBL in Section IV. The WBL hypothesis has two main practical consequences. First, k(E) and v(E) can be approximated by their values k and v at E = 0. This allows us to rewrite the scattering states in Eq.( 2) as

with e ikn / |v| ≈ i n / 2λ γ, and similarly for Ψ RE n̸ =0 (t) with r ′ (t, E) and d ′ (t, E).

Second, the scattering amplitudes can be evaluated in the WBL by doing a gauge transformation to move the time dependent pulse V (t) from the dot to the leads, and then by combining the scattering amplitudes of three elementary scatterers in series (the pulse in the left lead, the time-independent dot, and the pulse in the right lead). To proceed, we introduce the generic notation S αβ for the scattering amplitudes, with S LL = r, S RR = r ′ , S RL = d, and S LR = d ′ . In the stationary case (i.e. when

The stationary scattering amplitudes read

They satisfy

To calculate the scattering amplitudes in the time-dependent case, we start from the RLM model sketched in Fig. 1 and make a gauge transformation to move the time dependency into the leads. Then we assume that due to the WBL approximation, we can restrict the time dependency to the outermost parts of the lead (e.g. the left lead does not stop at site -1 but at site -i with large i).

The scattering problem can now be solved by combining the scattering amplitudes of a voltage pulse in an infinite lead and of the stationary dot. Importantly, in the WBL approximation, electrons are perfectly transmitted across the abrupt voltage drops in the leads (no reflections) but their energies are redistributed and we have andd ′ p (E ′ , E) are respectively the left-to-right and right-to-left transmission amplitudes associated to the pulse in the right lead, or by symmetry the rightto-left and left-to-right transmission amplitudes associated to the pulse in the left lead. Here K(U) = dt e iUt e iφ (t) with φ (t) = t 0 duV (u). Therefore 43

Eq.(A4c) linking d(t, E) and r(t, E) can be easily derived by writing down the time-dependent Schrödinger equation for the scattering states (2) upon neglecting the energy dependency of the velocity v(E) (WBL approximation). And similarly for Eq.(A4d). Eq.( A4a) is also trivial to show by writing the Fourier transform of Eq.(A3). By inserting Eqs.(A1) and (A4) into Eqs.( 4), ( 7) and ( 8), we deduce Eqs.( 10)-( 14).

Finally, it can be shown using Eqs.(A4a) and (A2a) that d(t, E) obeys

where

and the same equation holds by replacing r by r ′ . Using Eqs.(A4b)-(A4c), (A5a), and the fact that V (t) is supposed to be finite in time in Section IV B (so that dt ∂ t |d| 2 = 0), we deduce Eqs. (19) and (20). The fact that dE T dyn (E) = 0 follows from Eq.(A6). Then Eqs.( 10)- (12) give Eq.( 21). Finally, using on one hand Eqs.( 13), ( 19)-( 21) to express ∆Q α , and on the other hand the set of equations (A5) and Eq.( 14) to express W ext , as well as again the fact that V (t) is finite in time, we derive Eq.( 22).


Ph.D Clara Bled
email: clara.bled@univ-tlse2.fr

L Bouvet

Validation of the French version of Object Spatial Imagery and Verbal Questionnaire

Keywords: cognitive style, object imagery, spatial imagery, psychometric proprieties, French validation

Validation de la version française du

individual's manner of cognitive functioning, particularly with respect to acquiring and processing information [START_REF] Messick | Personality consistencies in cognition and creativity[END_REF]. Many forms of cognitive styles have been described but the one most frequently used is the visual-verbal style [START_REF] Kozhevnikov | Cognitive styles in the context of modern psychology: Toward an integrated framework of cognitive style[END_REF]. The visualverbal style is based on the dual-coding theory according to which information is processed and mentally represented in two ways: verbally and visually [START_REF] Paivio | Dual coding theory: Retrospect and current status[END_REF]. The Visual-Verbal cognitive style therefore describes consistencies and preferences in processing visual versus verbal information. [START_REF] Mayer | Three Facets of Visual and Verbal Learners: Cognitive Ability, Cognitive Style, and Learning Preference[END_REF] further define the "visualizer-verbalizer cognitive style" as thinking in pictures or words. These preferences reflect "heuristics and individual uses to process information about his or her environment" [START_REF] Kozhevnikov | Cognitive styles in the context of modern psychology: Toward an integrated framework of cognitive style[END_REF].

