Paul Eyméoud

Adeline Crépieux

Mixed electrical-heat noise spectrum in a quantum dot

Using the Keldysh Green function technique, we calculate the finite-frequency correlator between the electrical current and the heat current flowing through a quantum dot connected to reservoirs. At equilibrium, we find that this quantity, called mixed noise, is linked to the thermoelectric acconductance by the fluctuation-dissipation theorem. Out-of-equilibrium, we discuss its spectrum and find evidence of the close relationship between the mixed noise and the thermopower. We study the spectral coherence and identify the conditions to have a strong correlation between the electrical and heat currents. The change in the spectral coherence due to the presence of a temperature gradient between the reservoirs is also highlighted.

IV. DISCUSSION

A. At equilibrium

I. INTRODUCTION

The electrical noise spectrum in quantum systems is accessible experimentally through very sensitive techniques such as spectrum analyzer 1,2 , use of a superconductorinsulator-superconductor tunnel junction as an on-chip spectrum analyzers 3 and measurement of photon emission spectrum 4 . It exists also a proposal to measure the heat statistics in quantum devices 5 . Given the fast progress of detection techniques, it is not forbidden to imagine that in the next few years, the measurement of the noise correlator between the electrical current and the heat current (mixed noise) would be possible. In parallel, calculations of finite-frequency electrical-heat mixed noise are needed for quantum systems. There exist very few works on the zero-frequency mixed noise [6][7][8][9][10] and not even one concerning the finite-frequency mixed noise. Theoretically, the studies are limited to the electrical noise spectrum (see [11][START_REF] Martin | Les Houches Session LXXXI[END_REF][START_REF] Zamoum | [END_REF] and references therein), to the energy noise spectrum [14][15][16] , to the statistics of the energy current in the presence of time-dependent excitation 17,18 , and to the heat noise spectrum 19 . This is regrettable since it has been shown recently that the zerofrequency mixed noise contains information on the thermoelectric response of the system 9,10 : it gives the figure of merit in the linear response regime and it is related to the thermoelectric efficiency in the weak transmission regime (Schottky regime). At finite-frequency, the mixed noise should bring information on the dynamics of the thermoelectric conversion, in particular on the thermoelectric response of time-modulated systems, which is the study of an increasing number of works [20][21][22][23][24][25][26][27][28][29] . In this paper, we fill this lacuna by calculating the mixed noise spectrum of a quantum dot (QD) using the Keldysh outof-equilibrium Green function technique. We focus on the non-symmetrized noise spectrum since this is the quantity which is relevant for quantum systems, due to the fact that the current operators do not commute with each other 3,30,31 . The paper is organized as follows: We present the model and give the definition of electrical and heat currents in Sec. II. The results for the noise spectra are presented in Sec. III, and discussed in Sec. IV. We conclude in Sec. V.

The charge current, Î0

α , and heat current, Î1 α , flowing from the reservoir α to the QD, are given by the time derivatives [START_REF] Haug | Quantum Kinetics in Transport and Optics of Semiconductors[END_REF][33][34] of the operators number of electrons in the reservoir α, N α , and energy of electrons in the reservoir α, H α : Î0 α = -e Ṅα , and Î1 α = -Ḣα + µ α Ṅα , with N α = k∈α c † kα c kα . The time derivatives of these two quantities are equal to Ṅα = i -1 H, N α , and Ḣα = i -1 H, H α , which lead after calculation to

Ṅα = i k∈α -V kα c † kα d + V * kα d † c kα , (1)
Ḣα = i k∈α ε kα -V kα c † kα d + V * kα d † c kα . (2)
Injecting Eqs. ( 1) and (2) in the definitions of Î0 α and Î1 α , we obtain:

Î0 α = ie k∈α V kα c † kα d -V * kα d † c kα , Î1 α = i k∈α (ε kα -µ α ) V kα c † kα d -V * kα d † c kα ,
which give in a compact form

Îp α = ie 1-p k∈α ε kα -µ α p V kα c † kα d -V * kα d † c kα . (3)

III. NOISE SPECTRUM

The non-symmetrized noise spectrum is defined as

S pq αβ (ω) = ∞ -∞ S pq αβ (t, 0)e -iωt dt , (4)
where S pq αβ (t, 0) = ∆ Îp α (t)∆ Îq β (0) is the current-current time-correlator, and ∆ Îp α (t) = Îp α (t) -Îp α , with Îp α , the electrical (p = 0) or heat (p = 1) current operator from the reservoir α to the central region through the barrier α. The finite-frequency non-symmetrized noise S pq αβ (ω) quantifies the correlation between the currents Îp α and Îq β at finite frequency ω. Such a general definition embeds three types of noise: (i) the charge noise, S 00 αβ (ω), corresponding to the correlator between the electrical current and itself; (ii) the mixed noises, S 01 αβ (ω) and S 10 αβ (ω), corresponding to the correlators between the electrical current and the heat current, and (iii) the heat noise, S 11 αβ (ω), corresponding to the correlator between the heat current and itself. We first compute the time-correlator S pq αβ (t, t ′ ) using the Keldysh out-of-equilibrium formalism 35 , and next calculate its Fourier transform in order to get S pq αβ (ω). To achieve this task, we insert the current operator, given by Eq. ( 3), in the definition of the noise, given by Eq. ( 4), and perform the calculation of the average of the product of four creation/annihilation operators, making the following assumptions: non-interacting electrons, wideband approximation, and symmetrical coupling strength between the reservoirs and the QD (i.e., symmetrical left and right barriers). The details of the calculation are given in the Appendix A. The final expression of finitefrequency non-symmetrized noise we obtain is

S pq αβ (ω) = e 2-p-q h ∞ -∞ dε × (ε -µ α ) p (ε -µ β ) q A αβ (ε, ω) +(ε -µ α ) p (ε -ω -µ β ) q B αβ (ε, ω) +(ε -ω -µ α ) p (ε -µ β ) q B * βα (ε, ω) +(ε -ω -µ α ) p (ε -ω -µ β ) q C αβ (ε, ω) , (5) with A αβ (ε, ω) = T (ε -ω)f h M (ε -ω) T (ε)f e M (ε) +[δ αβ -t(ε)]f e α (ε) + [δ αβ -t * (ε)]f e β (ε) , (6)
B αβ (ε, ω) = t(ε)t(ε -ω) [f e α (ε) -t * (ε)f e M (ε)] × f h β (ε -ω) -t * (ε -ω)f h M (ε -ω) , (7)
and

C αβ (ε, ω) = T (ε)f e M (ε) T (ε -ω)f h M (ε -ω) +[δ αβ -t * (ε -ω)]f h α (ε -ω) +[δ αβ -t(ε -ω)]f h β (ε -ω) , (8)
where

f e α (ε) = [1 + exp((ε -µ α )/k B T α )] -1 is the Fermi- Dirac distribution function for electrons, f h α (ε) = 1 - f e α (ε) is the distribution function for holes, f e,h M (ε) = [f e,h L (ε) + f e,h R (ε)
]/2 is the average left and right distribution, t(ε) is the transmission amplitude, and T (ε) = |t(ε)| 2 is the transmission coefficient. The transmission amplitude is related to the retarded Green function of the QD, G r (ε), through the relation t(ε) = iΓG r (ε), where Γ is the coupling strength between the QD and the reservoirs [START_REF] Zamoum | [END_REF] . Equation (5) gives the electrical noise when p = q = 0, it gives the mixed noise when either p = 0 and q = 1, or vice versa, and it gives the heat noise when p = q = 1. S pq αβ (ω) is a real quantity when p = q and α = β (auto-correlator), but can be complex otherwise (cross-correlator). We have checked that the electrical noise S 00 αβ (ω) extracted from Eq. ( 5) coincides with the results of the literature [START_REF] Zamoum | [END_REF]36 , and that the heat noise S 11 αβ (ω) extracted from Eq. ( 5) coincides with the existing results of the literature in the limit of energy-independent transmission amplitude 14 . The expressions for the mixed noises S 01 αβ (ω) and S 10 αβ (ω) are novels. This is the central result of this paper. It is valid at any frequency ω, coupling strength Γ between the QD and the reservoirs, QD energy level ε d , and for any temperature and voltage gradients between the left and right reservoirs.

In the following, we choose first to restrict our study to the case where temperatures for the left and right reservoirs are equal, T L = T R = T , and for ε d = 0 (electronhole symmetry point), and we discuss the mixed noise spectrum in three situations: (i) at equilibrium, (ii) for energy-independent transmission amplitude, and (iii) for an Anderson-type transmission amplitude. In the latter case, we also discuss the spectral coherence in the presence of a temperature gradient between the two reservoirs.

At equilibrium, i.e., µ L = µ R = ε F , where ε F is the Fermi energy for electrons in the reservoirs, and for equal left and right reservoir temperatures, i.e., T L = T R = T , we have from Eqs. ( 5)-( 8)

S pq αβ (ω) = e 2-p-q h ∞ -∞ dεf e M (ε)f h M (ε -ω) × ε p+q A αβ (ε, ω) + ε p (ε -ω) q B αβ (ε, ω) +(ε -ω) p ε q B * βα (ε, ω) + (ε -ω) p+q C αβ (ε, ω) , (9)
with

A αβ (ε, ω) = T (ε -ω) [2δ αβ -T (ε)] , (10)
B αβ (ε, ω) = t(ε)t(ε -ω) × [1 -t * (ε)] [1 -t * (ε -ω)] , (11)
C αβ (ε, ω) = T (ε) [2δ αβ -T (ε -ω)] , (12)
since for isothermal reservoirs at equilibrium, we have

f e,h L (ε) = f e,h R (ε) = f e,h M (
ε). From Eq. ( 9), it can be shown using S pq αβ (-ω) = e ω/kBT S qp αβ (ω) that the noise spectrum obeys the relation

S pq αβ (ω) = N ( ω)[S qp αβ (-ω) -S pq αβ (ω)] , (13)
where

N ( ω) = [exp( ω/k B T ) -1] -1
is the Bose-Einstein distribution function. Removing the reservoirs' index and using the definitions 37 of the electrical ac-conductance, G(ω) = [S 00 (-ω) -S 00 (ω)]/2 ω, the thermal ac-conductance K(ω) = [S 11 (-ω) -S 11 (ω)]/2 ωT , and the thermoelectric ac-conductance, X(ω) = [S 10 (-ω) -S 01 (ω)]/2 ωT which is the product of the ac-thermopower (i.e., Seebeck coefficient) by the electrical ac-conductance, we establish that the noises are related at equilibrium to the ac-conductances through the following fluctuation-dissipation relations

S 00 (ω) = 2 ωN ( ω)G(ω) , ( 14
) S 01 (ω) = 2 ωT N ( ω)X(ω) , ( 15
) S 11 (ω) = 2 ωT N ( ω)K(ω) . ( 16
)
Through these relations, we can state that in a similar way that the finite-frequency electrical noise contains information on the dynamics of the charge transfer, the finite-frequency heat noise contains information on the dynamic of the heat transfer (since G(ω) and K(ω) are the response to an excitation modulated in time). Moreover, Eq. ( 15) confirms the key role played by the mixed noise S 01 (ω) to quantify the thermoelectric conversion. Note that in the limit of zero-frequency, Eqs. ( 14)-( 16) reduce to the relations given in Ref. 9, since we have in that limit N ( ω) → k B T / ω.

B. Energy-independent transmission

For an energy-independent transmission amplitude, t, the real parts of the electrical, mixed, and heat noise spectra are given by Fig. 2 in the low-temperature limit. We do not plot their imaginary parts whose magnitudes are smaller with a factor 100 comparing to the ones of the real parts. Let us now discuss the features appearing on Fig. 2. First, we notice that similarly to the electrical noise, which cancels when the frequency is larger than the voltage, ω > eV , the mixed and heat noises cancel as well. The reason is the following: knowing that the noise is called emission noise at positive frequency and absorption noise at negative frequency 38 , we understand that the system can not emit an energy larger than the energy provided to it, here the voltage since temperature is taken small. Second, we observe that the electrical noise varies linearly or by plateaus with both voltage and frequency, due to the fact that when transmission is energy independent, the system works in the linear regime. Third, the mixed noise can change its sign whereas the electrical and heat noises keep a single sign. Fourth, the electrical and mixed correlators between distinct reservoirs, S pq LR , are equal in absolute values to the correlators in the same reservoir, S pq LL , and nearly equal for the heat correlator [START_REF]For an energy independent transmission amplitude, we have S 00 αα (ω) = -S 00 α ᾱ(ω), and S 01 Lα[END_REF] . αβ (ω) is plotted in units of e 2-p-q Γ 1+p+q /h. The right reservoir is grounded (µR = 0).

In the limit of weak or perfect transmission, i.e., T ≪ 1 or T = 1 respectively, the integration over energy in Eq. ( 5) can be performed analytically (see Appendix B for the details of the calculation). The expressions of the noises, which are all real in these limits, are given in Table I. These expressions constitute a generalization of the fluctuation-dissipation theorem to an out-of-equilibrium situation since the Bose-Einstein distribution function is estimated at frequency shifted by ±eV / . Concerning the electrical noise, its expression at T ≪ 1 is in full agreement with the result of perturbative calculations 40 . Concerning the mixed and heat noises expressions, there is no previous work to compare in the literature. Note that at zero-voltage, the mixed noise cancels in both limits (T ≪ 1 and T = 1) but not in the intermediate regime: the mixed noise is then given by Eq. ( 15) which is a priori non-zero. It is also worth to notice that S 11 αβ (ω) contains a contribution which is proportional to S 00 αβ (ω) with a proportionality factor equal to LT 2 , where L = π 2 k 2 B /3e 2 is the Lorenz number. Since in certain limits, the heat noise is related to the thermal conductance and the electrical noise to the electrical conductance, as through Eqs. ( 16) and ( 14) at equilibrium for example, it is not surprising to find a relation which involves the Lorenz number between the heat noise and the electrical noise thanks to the Wiedemann-Franz law, or between the thermal conductance and the electrical noise as obtained in Ref. 41. Table I gives also the sum over reservoirs of the electrical, mixed, and heat noises, αβ S pq αβ (ω). Contrary to the total electrical and mixed noises, which are equal to zero in the limits we consider (no charging effect on the QD), the total heat noise takes a finite value which indeed corresponds to the heat power fluctuations. At zerofrequency, the power fluctuations are conserved, i.e., the heat power fluctuations are equal to the electrical power fluctuations 9 . It is also true at finite-frequency provided that T = 1, since in that limit we have from Table I,

αβ S 11 αβ (ω) = V 2 S 00 αα (ω). At zero-temperature, we get for T = 1: αβ S 11 αβ (ω) = 2| ω|(eV ) 2 Θ(-ω)/h,

Type of noise

Notation

T ≪ 1 T = 1 Electrical noise S 00 αβ (ω) e 2 T h (2δ αβ -1) ± ( ω ± eV )N ( ω ± eV ) e 2 h (2δ αβ -1)2 ωN ( ω) Total electrical noise αβ S 00 αβ (ω) 0 0 Mixed noise S 01 αβ (ω) eT h (2δαL -1) ± ∓ ( ω±eV ) 2 2 N ( ω ± eV ) e h (1 -2δαL)eV ωN ( ω) Total mixed noise αβ S 01 αβ (ω) 0 0 Heat noise S 11 αα (ω) LT 2 S 00 αα (ω) LT 2 + V 2 2 + 2 ω 2 12e 2 S 00 αα (ω) (auto-correlator) + T h 3 ω 3 N ( ω) + ± ( ω±eV ) 3 3 N ( ω ± eV ) + 2 ω 2 4h ± ( ω ± eV )N ( ω ± eV ) Heat noise S 11 α ᾱ(ω) LT 2 S 00 α ᾱ(ω) LT 2 S 00 α ᾱ(ω) - 2 ω 2 12e 2 S 00 αα (ω) (cross-correlator) + T h ± ( ω±eV ) 3 6 N ( ω ± eV ) - 2 ω 2 4h ± ( ω ± eV )N ( ω ± eV ) Total heat noise αβ S 11 αβ (ω) T h 2 3 ω 3 N ( ω) + ± ( ω ± eV ) 3 N ( ω ± eV ) V 2 S 00 αα (ω)
TABLE I: Expressions of the electrical, mixed, and heat noises in the energy-independent weak/perfect transmission limits [START_REF](-1) 1+p |eV -ω| 1+p+q Θ(eVω)+(-1) 1+q |eV + ω| 1+p+q Θ(-eVω)]/(1+p+q)!, with S 00 LL (ω) = -S 00 LR (ω), S 01 LL (ω) = S 01 LR (ω), and S 11 LL (ω) = 2S 11[END_REF] .

We have ᾱ = R when α = L, and vice versa. The total electrical, mixed and heat noises summed over both reservoirs are also given. The details of the calculations are performed in Appendix B.