Visual representation (or visual mental imagery) has long been considered to be of a unitary nature, but Koshevnikov and her team have provided evidence for two distinct types of visual mental imagery: object and spatial imagery [START_REF] Kozhevnikov | Revising the Visualizer-Verbalizer Dimension: Evidence for Two Types of Visualizers[END_REF][START_REF] Kozhevnikov | Spatial visualization in physics problem solving[END_REF]. Object imagery allows pictorial and analogous representations of objects and scenes, whereas spatial imagery is more a schematic representation of objects and patterns with their spatial relationships. This conception is supported by brain research showing the existence of two distinct neuronal pathways for the processing of different aspects of visual information [START_REF] Farah | Visual and spatial mental imagery: Dissociable systems of representation[END_REF][START_REF] Kosslyn | Mental images and the Brain[END_REF][START_REF] Kosslyn | The case for mental imagery[END_REF]. The ventral pathway is dedicated to processing the visual appearance of objects and scenes in terms of their shape, color and texture. The dorsal pathway processes object location, movement, spatial relationships and transformations [START_REF] Kosslyn | Neural foundations of imagery[END_REF][START_REF] Mazard | A PET meta-analysis of object and spatial mental imagery[END_REF]. Furthermore, visuo-spatial abilities are associated with a more efficient use of the dorsal pathway [START_REF] Mayer | Three Facets of Visual and Verbal Learners: Cognitive Ability, Cognitive Style, and Learning Preference[END_REF][START_REF] Lamm | Differences in the ability to process a visuo-spatial task are reflected in event-related slow cortical potentials of human subjects[END_REF] whereas visuo-object abilities are associated with a more efficient use of the ventral pathway [START_REF] Motes | Object-processing neural efficiency differentiates object from spatial visualizers[END_REF]. In this context, object versus spatial cognitive styles refer to an individual's preferences or habits of processing information in terms of object and spatial visual mental imagery.

Numerous self-reported questionnaires have been created to assess cognitive style, the most famous one being the Verbalizer-Visualizer Questionnaire (VVQ; [START_REF] Richardson | Verbalizer-visualizer: A cognitive style dimension[END_REF]. However, the VVQ has been criticized for poor internal reliability [START_REF] Boswell | A study of the internal consistency and factor structure of the Verbalizer-Visualizer Questionnaire[END_REF].

Moreover, the visual subscale of this questionnaire did not correlate with visual-spatial aptitude measures, which suggests a lack of predictive validity of the VVQ [START_REF] Mayer | Three Facets of Visual and Verbal Learners: Cognitive Ability, Cognitive Style, and Learning Preference[END_REF]. This lack of predictive validity is also observed in other cognitive style questionnaires as noted in Kozhevnikov's review on cognitive style [START_REF] Kozhevnikov | Cognitive styles in the context of modern psychology: Toward an integrated framework of cognitive style[END_REF]. As pre-existing questionnaires presented certain flaws and none of them evaluated the two dimensions of visual mental imagery, Blajenkova and Kozhevnikov developed the Object Spatial Imagery and Verbal Questionnaire (OSIVQ; [START_REF] Blazhenkova | The new object-spatial-verbal cognitive style model: Theory and measurement[END_REF]. The OSIVQ thus evaluates three dimensions of cognitive style: object imagery, spatial imagery and verbal. Blajenkova and Kozhevnikov (2009) evaluated the psychometric properties of the OSIVQ among 625 students. They found that the OSIVQ showed good internal construct validity (with a three-factor solution of a principal component analysis), good internal consistency (for the verbal scale, Cronbach alpha reached .74; for the object scale, α = .83 and for the spatial scale, α = .79) [START_REF] Blazhenkova | The new object-spatial-verbal cognitive style model: Theory and measurement[END_REF]. Furthermore, the OSIVQ demonstrated acceptable test-retest reliability with significant correlations between testing sessions (r = .73 for the verbal scale, r = .75 for the object scale and r = .84 for the spatial scale).

Moreover, unlike many previous questionnaires, OSIVQ scores significantly correlated with performances on different tasks, ensuring the convergent validity of the instrument. "Object visualizers" were found to have good performances in object imagery tasks that required vivid visualization of pictorial properties (e.g., recognizing degraded objects) whereas "spatial visualizers" had good performances during spatial imagery tasks that involved visualization of schematic representations and spatial transformations (e.g., mental rotation) [START_REF] Blajenkova | Object-spatial imagery: A new self-report imagery questionnaire[END_REF][START_REF] Kozhevnikov | Spatial versus object visualizers: A new characterization of visual cognitive style[END_REF]. Finally, OSIVQ scores predicted professional interests or course choices. Visual artists were generally "object visualizers" and had a high object imagery ability, while scientists and engineers were generally "spatial visualizers" and had a high spatial imagery ability [START_REF] Blajenkova | Object-spatial imagery: A new self-report imagery questionnaire[END_REF][START_REF] Blazhenkova | The new object-spatial-verbal cognitive style model: Theory and measurement[END_REF]Blazhenkova & Kozhevnikov, 2010;[START_REF] Kozhevnikov | Trade-off in object versus spatial visualization abilities: restriction in the development of visual-processing resources[END_REF]. The OSIVQ is therefore an ecologically valid scale with good predictive validity and psychometric properties.