C. Anderson-type energy transmission

For an Anderson-type transmission amplitude of the form

t(ε) = iΓ/[(ε -ε d ) + iΓ],
both the real and imaginary parts of the electrical, mixed, and heat noise spectra are given by Fig. 3 in the low temperature limit. Note that the imaginary parts of S 00 LL (ω) and S 11 LL (ω) are both zero since the auto-correlators are real quantities, and that the real and imaginary parts of the cross-correlators are of the same order of magnitude, contrary to the energy independent transmission case. The main observation is the dramatically distinct spectra that we have for the auto-correlators, S 00 LL (ω) and S 11 LL (ω), in comparison to the cross-correlators, S pq αβ (ω) with p = q or/and α = β. Whereas the auto-correlator spectra are quite similar to the ones obtained in the case of an energy-independent transmission amplitude (compare to Fig. 2), excepted an additional structure in the region of small positive frequency, the cross-correlator spectra exhibit the following features: (i) their sign can change, (ii) it appears a new region with specific behavior close to small frequency, but (iii) we still have a cancellation of the noises for ω > eV , again due to the fact that the system can not emit energy larger than the one provided to it. We remark that in any situations, those depicted in Figs. 2 and3 and those summarized in Table I, the mixed noise cancels at zero-voltage, meaning that the cancellation of the ratio -V /∆T , which is equal to the Seebeck coefficient S T for open circuit, causes the cancellation of the mixed noise. This is one evidence that thermopower and mixed noise

are closely connected.

To have a deeper insight in the electrical, mixed, and heat noises, we plot their real and imaginary parts as a function of frequency at weak coupling strength Γ, for increasing temperatures in Figs. 4 and5. All the types of noise exhibit an asymmetric spectrum at low temperature (red curves) and a nearly symmetrical spectrum at large temperature with a vanishing imaginary part (black curves), due to the fact that when the temperature increases we are leaving the quantum regime. Thus, at large temperature, it is no longer necessary to make the distinction between non-symmetrized and symmetrized noises since the currents are no longer operators but just scalars (classical regime). The electrical and heat auto-correlators (see Figs. 4(a) and 4(b)) are real and positive quantities. The electrical auto-correlator, S 00 LL (ω), is strongly frequency dependent at low temperature with a down staircase-like behavior starting from the value 2πΓe 2 /h and going to the value 0 (see red curve in Fig. 4(a)), but resembles to a white noise at large temperature 43 with a constant value equals to πΓe 2 /h, except in a narrow low frequency region (see black curve in Fig. 4(a)). At large temperature, the heat auto-correlator, S 11 LL (ω), presents a power-law variation with frequency, given by 2 ω 2 πΓ/h (see black curve in Fig. 4 trical and heat cross-correlators, depicted in Figs. 4(c)-4(f), are complex quantities whose imaginary parts cancel at large temperature (black curves), making the crosscorrelators real quantities in that limit. The electrical auto-correlator and the real part of the electrical crosscorrelator have distinct profiles but coincide at zerofrequency in absolute value since due to charge conservation we have S 00 LL (ω = 0) = -S 00 LR (ω = 0) = πΓe 2 /2h in that limit. We turn now our interest to the mixed noise depicted in Fig. 5. Similarly than for electrical and heat noises, increasing the temperature changes the mixed spectrum from an asymmetric profile to a symmetric profile with frequency, and cancels its imaginary part, again due to the fact that we are leaving the quantum regime. At low temperature, we also see that the imaginary parts of the mixed noises, S 01 LL (ω) and S 01 LR (ω), have a staircase-like profile which is a reminiscent of the electrical noise autocorrelator (compare Figs. 5(c) and 5(d) to Fig. 4(a)). Besides, the real parts of the mixed noises present quite particular profiles at low temperature: a linear profile in frequency for S 01 LL (ω) (see the red curve in Fig. 5(a)) and vanishing value when | ω| > |eV | for S 01 LR (ω) (see the red curve in Fig. 5(b)). At large temperature, S 01 LL (ω) becomes frequency independent with an asymptotic value equal to -πΓe 2 V /h, and S 01 LR (ω) cancels 43 . Here again, we find that the mixed noise is related to the Seebeck coefficient S T , since both quantities vary linearly with voltage. For completeness, we discuss the spectral coherence of the cross-correlators, defined as

C pq αβ (ω) = |S pq αβ (ω)| 2 /[S pp αα (ω)S qq ββ (ω)]
, and plot their profiles on Fig. 6. Thanks to Cauchy-Schwarz inequality, we have 0 ≤ C pq αβ (ω) ≤ 1, where the value zero for the spectral coherence means that the currents I p α and I q β are uncorrelated, whereas the value one means that the currents I p α and I q β are fully correlated. The plots on the left column of Fig. 6 are obtained for a weak coupling strength (Γ/eV = 0.02), whereas the plots on the right column correspond to an intermediate coupling strength (Γ/eV = 0.2). In the weak coupling strength limit, we remark that C pq α =β (ω) is equal to zero at negative frequency, meaning that the absorbed signals in distinct reservoirs are uncorrelated. Moreover, we see in Fig. 6(a) that the left and right electrical currents are well correlated only at zero-frequency, i.e., in the largetime limit, due to charge conservation which imposes

C 00 LR (ω = 0) = C 00 LL (ω = 0) = 1.
When the coupling strength increases, the spectral coherence C 00 LR (ω) is broadened to non-zero frequencies (see Fig. 6(b)) and can even reach 40% at positive frequencies, an effect which is amplified around ω = eV when a temperature gradient is applied (see red curve in Fig. 6(b)). The electrical and heat currents inside a single reservoir, here L, are well correlated at ∆T = 0 (see blue curves in Figs. 6(c) and 6(d)) provided that ω > -eV , excepted in a narrow region around ω = eV where a minimum of C 01 LL (ω) is observed. At the same frequency, C 01 LR (ω) exhibits a maximum (see blue curves in Figs. 6(e) and 6(f)) meaning that the left electrical current and the right heat current are maximally correlated in that region of frequency. The increasing of the coupling strength reinforces this effect with values of C 01 LR (ω) up to 80% (see blue curve in Fig. 6(f)). This result allows us to make the prediction that the thermoelectric conversion could be optimal when the voltage applied to the QD is time-modulated with a frequency equals to the dc-voltage, i.e., eV / . This effect is, however, suppressed in the presence of a temperature gradient (see green and red curves in Fig. 6(f)). On the contrary, C 11 LR (ω) can be increased at high frequency by the application of a temperature gradient (see Fig. 6(h)).

V. CONCLUSION

We have used the Keldysh out-of-equilibrium Green function technique to calculate the finite-frequency mixed noise, and have shown that its spectrum presents a rich and specific profile, which differs from the ones of the electrical and heat noises. At equilibrium, it is related to the thermoelectric ac-conductance, meaning that the finite-frequency mixed noise gives access to the dynamics of the thermoelectric conversion. Out-of-equilibrium, by a careful study of the spectral coherence, we find that the electrical current in one reservoir is strongly correlated to the heat current in the other reservoir when the frequency is of the order of the applied voltage. The method developed here constitutes an adequate framework which can be used in future works on this quantity in more complicated quantum systems, including multiterminals, multi-channels, and interactions in some ex-tent.

We use an approach analog to the one developed by Haug and Jauho for the calculation of electrical noise [START_REF] Haug | Quantum Kinetics in Transport and Optics of Semiconductors[END_REF] , but instead of calculating the symmetrized noise, we calculate the non-symmetrized one since it is this latter quantity which is accessible in the experiments measuring the electrical current noise. We perform this calculation for each type of noise, the electrical, mixed and heat ones, using the general framework exposed in this Appendix.

a. Expression of S pq αβ (t, t ′ ) in terms of the two-particle Green function of the QD, G dd i We report Eq. (3) in Eq. ( 4) and get

S pq αβ (t, t ′ ) = - e 2-p-q 2 k∈α,k ′ ∈β (ε kα -µ α ) p (ε k ′ β -µ β ) q × V kα V k ′ β c † kα (t)d(t)c † k ′ β (t ′ )d(t ′ ) -V kα V * k ′ β c † kα (t)d(t)d † (t ′ )c k ′ β (t ′ ) -V * kα V k ′ β d † (t)c kα (t)c † k ′ β (t ′ )d(t ′ ) +V * kα V * k ′ β d † (t)c kα (t)d † (t ′ )c k ′ β (t ′ ) -Îp α Îq β .
(A1) Defining the following greater two-particle Green functions 32