However, it is worth noting that [START_REF] Blazhenkova | The new object-spatial-verbal cognitive style model: Theory and measurement[END_REF] reported a significant gender effect on the OSIVQ scores. Women obtained higher scores than men on the Object scale. In contrast, men obtained higher scores than women on the spatial scale. As for the Verbal scale, no gender differences were observed [START_REF] Blajenkova | Object-spatial imagery: A new self-report imagery questionnaire[END_REF][START_REF] Blazhenkova | The new object-spatial-verbal cognitive style model: Theory and measurement[END_REF], 2010;[START_REF] Blazhenkova | Object-spatial imagery and verbal cognitive styles in children and adolescents: Developmental trajectories in relation to ability[END_REF][START_REF] Campos | Gender differences in imagery[END_REF]. Gender differences are frequently found in spatial mental imagery. For example, studies using the Mental Rotation Test (MRT; [START_REF] Vandenberg | Mental rotations, a group test of three-dimensional spatial visualization[END_REF] found significantly higher scores for men [START_REF] Blajenkova | Object-spatial imagery: A new self-report imagery questionnaire[END_REF]Blazhenkova & Kozhevnikov, 2010;[START_REF] Campos | Gender differences in imagery[END_REF][START_REF] Dean | The relationship between self-reports of imagery and spatial ability[END_REF][START_REF] Vandenberg | Mental rotations, a group test of three-dimensional spatial visualization[END_REF]. [START_REF] Hegarty | Ability and sex differences in spatial thinking: What does the mental rotation test really measure?[END_REF] established that the use of an analytic strategy ("global-shape" strategy) in this test, contrary to the use of imagery transformation, was positively correlated with accuracy. The difference between men and women in the MRT might be due to a difference in the use of strategies and not in spatial imagery per se. To test this hypothesis, [START_REF] Campos | Gender and age differences in spatial imagery and image rotation[END_REF] compared performance between men and women with tests measuring spatial imagery ability and mental rotation of images ability. Their results showed significant gender differences in image rotation (men obtained higher scores) but not in spatial imagery. Regarding the gender differences in the vividness of mental representations, the literature is more contradictory. In his review of the literature on the Vividness of Visual Imagery Questionnaire (VVIQ; [START_REF] Marks | Visual imagery differences in the recall of pictures[END_REF] [START_REF] Richardson | Gender differences in the vividness of visual imagery questionnaire: A meta-analysis[END_REF] concluded that women obtained higher scores than men while other authors found no gender differences with the VVIQ 2 [START_REF] Campos | Gender differences in imagery[END_REF][START_REF] Mckelvie | The VVIQ as a psychometric test of individual differences in visual imagery vividness: A critical quantitative review and plea for direction[END_REF]. With another questionnaire on the vividness of mental images, Betts' Questionnaire Upon Mental Imagery (Betts' QMI; [START_REF] Sheehan | A shortened form of Betts' questionnaire upon mental imagery[END_REF], [START_REF] White | The measurement of imagery vividness: Normative data and their relationship to sex, age, and modality differences[END_REF] found higher scores for women whereas other studies found no significant differences [START_REF] Campos | Gender differences in imagery[END_REF][START_REF] Campos | The Spanish version of the Betts' questionnaire upon mental imagery[END_REF][START_REF] Sacco | The Italian form of the questionnaire upon mental imagery (QMI)[END_REF].

To conclude, the OSIVQ evaluates the visual/verbal cognitive style allowing also to make the difference between object and spatial imagery and possesses robust psychometric properties.

Therefore, we aimed to validate a French version of the OSIVQ to provide an equivalent tool to assess cognitive style in the French community. To do so, we formulated several hypotheses in order to evidence the same structure as the original version of the OSIVQ [START_REF] Blazhenkova | The new object-spatial-verbal cognitive style model: Theory and measurement[END_REF] which is a three factors questionnaire that evaluates the three dimensions of cognitive style: object imagery, spatial imagery and verbal. First, the principal component analysis should reveal three distinct factors that will correspond to three dimensions of cognitive style. Second, confirmatory analysis should show that a three-factor model best fits the data. Finally, we should observe a positive correlation between participants' ratings on the object, spatial and verbal scales of the OSIVQ and the corresponding behavioral task.

Material and method

Participants

For the exploratory study, 450 participants (121 males and 329 females; age range 18 to 50, Mean = 31.51, SD = 8.90) completed the questionnaire. According to the formula N = Z*10 (with N = number of subjects and Z = number of items) a minimum of 450 subjects is necessary for accurate validation of a questionnaire with 45 items [START_REF] Hair | Multivariate data analysis[END_REF].

For the confirmatory study, 302 participants, different from those in the exploratory study, (63 males and 239 females; age range 18 to 50, Mean = 27.84, SD = 8.85) were recruited. 144 of them also took a paper-and-pencil assessment to evaluate spatial imagery, object imagery and verbal abilities in addition to the questionnaire. This number of participants is slightly below the number required according to the formula (N = 34*10 = 340).