G cd,> 1 (t, t ′ ) = i 2 T c † kα (t)d(t)c † k ′ β (t ′ )d(t ′ ) , G cd,> 2 (t, t ′ ) = i 2 T c † kα (t)d(t)d † (t ′ )c k ′ β (t ′ ) , G cd,> 3 (t, t ′ ) = i 2 T d † (t)c kα (t)c † k ′ β (t ′ )d(t ′ ) , G cd,> 4 (t, t ′ ) = i 2 T d † (t)c kα (t)d † (t ′ )c k ′ β (t ′ ) ,
and using the Keldysh formalism 35 , the non-equilibrium contour-ordered counterparts of the correlation function can be expressed in terms of G cd i (τ, τ ′ ), the contourordered counterparts of the two-particle Green functions, G cd,> i (t, t ′ ), through

S pq αβ (τ, τ ′ ) = e 2-p-q 2 k∈α,k ′ ∈β (ε kα -µ α ) p (ε k ′ β -µ β ) q × V kα V k ′ β G cd 1 (τ, τ ′ ) -V kα V * k ′ β G cd 2 (τ, τ ′ ) -V * kα V k ′ β G cd 3 (τ, τ ′ ) + V * kα V * k ′ β G cd 4 (τ, τ ′ ) -Îp α Îq β . (A2)
The next step is to express the two-particle Green functions, G cd i , mixing c and d operators in terms of the twoparticle Green functions of the QD, G dd i , and of the bare Green function of the reservoirs, g kα . We have 32

G cd 1 (τ, τ ′ ) = - V * kα V * k ′ β 2 dτ 1 dτ 2 ×g kα (τ 1 , τ )g k ′ β (τ 2 , τ ′ )G dd 1 (τ, τ ′ , τ 1 , τ 2 ) , G cd 2 (τ, τ ′ ) = -δ kk ′ δ αβ g kα (τ ′ , τ )G(τ, τ ′ ) - V * kα V k ′ β 2 × dτ 1 dτ 2 g kα (τ 2 , τ )g k ′ β (τ ′ , τ 1 )G dd 2 (τ, τ ′ , τ 1 , τ 2 ) , G cd 3 (τ, τ ′ ) = -δ kk ′ δ αβ g kα (τ, τ ′ )G(τ ′ , τ ) + V kα V * k ′ β 2 × dτ 1 dτ 2 g kα (τ, τ 1 )g k ′ β (τ 2 , τ ′ )G dd 3 (τ, τ ′ , τ 1 , τ 2 ) , G cd 4 (τ, τ ′ ) = - V kα V k ′ β 2 dτ 1 dτ 2 ×g kα (τ, τ 1 )g k ′ β (τ ′ , τ 2 )G dd 4 (τ, τ ′ , τ 1 , τ 2 ) , with G dd 1 (τ, τ ′ , τ 1 , τ 2 ) = i 2 T C d(τ )d(τ ′ )d † (τ 1 )d † (τ 2 ) , G dd 2 (τ, τ ′ , τ 1 , τ 2 ) = i 2 T C d(τ )d † (τ ′ )d(τ 1 )d † (τ 2 ) , G dd 3 (τ, τ ′ , τ 1 , τ 2 ) = i 2 T C d † (τ )d(τ ′ )d(τ 1 )d † (τ 2 ) , G dd 4 (τ, τ ′ , τ 1 , τ 2 ) = i 2 T C d † (τ )d † (τ ′ )d(τ 1 )d(τ 2 ) .
Injecting the above expressions of G cd i in Eq. (A2), we get

S pq αβ (τ, τ ′ ) = e 2-p-q 2 δ αβ k∈α (ε kα -µ α ) p+q ×|V kα | 2 g kα (τ ′ , τ )G(τ, τ ′ ) + g kα (τ, τ ′ )G(τ ′ , τ ) + k∈α,k ′ ∈β (ε kα -µ α ) p (ε k ′ β -µ β ) q |V kα V k ′ β | 2 2 × dτ 1 dτ 2 -g kα (τ 1 , τ )g k ′ β (τ 2 , τ ′ )G dd 1 (τ, τ ′ , τ 1 , τ 2 ) +g kα (τ 2 , τ )g k ′ β (τ ′ , τ 1 )G dd 2 (τ, τ ′ , τ 1 , τ 2 ) -g kα (τ, τ 1 )g k ′ β (τ 2 , τ ′ )G dd 3 (τ, τ ′ , τ 1 , τ 2 ) -g kα (τ, τ 1 )g k ′ β (τ ′ , τ 2 )G dd 4 (τ, τ ′ , τ 1 , τ 2 ) -Îp α Îq β , (A3)
where

G(τ, τ ′ ) = -i T C d(τ )d † (τ ′
) is the one-particle Green function of the QD.

P0(t, t ′ ) g < kα (t ′ , t)G > (t, t ′ ) + g > kα (t, t ′ )G < (t ′ , t) P1(t, t ′ ) -dt1 G r (t ′ , t1)g < kα (t1, t) + G < (t ′ , t1)g a kα (t1, t) dt2 G r (t, t2)g > k ′ β (t2, t ′ ) + G > (t, t2)g a k ′ β (t2, t ′ ) P2(t, t ′ ) G > (t, t ′ ) dt1dt2 g r k ′ β (t ′ , t1)G r (t1, t2)g < kα (t2, t) +g r k ′ β (t ′ , t1)G < (t1, t2)g a kα (t2, t) + g < k ′ β (t ′ , t1)G a (t1, t2)g a kα (t2, t) P3(t, t ′ ) G < (t ′ , t) dt1dt2 g r kα (t, t1)G r (t1, t2)g > k ′ β (t2, t ′ ) +g r kα (t, t1)G > (t1, t2)g a k ′ β (t2, t ′ ) + g > kα (t, t1)G a (t1, t2)g a k ′ β (t2, t ′ ) P4(t, t ′ ) -dt1 g r kα (t, t1)G > (t1, t ′ ) + g > kα (t, t1)G a (t1, t ′ ) × dt2 g r k ′ β (t ′ , t2)G < (t2, t) + g < k ′ β (t ′ , t2)G a (t2, t)
TABLE II: Expressions of the Pi(t, t ′ ) coefficients appearing in Eq. (A7). b. Evaluation of the two-particle Green functions G dd i using decoupling procedure

P0(ω) dε g < kα (ε)G > (ε -ω) + g > kα (ε -ω)G < (ε) P1(ω) -dε G r (ε)g < kα (ε)G r (ε -ω)g > k ′ β (ε -ω) + G r (ε)g < kα (ε)G > (ε -ω)g a k ′ β (ε -ω) +G < (ε)g a kα (ε)G r (ε -ω)g > k ′ β (ε -ω) + G < (ε)g a kα (ε)G > (ε -ω)g a k ′ β (ε -ω) P2(ω) dεG > (ε -ω) g r k ′ β (ε)G r (ε)g < kα (ε) + g r k ′ β (ε)G < (ε)g a kα (ε) + g < k ′ β (ε)G a (ε)g a kα (ε) P3(ω) dεG < (ε) g r kα (ε -ω)G r (ε -ω)g > k ′ β (ε -ω) +g r kα (ε -ω)G > (ε -ω)g a k ′ β (ε -ω) + g > kα (ε -ω)G a (ε -ω)g a k ′ β (ε -ω) P4(ω) -dε g r kα (ε -ω)G > (ε -ω) + g > kα (ε -ω)G a (ε -ω) g r k ′ β (ε)G < (ε) + g < k ′ β (ε)G a (ε)
A αβ (ε, ω) -1 2 f h α (ε -ω) + f h ᾱ (ε -ω) [f e α (ε) -f e β (ε)]t(ε)T (ε -ω) -1 2 [f e β (ε) -3f e β (ε)]T (ε)T (ε -ω) B αβ (ε, ω) f e α (ε)f h β (ε -ω)t(ε)t(ε -ω) -1 2 f e α (ε) f h α (ε -ω) + f h ᾱ (ε -ω) t(ε)T (ε -ω) -1 2 f h β (ε -ω) f e α (ε) + f e ᾱ(ε) T (ε)t(ε -ω) + 1 4 f e α (ε) + f e ᾱ(ε) f h α (ε -ω) + f h ᾱ (ε -ω) T (ε)T (ε -ω) C αβ (ε, ω) -1 2 f h β (ε -ω) f e β (ε) + f e β (ε) T (ε)t(ε -ω) -1 2 f h α (ε -ω) f e β (ε) + f e β (ε) T (ε)t * (ε -ω) + 1 4 f e β (ε) + f e β (ε) f h β (ε -ω) + f h β (ε -ω) T (ε)T (ε -ω) Dα(ε, ω) f e α (ε) f h α (ε -ω) + f h ᾱ (ε -ω) T (ε -ω) Eα(ε, ω) f e α (ε) + f e ᾱ(ε) f h α (ε -ω)T (ε)
In the absence of Coulomb interactions, we can express fully the two-particle Green functions of the QD, G dd i (τ, τ ′ , τ 1 , τ 2 ), in terms of the one-particle Green func-tion of the QD, G(τ, τ ′ ), through