Participants were recruited by posting an advertisement on social media (Facebook ®) and among undergraduate students in the University community. Participants were invited to participate in a study which evaluates difference in cognitive style.

General procedure

The original English version of the Object Spatial Imagery and Verbal Questionnaire (OSIVQ; [START_REF] Blazhenkova | The new object-spatial-verbal cognitive style model: Theory and measurement[END_REF] was translated into French using a classical translation-back translation procedure: the original version of the questionnaire was first translated from English into French by a native English-speaking person who is fluent in French. This French version was then translated back into English by a native Frenchspeaking English teacher. The two English versions were then compared in order to validate the French translation. The translated version of the questionnaire is presented in supplementary material.

For the exploratory study, participants responded to our French version of the Object Spatial Imagery and Verbal Questionnaire (OSIVQ) online. The online questionnaire was created using LimeSurvey©, and securely hosted by the University of Toulouse.

For the confirmatory study, participants were administered the French version of the OSIVQ and the following paper-and-pencil assessments: one spatial imagery test (Mental Rotation Test; [START_REF] Vandenberg | Mental rotations, a group test of three-dimensional spatial visualization[END_REF], one object imagery assessment (Vividness of Visual Imagery Questionnaire; [START_REF] Marks | Visual imagery differences in the recall of pictures[END_REF] and one verbal test (Mill Hill vocabulary test; [START_REF] Raven | Manual for Raven's Progressive Matrices and Vocabulary Scales[END_REF] described below. Each participant was given an anonymous code. This specific code was used for the online questionnaire and the paperpencil tasks. The order of tests was randomized.

All participants read an information sheet with explanations concerning the study and indicated their consent before beginning.

This study was approved by the Research Ethics Committee of Toulouse University and received authorization from the French National Commission for Informatics and Freedoms (CNIL) regarding the protection and anonymization of personal data.

Measures

Object Spatial Imagery and Verbal Questionnaire (OSIVQ; [START_REF] Blazhenkova | The new object-spatial-verbal cognitive style model: Theory and measurement[END_REF] French translation: OSIVQ is a 45-item self-report questionnaire designed to assess separately an individual's propensities for three different cognitive style dimensions (object imagery, spatial processing and verbal processing). The following instructions were provided to the participants: "This is a questionnaire about the way you think. Please read the following statements and rate each of them on a 5-point scale. Select 5 to indicate that you absolutely agree that the statement describes you, and select 1 to indicate that you totally disagree with the statement. Select 3 if you are not sure, but try to make a choice. It is very important that you answer all the items in the questionnaire". There was no time limit for the completion of the questionnaire and participants could go back to correct their answers if they wanted. Vividness of Visual Imagery Questionnaire (VVIQ; [START_REF] Marks | Visual imagery differences in the recall of pictures[END_REF] French translation by [START_REF] Santarpia | Évaluer la vivacité des images mentales dans différentes populations françaises[END_REF]: The VVIQ is a frequently used self-report measure of the vividness of visual mental images. Participants had to rate the vividness of mental images they were asked to create (e.g. "The sun is rising above the horizon into a hazy sky and a strong wind blows on the trees and on the lake, causing waves"). Participants responded on a 5-point scale (from "no picture at all" to "a perfectly clear and vivid picture"). There are 16 items and each item is scored from 1 to 5 and the total score (sum of the scores of all items) gives an indication of the vividness of mental images.

Mill Hill vocabulary test [START_REF] Raven | Manual for Raven's Progressive Matrices and Vocabulary Scales[END_REF] French adaptation by [START_REF] Deltour | Echelle de vocabulaire de Mill Hill de JC Raven[END_REF]: This test assesses vocabulary knowledge and verbal skills. In this paper and pencil test, participants have to choose and circle the word that has either the same meaning or nearly the same meaning (synonym) as the target word, among six others in a printed list. There are 34 items.

Data analysis

One spatial and three verbal items of the OSIVQ were negatively formulated and therefore reversed for the analysis. The internal consistency was measured with Cronbach alphas for each subscale of the OSIVQ. [START_REF] Mckelvie | Guidelines for judging psychometric properties of imagery questionnaires as research instruments: A quantitative proposal[END_REF] recommended alpha coefficients of .85 in order to accept the validity of scales, and the minimum acceptable coefficient should not be less than .70 [START_REF] Nunnally | Psychometric theory (2nd edit[END_REF]. Exploratory analyses were conducted using principal component factor analysis with Varimax rotation using IBM SPSS (Version 23.0).

For descriptive statistics and in order to evaluate a possible gender effect, Student's t-test and repeated measures ANOVAs were performed.

In order to assess the psychometric properties of the OSIVQ and its factorial validity, confirmatory factor analysis (CFA) and path analysis were used and computed using IBM

AMOS (Version 22).

We compared the two models: a two-factor model, as in the traditional model, and a threefactor model, as in Blazhenkova and Kozhevnikov's model (2009), with a second order CFA analysis. We compared the different models by using fit indices: chi-square test (χ²/df), root mean square error of approximation (RMSEA), Bentler's comparative fit index (CFI), goodness of fit index (GFI) and Akaike information criterion (AIC). It was assumed that the model could be considered satisfactory when the χ²/df was lower than 3, RMSEA lower than .10, CFI higher than .80 and GFI higher than .90 [START_REF] Tabachnick | Principal components and factor analysis[END_REF]. The AIC is a comparative measure of fit and so it is meaningful when two different models are estimated: the lower the value is, the better the fit [START_REF] Tabachnick | Using multivariate statistics Boston[END_REF].

Convergent validity of the OSIVQ was measured with correlations between the OSIVQ subscale scores and the paper-and-pencil assessment scores. A correlation with .30 < r < .50 is considered as low, moderate with .50 < r < .70 and strong with r > .70 [START_REF] Tabachnick | Principal components and factor analysis[END_REF].

Results

Exploratory study

Principal component analysis

In their principal components analysis, [START_REF] Blazhenkova | The new object-spatial-verbal cognitive style model: Theory and measurement[END_REF] found 18 factors with eigenvalues above 1.0. Only three factors had eigenvalues greater than the others and explained a total of 31.95% of the variance. Their second three-factor principal components analysis confirmed that the first factor (12.8% of variance) was loaded only by object imagery items, the second factor (12.3% of variance) only by spatial imagery items, and the third (6.8% of variance) was loaded only by verbal items. Thus, we expected that, as for the original OSIVQ, findings would support three distinct factors corresponding to the three dimensions of cognitive style: object imagery, spatial imagery and verbal.

Our initial principal component analysis revealed an 11-factor solution with eigenvalues above 1. Only three of them had eigenvalues distinctly higher (6.23, 5.73 and 3.29) than the others (ranging from 2.05 to 1.02). These first three factors explained 33.88% of the variance in total (13.84%, 12.73% and 7.31%). None of the other factors met the recommendations for component saturation [START_REF] Guadagnoli | Relation of sample size to the stability of component patterns[END_REF] and the analysis of the decreasing eigenvalue curve ("scree plot") also revealed a three-factor solution. Therefore, other solutions were not considered further.

Based on these results, a second principal component analysis with Varimax rotation was performed with 3 factors. First of all, the test conditions of use were checked using the Bartlett test. The sphericity test was significant (χ²(990) = 7321.69, p < .001), and the Kaiser-Meyer-Olkin test (KMO) was greater than 0.80 (KMO = 0.839), indicating sufficient correlations for component analysis. Items that loaded significantly on multiple factors were removed from the analysis. Two items were considered to "cross-load" if the difference between two factors was less than .20 [START_REF] Tabachnick | Principal components and factor analysis[END_REF]. Items that did not load significantly (below the critical value of .35) and therefore did not correlate with any factor were also eliminated [START_REF] Hair | Multivariate data analysis[END_REF]. Thus,11 items (3,10,15,19,21,22,24,25,28,36 and 38) were removed from the original questionnaire in order to obtain a coherent structure.

The 34 remaining items of the OSIVQ (14 items for the object scale, 11 items for the spatial scale and 9 items for the verbal scale) and their loadings on the 3 factors are presented in Table 1.

INSERT TABLE 1 HERE

The first factor was identified as the object imagery factor, since all of the items designed to assess object imagery loaded positively on this factor, whereas all of the items designed to assess spatial imagery or verbal preferences did not load. The second factor was identified as the spatial imagery factor, since all of the items designed to assess spatial imagery loaded positively on the second factor, whereas none of other items did. Finally, the third factor was identified as the verbal factor, since all of the items designed to assess verbal preferences loaded positively on the third factor and other items did not load on this factor. 1

Internal consistency

The internal consistency was measured with Cronbach's alpha. For the verbal scale, α = .76; for the object scale, α = .84 and for the spatial scale, α = .85. Cronbach alphas showed a good internal consistency for the three sub-scales, with the selected items.

1

As there is a gender imbalance in our participants, an exploratory principal component analysis was also performed with a smaller group containing the same number of women and men. With this group, the same results were obtained. As no difference was found and in order to have a satisfactory statistical power, analyses were conducted on all participants.

In the original version of [START_REF] Blazhenkova | The new object-spatial-verbal cognitive style model: Theory and measurement[END_REF] Cronbach alphas reached .74 for the verbal scale; α = .83 for the object scale and α = .79 for the spatial scale.

Descriptive statistics and gender differences

As a result of the selection of the 34 items, the number of items per subscale is not equivalent.

Thus, for each participant, the scores for each subscale were transformed into percentages.

For the three subscales, the mean Spatial score was 60.65 (± 17.09); the mean Object score was 72.10 (± 14.39) and the mean Verbal score was 66.64 (± 15.48).