G dd 1 (τ, τ ′ , τ 1 , τ 2 ) = G(τ, τ 2 )G(τ ′ , τ 1 ) -G(τ, τ 1 )G(τ ′ , τ 2 ) , G dd 2 (τ, τ ′ , τ 1 , τ 2 ) = G(τ, τ ′ )G(τ 1 , τ 2 ) -G(τ, τ 2 )G(τ 1 , τ ′ ) , G dd 3 (τ, τ ′ , τ 1 , τ 2 ) = G(τ 1 , τ )G(τ ′ , τ 2 ) -G(τ ′ , τ )G(τ 1 , τ 2 ) , G dd 4 (τ, τ ′ , τ 1 , τ 2 ) = G(τ 2 , τ )G(τ 1 , τ ′ ) -G(τ 1 , τ )G(τ 2 , τ ′ ) .
Injecting these four expressions in Eq. (A3), we obtain a result which can be separated into two parts, a connected part and a disconnected part 32

S pq αβ (τ, τ ′ ) = S pq αβ,disc (τ, τ ′ ) + S pq αβ,conn (τ, τ ′ ) -Îp α Îq β , (A4)
with

S pq αβ,disc (τ, τ ′ ) = e 2-p-q 2 × k∈α,k ′ ∈β (ε kα -µ α ) p (ε k ′ β -µ β ) q |V kα V k ′ β | 2 2 × dτ 1 g kα (τ 1 , τ )G(τ, τ 1 ) dτ 2 g k ′ β (τ 2 , τ ′ )G(τ ′ , τ 2 ) -dτ 2 g kα (τ 2 , τ )G(τ, τ 2 ) dτ 1 g k ′ β (τ ′ , τ 1 )G(τ 1 , τ ′ ) -dτ 1 g kα (τ, τ 1 )G(τ 1 , τ ) dτ 2 g k ′ β (τ 2 , τ ′ )G(τ ′ , τ 2 ) + dτ 1 g kα (τ, τ 1 )G(τ 1 , τ ) dτ 2 g k ′ β (τ ′ , τ 2 )G(τ 2 , τ ′ ) , (A5)
and

S pq αβ,conn (τ, τ ′ ) = e 2-p-q 2 δ αβ k∈α (ε kα -µ α ) p+q |V kα | 2 × g kα (τ ′ , τ )G(τ, τ ′ ) + g kα (τ, τ ′ )G(τ ′ , τ ) + k∈α,k ′ ∈β (ε kα -µ α ) p (ε k ′ β -µ β ) q |V kα V k ′ β | 2 2 × dτ 1 dτ 2 -g kα (τ 1 , τ )g k ′ β (τ 2 , τ ′ )G(τ, τ 2 )G(τ ′ , τ 1 ) +g kα (τ 2 , τ )g k ′ β (τ ′ , τ 1 )G(τ, τ ′ )G(τ 1 , τ 2 ) +g kα (τ, τ 1 )g k ′ β (τ 2 , τ ′ )G(τ ′ , τ )G(τ 1 , τ 2 ) -g kα (τ, τ 1 )g k ′ β (τ ′ , τ 2 )G(τ 2 , τ )G(τ 1 , τ ′ ) . (A6)
The disconnected part can be calculated directly. We obtain S pq αβ,disc (τ, τ ′ ) = Îp α Îq β , thus finally S pq αβ (τ, τ ′ ) = S pq αβ,conn (τ, τ ′ ).

c. Analytic continuation of the connected part

We perform now the analytic continuation of Eq. (A6) to get its τ > τ ′ component

S pq αβ (t, t ′ ) = e 2-p-q 2 δ αβ k∈α (ε kα -µ α ) p+q |V kα | 2 × g kα (τ ′ , τ )G(τ, τ ′ ) + g kα (τ, τ ′ )G(τ ′ , τ ) τ >τ ′ P0(t,t ′ ) + k∈α,k ′ ∈β (ε kα -µ α ) p (ε k ′ β -µ β ) q |V kα V k ′ β | 2 2 × - dτ 1 dτ 2 G(τ ′ , τ 1 )g kα (τ 1 , τ )G(τ, τ 2 )g k ′ β (τ 2 , τ ′ ) τ >τ ′ P1(t,t ′ ) + G(τ, τ ′ ) dτ 1 dτ 2 g k ′ β (τ ′ , τ 1 )G(τ 1 , τ 2 )g kα (τ 2 , τ ) τ >τ ′ P2(t,t ′ ) + G(τ ′ , τ ) dτ 1 dτ 2 g kα (τ, τ 1 )G(τ 1 , τ 2 )g k ′ β (τ 2 , τ ′ ) τ >τ ′ P3(t,t ′ ) - dτ 1 dτ 2 g kα (τ, τ 1 )G(τ 1 , τ ′ )g k ′ β (τ ′ , τ 2 )G(τ 2 , τ ) τ >τ ′ P4(t,t ′ )
.

The five P i (t, t ′ ) contributions are computed using analytic continuation rules [START_REF] Haug | Quantum Kinetics in Transport and Optics of Semiconductors[END_REF] . Their expressions are given in Table II. Using these notations, the noise reads as

S pq αβ (t, t ′ ) = e 2-p-q 2 δ αβ k∈α (ε kα -µ α ) p+q |V kα | 2 P 0 (t, t ′ ) + k∈α k ′ ∈β (ε kα -µ α ) p (ε k ′ β -µ β ) q |V kα V k ′ β | 2 2 4 i=1 P i (t, t ′ ) .
(A7) Performing the Fourier transform of Eq. (A7) and using the fact that in the stationary case the Green functions depend on the difference of their time arguments only, we obtain:

S pq αβ (ω) = e 2-p-q δ αβ k∈α (ε kα -µ α ) p+q |V kα | 2 P 0 (ω) + k∈α k ′ ∈β (ε kα -µ α ) p (ε k ′ β -µ β ) q |V kα V k ′ β | 2 4 i=1 P i (ω) , (A8)
where the expressions of P i (ω) are given in Table III.

Using the expressions of the bare Green functions of the reservoirs, g > kα (ε) and g < kα (ε), in terms of the Fermi-Dirac distribution function for electrons, f e α (ε), and Fermi-Dirac distribution function for holes,

f h α (ε) = 1 -f e α (ε), g < kα (ε) = 2πif e α (ε)δ(ε -ε kα ) , (A9) g > kα (ε) = -2πif h α (ε)δ(ε -ε kα ) , (A10)
we can rewrite Eq. (A8) under the form

S pq αβ (ω) = e 2-p-q h dε δ αβ iF p+q α (ε)f e α (ε) × G < (ε -ω) + G r (ε -ω) -G a (ε -ω) -δ αβ iF p+q α (ε -ω)f h α (ε -ω)G < (ε) -G r (ε)G r (ε -ω)f e α (ε)f h β (ε -ω)F p α (ε)F q β (ε -ω) -G a (ε)G a (ε -ω)f e β (ε)f h α (ε -ω)F p α (ε -ω)F q β (ε) +iG r (ε)G > (ε -ω)f e α (ε)F p α (ε)H q * β (ε) +iG < (ε)G r (ε -ω)f h β (ε -ω)F q β (ε -ω)H p α (ε) +iG a (ε)G > (ε -ω)f e β (ε)F q β (ε)H p α (ε) +iG < (ε)G a (ε -ω)f h α (ε -ω)F p α (ε -ω)H q * β (ε) +G < (ε)G > (ε -ω)H p α (ε, p)H q * β (ε) , (A11)
where we have introduced the two following functions

F p α (ε) = 2π k∈α (ε kα -µ α ) p |V kα | 2 δ(ε -ε kα ) = (ε -µ α ) p 2π|V α (ε)| 2 ρ α (ε) =Γα(ε) , (A12)
and