We then examined the effect of gender on object-spatial-verbal preferences for information processing, using a repeated measures ANOVA. Our results demonstrated a significant difference between the three scales of the OSIVQ, F(2, 896) = 21.375, p < .001, η 2 = .046.

Participants were inclined to rate themselves higher on Object imagery than on Spatial imagery (p < .001) or Verbal dimensions (p < .001) but there was no difference between Spatial imagery scores and Verbal scores (p = .092).

The repeated measures ANOVA showed no effect of gender, F(1, 448) = 3.396, p = .066, η 2 = .008, but a significant Gender*OSIVQ interaction, F(2, 896) = 34.968, p < .001, η 2 = .072.

Male participants scored higher than female participants on the OSIVQ spatial subscale [Mean male participants = 69.80, SD = 13.80, Mean female participants = 57.29, SD = 16.96, t(448) = 7.27, p < .001]. Female participants had higher OSIVQ object imagery scores than male participants [Mean female participants = 73.56, SD = 14.22, Mean male participants = 68.16, SD = 14.15, t(448) = -3.58, p < .001] . We found no significant gender difference on OSIVQ verbal scores, t(448) = -1.06, p = .288.

Confirmatory study

Confirmatory factor analysis

According to the traditional model of Visual-Verbal style, Verbal and Visual styles constitute two opposite dimensions whereas Blazhenkova and Kozhevnikov's theoretical approach distinguishes two types of imagery (object and spatial). According to Blazhenkova and Kozhevnikov's revised model of cognitive style, there should be three separate dimensions (object, spatial and verbal). In their original version of the questionnaire, Blazhenkova and Kozhevnikov indeed found that a three-factor model was the one that best fitted the data.

Following our exploratory analysis, some items were eliminated. As the structure of the questionnaire was modified, we decided, to also test a three-factor model and a two-factor model in our confirmatory factor analysis.

Figure 1 shows the results of the estimated three-factor model, and the values of fit for the different models are reported in Table 2.

INSERT FIGURE 1 HERE

As shown in Figure 1, items from the verbal scale scores loaded on a "verbal" factor, items from the spatial imagery scale scores loaded on a "spatial" factor and items from the object imagery scale scores loaded on an "object" factor. Interestingly, results of the confirmatory factor analysis indicated no significant correlation between our three factors (r = -.16 between spatial and verbal factors; r = .08 between spatial and object factors and r = .19 between object and verbal factors) (see Figure 1).

INSERT TABLE 2 HERE

In order to confirm our item selection, we tested the fit of the three-factor model with all 45 original items of our translation. The values showed that the fit of our 34 selected items scale was better than the one with the 45 original items (see Table 2).

Secondly, we compared the three-factor and the two-factor models using our 34 selected items. The two-factor model showed poor fit indices and the AIC score was lower than with the three-factor model. Overall, indices suggested that the three-factor model fits the data well which indicates that this model does explain the data. This is why the three-factor solution was retained.

Convergent validity

Our hypotheses were in line with those of [START_REF] Blazhenkova | The new object-spatial-verbal cognitive style model: Theory and measurement[END_REF] in the original version of the OSIVQ. We expected positive correlations between the object scale scores of the OSIVQ and the VVIQ scores (as the VVIQ measures the vividness of mental imagery), the spatial scale scores and the MRT scores (as the MRT requires spatial manipulations), and the verbal scale scores and the Mill Hill scores (as the Mill Hill is a verbal measure).

Correlations among the measures of verbal, spatial and object ability are presented in Table 3.

Higher OSIVQ verbal scores were associated with higher scores on the Mill Hill measure of verbal capacity (p < .001). Higher OSIVQ spatial scores were associated with higher scores on the MRT measure of spatial imagery (p < .001). Finally, higher OSIVQ object imagery scores were associated with higher scores on the VVIQ measure of object imagery (p < .001).

Thus, all three scales of the OSIVQ demonstrated acceptable convergent validity.

Regarding the discriminant validity of the three scales, although none of the OSIVQ scales correlated positively with any task corresponding to the other scales, there was a significant but weak negative correlation between the OSIVQ verbal scores and the MRT scores (r = -.29, p = .001).

INSERT TABLE 3 HERE

Discussion

We developed a French version of the Object Spatial Imagery and Verbal Questionnaire (OSIVQ), a self-report instrument that allows the evaluation of the visual-verbal cognitive style in three dimensions with two imagery (object and spatial) scales and one verbal scale.

The first important outcome of the exploratory study is that, as hypothesized, our translated version of the OSIVQ (with the 34 selected items) demonstrated acceptable internal consistency for all three scales. Furthermore, our principal component analysis also revealed three distinct factors (object, spatial and verbal). These results are thus consistent with Blazhenkova and Kozhevnikov's model of cognitive style [START_REF] Blazhenkova | The new object-spatial-verbal cognitive style model: Theory and measurement[END_REF].

The confirmatory study provided further validation of our French version of the OSIVQ. The traditional two-factor model that describes Visual-Verbal cognitive style as a bipolar dimension was compared to the three-factor model that comprises three independent dimensions (verbal, object and spatial). The results of our confirmatory factor analysis indicate that the overall fit of the three-factor model was indeed significantly greater than that of the two-factor model. It should be noted that the number of participants in the confirmatory study is a little below the recommended minimal number needed, which might explain why the fit of our three-factor model is satisfactory but could be improved if a larger sample was tested. Overall, these findings again provide support for the Blazhenkova and Kozhevnikov three-dimensional model of cognitive style.

Lastly, in order to validate our translation, our data showed acceptable convergent validity.

Indeed, participants' ratings on the object, spatial and verbal scales correlated positively with the corresponding task only. Although we did not find significant correlations between the verbal and spatial factors nor between OSIVQ verbal and spatial scores, we did find a weak but significant negative correlation between the OSIVQ verbal scores and MRT scores. This suggests that there might be some interference between spatial and verbal information processing. It is worth noting that a negative correlation between verbal and spatial factors was similarly found by [START_REF] Blazhenkova | The new object-spatial-verbal cognitive style model: Theory and measurement[END_REF]. They also found negative correlations between OSIVQ object and spatial imagery measures which we did not find.

Rather, our results indicate that object and spatial imagery are independent. The difference might be due to the number of items used in the two studies.

Another important finding is that the distribution of object, spatial and verbal scores differs among the three subscales. Participants tend to rate themselves higher on the object imagery scale than on other scales. This trend is consistent with previous findings [START_REF] Blazhenkova | The new object-spatial-verbal cognitive style model: Theory and measurement[END_REF][START_REF] Chabris | Spatial and object visualization cognitive styles: Validation studies in 3800 individuals[END_REF]. In addition, we also reported a significant gender effect, with females tending to report themselves as object visualizers and males as spatial visualizers. This difference is not surprising as previous studies reported a gender difference on spatial tasks with males performing better than females [START_REF] Linn | Emergence and characterization of sex differences in spatial ability: A meta-analysis[END_REF] and higher spatial imagery ratings for males, while females tend to report higher object imagery and imagery vividness [START_REF] Blajenkova | Object-spatial imagery: A new self-report imagery questionnaire[END_REF]. According to Blajenkova et al., the difference between spatial and object visualizers, however, cannot be reduced to gender differences [START_REF] Blajenkova | Object-spatial imagery: A new self-report imagery questionnaire[END_REF][START_REF] Kozhevnikov | Spatial versus object visualizers: A new characterization of visual cognitive style[END_REF]. In fact, 40.42% of our female participants reported above average spatial imagery preferences and 38.02% of male participants reported above average object imagery preferences. Furthermore, using partial correlations, Blazhenkova and Kozhevnikov found that all the significant correlations between different object, spatial and verbal scales and corresponding objective measures remained significant even after the effect of gender was partialled out [START_REF] Blazhenkova | The new object-spatial-verbal cognitive style model: Theory and measurement[END_REF].

One limitation of our study is that we were not able to evaluate the test-retest reliability and the ecological validity of this translated questionnaire. It would be interesting in the future to reevaluate some of our participants. Similarly, we were unable to assess cognitive style in different professional domains or university courses as did [START_REF] Blazhenkova | The new object-spatial-verbal cognitive style model: Theory and measurement[END_REF]. Nevertheless, our translated questionnaire showed good psychometric properties. We believe that it was necessary to translate this questionnaire in order to provide a French tool that evaluates cognitive style while making the distinction between object and spatial imagery since cognitive styles have important every-day life implications that need to be understood.

In fact, the original version of the OSIVQ has allowed many psychology and cognition studies to be conducted that showed the importance of cognitive style. For instance, when learning with texts and pictures, the learner's cognitive style might have a direct influence on learning behavior and preferences. Indeed, when learning with text-picture combinations, participants scoring high on the object scale and/or the spatial scale of the OSIVQ relied more heavily on pictures than on texts, while participants scoring high on the verbal scale tended to rely more on text [START_REF] Höffler | More Evidence for Three Types of Cognitive Style: Validating the Object-Spatial Imagery and Verbal Questionnaire Using Eye Tracking when Learning with Texts and Pictures: Evidence for three types of cognitive style[END_REF]. Moreover, cognitive style also seems be linked with mathematical performances. "Spatial visualizers" had significantly higher spatial ability and performed better in mathematical tests than "object visualizers" [START_REF] Haciomeroglu | Object-spatial Visualization and Verbal Cognitive Styles, and Their Relation to Cognitive Abilities and Mathematical Performance[END_REF].