H p α (ε) = k∈α (ε kα -µ α ) p |V kα | 2 g a kα (ε) -g r kα (ε -ω) , (A13)
with ρ α (ε) the density of states associated to the reservoirs α, and Γ α = 2π|V α (ε)| 2 ρ α (ε), the coupling strength between the QD and the reservoir α. Note that Eq. (A11), given S pq αβ (ω), has been obtained without making any approximation at this stage: it is the exact result for a non-interacting QD.

b. S pq αβ (ω) for symmetrical barriers in the wide-band limit

To continue further the calculation, we make two simplifying assumptions: (i) wide-band limit, i.e., we are working on an interval of energy in which the density of states is constant, ρ α (ε) = cst, and we assume as well that V α (ε) is energy independent, consequently, we have

Γ α (ε) = 2π|V α (ε)| 2 ρ α (ε) ≡ Γ α
, and (ii) symmetrical barriers, i.e., we assume that the left and right barriers are symmetrical (Γ L = Γ R ≡ Γ). In that case, we have the remarkable relation 13 : t(ε) + t * (ε) = 2T (ε), with t(ε) = iΓG r (ε), the transmission amplitude and T (ε), the transmission coefficient. Within these two simplifying assumptions, we have

F p α (ε) = Γ(ε -µ α ) p , and
H p α (ε) = iΓ 2 [(ε -µ α ) p + (ε -ω -µ α ) p ] .
Injecting these two last expressions in Eq. (A11), rearranging the terms, and using the relations 32,44

G > (ε) -G < (ε) = G r (ε) -G a (ε) , G < (ε) = iΓG r (ε)G a (ε) f e α (ε) + f e ᾱ(ε) , G r (ε) -G a (ε) = -2iΓG r (ε)G a (ε) , T (ε) = Γ 2 G r (ε)G a (ε) .
We finally get

S pq αβ (ω) = e 2-p-q h dε δ αβ (ε -µ α ) p+q D α (ε, ω) +δ αβ (ε -ω -µ α ) p+q E α (ε, ω) + (ε -µ α ) p (ε -µ β ) q A αβ (ε, ω) + (ε -µ α ) p (ε -ω -µ β ) q B αβ (ε, ω) + (ε -ω -µ α ) p (ε -µ β ) q B * βα (ε, ω) + (ε -ω -µ α ) p (ε -ω -µ β ) q C αβ (ε, ω) , (A14)
where the coefficients A α (ε, ω), B α (ε, ω), C α (ε, ω), D α (ε, ω), and E α (ε, ω) are given in Table IV. Equation (A14) leads to the Eqs. ( 5)-( 8) once we define

A αβ (ε, ω) = A αβ (ε, ω) + δ αβ D α (ε, ω) , B αβ (ε, ω) = B αβ (ε, ω) , C αβ (ε, ω) = C αβ (ε, ω) + δ αβ E α (ε, ω) , f e,h M (ε) = 1 2 f e,h α (ε) + f e,h ᾱ (ε) ,
where ᾱ = R when α = L, and ᾱ = L when α = R.

S 01 αβ (ω) = e h γδ ∞ -∞ dε (ε -µ β )M γδ αβ -ωN γδ αβ ×f e γ (ε)f h δ (ε -ω) , (B2) S 10 αβ (ω) = e h γδ ∞ -∞ dε (ε -µ α )M γδ αβ -ω(N γδ βα ) * ×f e γ (ε)f h δ (ε -ω) , (B3)
and

S 11 αβ (ω) = 1 h γδ ∞ -∞ dε (ε -µ α )(ε -µ β )M γδ αβ -ω(ε -µ α )N γδ αβ -ω(ε -µ β )(N γδ βα ) * + 2 ω 2 O γδ αβ f e γ (ε)f h δ (ε -ω) , (B4)
with the coefficients M γδ αβ , N γδ αβ , and O γδ αβ given in Tables V-VII, where Z = [T (1-T )] 1/2 is the imaginary part of t, T being the real part of t. These real and imaginary parts are extracted from the two relations: tt * = T and t + t * = 2T .

M γδ αβ γ = δ = L γ = δ = R γ = L, δ = R γ = R, δ = L α = β = L T 2 T 2 T (1 -T ) T (1 -T ) α = β = R T 2 T 2 T (1 -T ) T (1 -T ) α = L, β = R -T 2 -T 2 -T (1 -T ) -T (1 -T ) α = R,β = L -T 2 -T 2 -T (1 -T ) -T (1 -T )
N γδ αβ γ = δ = L γ = δ = R γ = L, δ = R γ = R, δ = L α = β = L T 2 2 + iZT T 2 2 -iZT 2 T (1 -T ) -iZT 2 α = β = R T 2 2 T 2 2 + iZT T (1 -T ) -iZT 2 -iZT 2 α = L, β = R -T 2 2 -T 2 2 -iZT -T (1 -T ) + iZT 2 iZT 2 α = R,β = L -T 2 2 -iZT -T 2 2 iZT 2 -T (1 -T ) + iZT 2
O γδ αβ γ = δ = L γ = δ = R γ = L, δ = R γ = R, δ = L α = β = L T 2 4 + T (1 -T ) T 2 4 T 2 4 T 2 4 + T (1 -T ) α = β = R T 2 4 T 2 4 + T (1 -T ) T 2 4 + T (1 -T ) T 2 4 α = L, β = R -T 2 4 + iZT 2 -T 2 4 -iZT 2 -T 2 4 -iZT 2 -T 2 4 + iZT 2 α = R,β = L -T 2 4 -iZT 2 -T 2 4 + iZT 2 -T 2 4 + iZT 2 -T 2 4 -iZT 2
TABLE VII: Expressions of the coefficients O γδ αβ appearing in Eq. (B4).

Preliminary calculations

In the following sections, we will meet the integral

I (n) γδ = ∞ -∞ ε n dεf e γ (ε)f h δ (ε -ω) , (B5)
with n = 0, 1 or 2. Here, we calculate this integral considering the isothermal case, T L = T R ≡ T , thus

I (n) γδ = N ( ω + µ δ -µ γ ) × ∞ -∞ ε n dε sinh ω+µ δ -µγ 2kB T 2 cosh ε-µγ 2kB T cosh ε-µ δ -ω 2kB T , (
I (n) γδ = N ( ω + µ δ -µ γ ) 2 × ∞ -∞ ε n dε tanh ε -µ γ 2k B T -tanh ε -µ δ -ω 2k B T . (B7)
To go further, we perform a Taylor expansion up to the third order with x = ω, µ γ or µ δ . It leads to

I (n) γδ = N ( ω + µ δ -µ γ ) 2 µ γ -µ δ -ω 2k B T L (n) 1 + µ 2 γ -(µ δ + ω) 2 4k 2 B T 2 L (n) 2 + µ 3 γ -(µ δ + ω) 3 24k 3 B T 3 L (n) 3 , (B8) with L
(n) 1 = ∞ -∞ ε n dε[tanh 2 (ε/2k B T ) -1], L (n) 2 = ∞ -∞ ε n dε[tanh 3 (ε/2k B T ) -tanh(ε/2k B T )], and
L (n) 3 = ∞ -∞ ε n dε[1 + 3 tanh 4 (ε/2k B T ) -4 tanh 2 (ε/2k B T )]
. The calculation of these integrals gives

L (n) 1 =        -4k B T (n = 0) 0 (n = 1) -4π 2 k 3 B T 3 /3 (n = 2) , (B9) L (n) 2 =        0 (n = 0) -4k 2 B T 2 (n = 1) 0 (n = 2) , (B10)
and

L (n) 3 =        0 (n = 0) 0 (n = 1) -16k 3 B T 3 (n = 2) . (B11)
Finally, we get

I (0) γδ = ( ω + µ δ -µ γ )N ( ω + µ δ -µ γ ) , (B12) I (1) γδ = ( ω + µ δ ) 2 -µ 2 γ 2 N ( ω + µ δ -µ γ ) , (B13) I (2) γδ = ( ω + µ δ -µ γ )π 2 k 2 B T 2 3 + ( ω + µ δ ) 3 -µ 3 γ 3 N ( ω + µ δ -µ γ ) . (B14)
2. Limit of weak transmission T ≪ 1

In this subsection, we give the calculation of the expressions appearing in the first column of Table I.

a. Electrical noise spectrum

We calculate only S 00 LL (ω) since when t and T are independent of energy, we have the relations S 00 LL (ω) = S 00 RR (ω) = -S 00 LR (ω) = -S 00 RL (ω) . At weak T , we have

S 00 LL (ω) = e 2 h γδ ∞ -∞ dεM γδ LL f e γ (ε)f h δ (ε -ω) = e 2 T h ∞ -∞ [f e L (ε)f h R (ε -ω) + f e R (ε)f h L (ε -ω)]dε = e 2 T h I (0) LR + I (0) RL , (B15)
which gives

S 00 LL (ω) = e 2 T h ( ω -eV )N ( ω -eV ) +( ω + eV )N ( ω + eV ) . (B16)
It reduces at equilibrium (zero-voltage) to

S 00 LL (ω) = 2e 2 T h ωN ( ω) , (B17)
and at zero-temperature to

S 00 LL (ω) = e 2 T h (eV -ω)Θ(eV -ω) -( ω + eV )Θ(-eV -ω) , (B18)
since we have N (x) = -Θ(-x) when T → 0.

b. Mixed noise spectrum

We start from

S 01 αβ (ω) = e h γδ ∞ -∞ dε (ε -µ β )M γδ αβ -ωN γδ αβ f e γ (ε)f h δ (ε -ω) . (B19) We calculate only S 01
LL (ω) since at weak T , we have:

S 01 LL (ω) = -S 01 RR (ω) = S 01 LR (ω) = -S 01 RL (ω) . We have S 01 LL (ω) = eT h ∞ -∞ dε (ε -µ L )[f e L (ε)f h R (ε -ω) +f e R (ε)f h L (ε -ω)] -ωf e R (ε)f h L (ε -ω) = eT h I (1)
LRµ L I (0)

LR + I (1) RL -(µ L + ω)I (0) RL , (B20)
which gives

S 01 LL (ω) = eT 2h ( ω -eV ) 2 N ( ω -eV ) -( ω + eV ) 2 N ( ω + eV ) . (B21)
It reduces to S 01 LL (ω) = 0 at equilibrium (zero-voltage) and to

S 01 LL (ω) = eT 2h -( ω -eV ) 2 Θ(eV -ω) +( ω + eV ) 2 Θ(-eV -ω) , (B22)
at zero-temperature.

c. Heat noise spectrum

We start from

S 11 αβ (ω) = 1 h γδ ∞ -∞ dε (ε -µ α )(ε -µ β )M γδ αβ -ω(ε -µ α )N γδ αβ -ω(ε -µ β )(N γδ βα ) * + 2 ω 2 O γδ αβ f e γ (ε)f h δ (ε -ω) , (B23)
which gives for S 11 LL (ω)

S 11 LL (ω) = T h ∞ -∞ dε (ε -µ L ) 2 [f e L (ε)f h R (ε -ω) +f e R (ε)f h L (ε -ω)] -2 ω(ε -µ L )f e R (ε)f h L (ε -ω) + 2 ω 2 [f e L (ε)f h L (ε -ω) + f e R (ε)f h L (ε -ω)] = T h I (2)
LR + I RR (ω) = S 11 LL (ω). We now calculate

S 11 LR (ω) = T h ∞ -∞ dε -(ε -µ L )(ε -µ R ) ×[f e L (ε)f h R (ε -ω) + f e R (ε)f h L (ε -ω)] + ω(ε -µ L )f e L (ε)f h R (ε -ω) + ω(ε -µ R )f e R (ε)f h L (ε -ω) = T h -I (2)
LR -I

RL + (µ L + µ R ) I In this subsection, we give the calculation of the expressions appearing in the second column of Table I. For T = 1, we have

S 01 LL (ω) = e h ∞ -∞ dε ε -µ L - ω 2 × f e L (ε)f h L (ε -ω) + f e R (ε)f h R (ε -ω) = e h I (1)
LL + I For T = 1, we have

S 11 LL (ω) = 1 h ∞ -∞ dε [(ε -µ L ) 2 -ω(ε -µ L )] ×[f e L (ε)f h L (ε -ω) + f e R (ε)f h R (ε -ω)] + 2 ω 2 4 γδ f e γ (ε)f h δ (ε -ω) = 1 h I (2)
LL + I

  FIG. 1: Picture of the QD connected to left and right reservoirs with distinct temperatures and chemical potentials (we take eV = µL -µR and ∆T = TL -TR). The black arrows indicate the convention chosen for the currents' direction.

FIG. 2 :

 2 FIG. 2: Noises spectra as a function of frequency ω/Γ ∈ [-10, 10] and voltage eV /Γ ∈ [-10, 10] for T = 0.01 and t = T +i[T (1-T )] 1/2 , at low temperature kBT /Γ = 0.01. S pq αβ (ω) is plotted in units of e 2-p-q Γ 1+p+q /h. The right reservoir is grounded (µR = 0).

FIG. 3 :

 3 FIG. 3: Noises spectra as a function of frequency ω/Γ ∈ [-10, 10] and voltage eV /Γ ∈ [-10, 10] for T (ε) = Γ 2 /(ε 2 + Γ 2 ), at low temperature kBT /Γ = 0.01. S pq αβ (ω) is plotted in units of e 2-p-q Γ 1+p+q /h. The right reservoir is grounded (µR = 0).

  FIG. 4: Electrical noises S 00 LL (ω) and S 00 LR (ω) (left column) and heat noises S 11 LL (ω) and S 11 LR (ω) (right column), as a function of frequency ω/eV , for T (ε) = Γ 2 /(ε 2 + Γ 2 ), with Γ/eV = 0.01, and for varying values of the temperature kBT /eV : 0.01 (red line), 0.5 (orange line), 1 (blue line), 2 (brown line), and 10 (black line). S pq αβ (ω) is plotted in units of e 2-p-q (eV ) 1+p+q /h. The right reservoir is grounded (µR = 0).

FIG. 5 :

 5 FIG. 5: Mixed noises S 01 LL (ω) (left column) and S 01 LR (ω) (right column), as a function of frequency ω/eV . Same parameters as in Fig. 4.

11 ΩFIG. 6 :

 116 FIG.6: Spectral coherence for T (ε) = Γ 2 /(ε 2 + Γ 2 ), for Γ/eV = 0.02 (left column) and Γ/eV = 0.2 (right column), at kBTR/eV = 0.1 and temperature gradient equals to: ∆T /eV = 0 (blue curve), ∆T /eV = 0.5 (green curve), and ∆T /eV = 1 (red curve). The dashed black line shows the maximal possible value for the spectral coherence, i.e., 1.

  B6) where we have introduced the Bose-Einstein distribution function: N ( ω) = [exp( ω/k B T )-1] -1 . Using the identity: sinh(a-b)/[cosh(a) cosh(b)] = tanh(a)-tanh(b), we end up with

  expressions of the integrals given by Eqs. (B12)-(B14) and factorize the various contributions. eV )N ( ω -eV )+( ω + eV )N ( ω + eV ) + ( ω -eV ) 3 3 N ( ω -eV ) + ( ω + eV ) 3 3 N ( ω + eV ) ,(B25)which reduces at equilibrium (zero-voltage) toS 11 LL (ω) = eVω) , (B27)since we have N (x) = -Θ(-x) when T → 0. Note that S 11 RR (ω) is obtained from the expression of S 11 LL (ω) by inverting the voltage V → -V , as a consequence we have S11

  expressions of the integrals given by Eqs. (B12)-(B14) and factorize the various contributions. eV )N ( ω -eV )+( ω + eV )N ( ω + eV ) + ( ω -eV ) 3 6 N ( ω -eV ) + ( ω + eV ) 3 6 N ( ω + eV ) ,(B29)which reduces at equilibrium (zero-voltage) to ω) 3 Θ(eVω)-( ω + eV ) 3 Θ(-eVω) , (B31)since we have N (x) = -Θ(-x) when T → 0. Note that S 11 RL (ω) is obtained from the expression of S 11 LR (ω) by inverting the voltage V → -V , as a consequence we have S 11 RL (ω) = S 11 LR (ω).

3 .

 3 Limit of perfect transmission T = 1

  αβ -1)2 ωN ( ω) . (B32) b. Mixed noise spectrum

  leads to S 01 RR (ω) = e h ωeV N ( ω). Moreover, we have S 01 LR (ω) = -S 01 RR (ω), and S 01 RL (ω) = -S 01 LL (ω).

6 ++ 2 ω 2 4 ± 4 ±

 644 ωe 2 V 2 N ( ω) ( ω ± eV )N ( ω ± eV ) . ( ω ± eV )N ( ω ± eV ) . (B37)In addition, we have S 11 RR (ω) = S 11 LL (ω) and S 11 RL (ω) = S 11 LR (ω).

  and for T ≪ 1, αβ S 11 αβ (ω) = 4T | ω| 3 Θ(-ω)/h at zero-voltage, and αβ S 11 αβ (ω) = T |eV | 3 /h at zero-frequency in good agreement with Ref. 14.