Finally, cognitive style has been shown to influence spatial orientation. "Object visualizers" were better than "spatial visualizers" at recognizing visual clues in an orientation task whereas "spatial visualizers" were better than "object visualizers" at judging relative positions between the visual clues [START_REF] Kraemer | Verbalizing, visualizing, and navigating: The effect of strategies on encoding a large-scale virtual environment[END_REF]. A link between cognitive style and creativity was also established. Object visualization seems to be linked to artistic creativity (physical creativity), spatial visualization is linked to scientific creativity (creativity in math and science), while both are distinct from verbal creativity (creativity in interpersonal relationships, communication, and writing) [START_REF] Kozhevnikov | Creativity, visualization abilities, and visual cognitive style[END_REF].

In conclusion, our French translation of the OSIVQ presents good psychometric properties and is in agreement with the latest theories on cognitive style. Furthermore, this French version of the OSIVQ can certainly be a valuable tool for future psychological research. 34 I can close my eyes and easily picture a scene that I have experienced .69

33 Sometimes my images are so vivid and persistent that it is difficult to ignore them .62

6 My images are very colorful and bright

.62

13 I have a photographic memory .62

43 My visual images are in my head all the time. They are just right there .61

26 When I imagine the face of a friend, I have a perfectly clear and bright image .57

11 When reading fiction, I usually form a clear and detailed mental picture of a scene or room that has been described .53

45 When I hear a radio announcer or a DJ I've never actually seen

  questionnaire d'imagerie objet spatiale et verbal (OSIVQ) mesure les différences individuelles dans le style cognitif Visuel-Verbal et fait la distinction entre l'imagerie visuelle spatiale et l'imagerie visuelle objet. Objectif : Le but de cette étude est d'évaluer les propriétés psychométriques d'une version française de l'OSIVQ. Méthode : Sept cents cinquante-deux participants ont complété l'OSIVQ français et 144 d'entre eux ont également passé des tests papier-crayons pour évaluer l'imagerie spatiale, l'imagerie objet et les capacités verbales. Résultats : Les analyses en composantes principales de l'OSIVQ ont démontré une bonne construction interne. Les résultats ont également révélé d'importantes différences individuelles et un effet significatif du genre. Les analyses factorielles confirmatoires sont en accord avec un modèle tridimensionnel du style cognitif. De plus, des corrélations ont été observées entre les différentes dimensions du questionnaire et les tâches papier-crayons spécifiques. Conclusion : Nos résultats permettent de valider une version française de l'OSIVQ. Mots clés : style cognitif, imagerie objet, imagerie spatiale, propriétés psychométriques, validation française Cognitive styles refer to psychological dimensions representing consistencies in an

  figure, and the other two comparison figures were rotated mirror images of the target figure.

  à un concept abstrait (ou à une construction), j'imagine mentalement une construction abstraite schématique ou son plan plutôt quentre dans un magasin familier pour obtenir un article spécifique, je peux facilement visualiser la localisation exacte de cet article, l'étagère sur laquelle il se trouve, comment il est arrangé et les articles environnants 1 -2 -3 -4 -5 19 Il est beaucoup plus facile pour moi d'assembler des meubles (par exemple, un meuble de télévision ou une chaise) lorsque j'ai des instructions verbales détaillées que lorsque je n'ai qu'un schéma ou une image 1 explique quelque chose, je préfère donner des explications verbales plutôt que de faire des dessins ou des croquis 1 un devait me donner des nombres à deux chiffres à additionner (par exemple, 43 et 32), je ferais simplement l'addition sans visualiser les nombres 1 -2 -3 -4 -523Mes images mentales de différents objets ressemblent beaucoup, en termes de taille, de forme et de couleur, aux vrais objets que j'ai vus 1 -2 -3 -4 -5 24 Habituellement, je n'essaie pas de visualiser ou de dessiner des diagrammes lors de la lecture d'un manuel 1 -2 -3 -4 -5 25 Normalement, je n'ai pas l'expérience de beaucoup d'images vives spontanées. J'utilise mon imagerie mentale surtout lorsque j'essaie de résoudre certains problèmes comme ceux en mathématiques 1 rappelle d'une scène, j'utilise des descriptions verbales plutôt que des images mentales 1 -2 -3 -4 -5

Table 1 .

 1 Principal component loadings, after Varimax rotation, for the translated OSIVQ items.

	Factor 1	Factor 2	Factor 3
	Object	Spatial	Verbal

Table 2 .

 2 Values of fit for the different models

	, I usually find

Table 3 .

 3 Matrix of correlations among the OSIVQ spatial, object and verbal scores and paperand-pencil measures

Acknowledgments

We thank Teresa Blicharska Strayer for her help in the translation of this questionnaire and Patrick Raynal for his insightful comments on this article. We also thank the undergraduate students for their assistance in data collection. We are grateful for the participants' time and commitment.

Funding

This work was funded by the French National Research Agency (ANR-19-CE28-0012).

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflict of interest

The authors have no conflict of interest to declare.

Supplementary material

Questionnaire d'Imagerie Spatiale-Objet et Verbal (OSIVQ)

Ceci est un questionnaire sur votre façon de penser. Merci de lire attentivement les