TABLE III

 III

: Expressions of the Pi(ω) coefficients appearing in Eq. (A8).

TABLE IV :

 IV Expressions of the coefficients appearing in Eq. (A14).

TABLE V :

 V Expressions of the coefficients M γδ αβ appearing in Eqs. (B1)-(B4).

TABLE VI :

 VI Expressions of the coefficients N γδ αβ appearing in Eqs. (B2)-(B4).

Acknowledgments

We would like to thank M. Guigou, M. Lavagna, T. Martin, F. Michelini, and R. Zamoum for useful discussions. We acknowledge financial support from the CNRS Cellule Energie funding project ICARE, and from A*Midex.

In case of independent energy transmission amplitude and coefficient, Eqs. ( 5)-( 8) reduce to


Marianne Graber
email: mgraber@univ-lr.fr

Hervé Rouillard
email: hrouilla@univ-lr.fr

Régis Delatouche
email: regis.delatouche@univ-lr.fr

Najla Fniter
email: najla.fniter@gmail.com

Belsem Belkhiria
email: belsem.belkhiria@gmail.com

Antoine Bonnet
email: antoine.bonnet@univ-lr.fr

Lisiannedomon

Andvalérie Thiéry
email: vthiery@univ-lr.fr

Improved racemateresolution of pentan-2-ol and trans-(Z)-cyclooct-5-ene-1,2-diol by lipase catalysis

Keywords: Biocatalysis, Kinetic Resolution, Lipase, Immobilized Enzymes, Substrate Engineering, Bioreactors

Lipases are important catalysts in chiral synthesis due to their wide substrate recognition combined with a high stereoselectivity. We demonstrate here that the state, free or immobilized, of Candida antarctica lipase B (CaLB) affects enantioselectivity and also alters the temperature dependancy of the enzyme. This indicates that CaLB undergoes various conformations induced by its interaction with the different immobilization supports studied.

Molecular imprinting experiments, using immobilized enzyme co-dried with mimic substrate molecules, enhanced the enantiomeric ratio two-fold or three-fold, depending on the immobilization support. The structure of the acyl donor has a pronounced effect on CaLBcatalyzed resolution, due to the proximity of the acyl and alcohol moieties during catalysis. When the acylation of pentan-2-ol was examined, we found that the 3C methyl propanoate donor afforded the highest resolution.Trans-(Z)-cyclooct-5-en-1,2-diolwas used as a modelracemicsubstrate to study the ability of lipase to catalyze the resolution of difunctionalized compounds. There was a clear enhancement in the enantiomer selectivity of the biotransformation of the diol when vinyl butanoateis used as the acyl donor.The conversion and enantiomeric excess of (1R,2R)-monoacetates were enhanced, using immobilized CaLB, when the chain length of the donors increased from C2 to C4.

1.Introduction

Chiral synthesis is a key process in synthetic organicchemistry and, in particular, in medicinal chemistry because the different enantiomers or diastereomers of a molecule frequently have substantially different biological activity.Enzymes with very high regio-and stereo-selectivity are catalysts of choice for chiral resolution and asymmetric synthesis. The requirement forcompounds with high stereochemical purity is increasing and there is a growing need for enzymes and reaction conditions with improved and preciseperformance.

Lipases are often used because they are available from various sources, they do not need cofactors and they arerelatively thermostable. In addition,they also accept a broad range of substrates, yet exhibit high selectivity [START_REF] Bornscheuer | Hydrolases in Organic Synthesis: Regio and Stereoselective Transformations[END_REF]. Candida antarctica Lipase B (CaLB) [START_REF] Noël | Kinetic resolution of rac-2-pentanol catalyzed by Candida antarctica lipase B in the ionic liquid, 1-butyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]amide[END_REF][START_REF] Léonard | Enantioselective acylation of R-2pentanol in a solid/gas reactor catalysed by lipase B from Candida antarctica[END_REF]and Burkholderiacepacia Lipase (BC)are relatively thermostable enzymes that are often used as catalysts in various biotransformations in non-aqueous media and demonstrate high enantioselectivity toward many chiral substrates.

Catalytic activity, selectivity, specificity and enzyme stability are key factors affecting the efficiency of biocatalysts and enzyme immobilization may improve these properties. In addition, immobilization of lipases confers good thermal and operational stabilities, enabling the development of continuous processes.Immobilization of lipases by sol-gel encapsulation is an easy and effective method that can also increase enzyme activity and selectivity [START_REF] Reetz | Second Generation Sol-Gel Encapsulated Lipases: Robust Heterogeneous Biocatalysts[END_REF][START_REF] Hellner | Novel Sol-Gel Lipases by Designed Bioimprinting for Continuous-Flow Kinetic Resolutions[END_REF]. Another successful strategy for enhancing the activity and selectivity of lipases involves the tuningof the enzyme active site by molecular imprinting.

The "molecular imprinting effect" theorywas proposed by several authors to explain the modification of enzyme selectivity or activity in the presence of molecules thatmimic substrates [START_REF] Hellner | Novel Sol-Gel Lipases by Designed Bioimprinting for Continuous-Flow Kinetic Resolutions[END_REF][START_REF] Yan | Combined strategy for preparation of a bioimprintedGeotrichum sp. lipase biocatalyst effective in non-aqueous media[END_REF][START_REF] Yang | Combination of bioimprinting and silane precursor alkyls improved the activity of sol-gel-encapsulated lipase[END_REF][START_REF] Chaput | Enhancing the enantioselectivity of CALB by substrate imprinting: a combined experimental and molecular dynamics simulation model study[END_REF]. The theory is that the enzyme slightly adjusts its active site structure around the imprint molecule and remains "trapped" in this conformation until the substrate enters.For example, pretreatment of a lipase with a chiral substrate such as (R)-2-octanol results in enantioselective activation [START_REF] Furukawa | Effect of imprinting sol-gel immobilized lipase with chiral template substrates in esterification of (R)-(+)-and (S)-(-)glycidol[END_REF]. The dualuse of sol-gel entrapment and imprinting has been successfully performed with lipases,to enhancetheir activity or selectivity [START_REF] Reetz | Second Generation Sol-Gel Encapsulated Lipases: Robust Heterogeneous Biocatalysts[END_REF][START_REF] Hellner | Novel Sol-Gel Lipases by Designed Bioimprinting for Continuous-Flow Kinetic Resolutions[END_REF][START_REF] Cao | Improving esterification activity of Burkholderiacepacia lipase encapsulated in silica by bioimprinting with substrate analogues[END_REF]. In these studies the enzyme is brought together with an "imprinting" molecule and then co-immobilized by solgel encapsulation.

The reaction efficiency can be enhanced by modifying the biocatalysts andalso by changing the reaction conditions. The influenceof temperature on selectivity is one of the major concerns in enzymatic reactions;in most cases, stereoselectivitydecreases with increasing temperature. Very few groupshave studied the temperature effect on lipase-catalyzed kinetic resolutions in continuous-flow mode;most datause batch mode (Abahazi et al., 2014).

Acyl donors also have the ability tochange lipase activity and selectivity during kinetic resolution of chiral molecules through acylation [START_REF] Devendran | Microwave Assisted Enzymatic Kinetic Resolution of (±)-1-Phenyl-2-propyn-1-ol in Nonaqueous Media[END_REF].By changing the acyl group of acylation agents, the lipase is transformed into various acyl-enzyme intermediates, with structurally different active sites. This exertsa variety of chiral discrimination abilities duringthe enantioselective acylation of chiral molecules [START_REF] Ema | Significant Effect of Acyl Groups on Enantioselectivity in Lipase-Catalyzed Transesterifications[END_REF].

In the present study, different strategies -engineering of the substrate molecule, reaction temperature, use of imprinting molecules andimmobilization of the enzyme-have been investigated toexplore lipase activity and enantioselectivity toward pentan-2-ol, through the acylation reaction. (S)-pentan-2-ol is a chiral intermediate in the synthesis of several potential anti-Alzheimer drugs that inhibit -amyloid peptide release and/or its synthesis (Audia et al., 1998). In addition, lipase-mediateddesymmetrization of trans-(Z)-cyclooct-5-ene-1,2diols [START_REF] Rouillard | Microwave-Assisted Kinetic Resolution of Homochiral (Z)-Cyclooct-5-ene-1,2-diol and (Z)-2-Acetoxycyclooct-4-enyl Acetate Using Lipases[END_REF] was examined. Indeed,enantiomerically pure vicinal diols are interesting building blocks for the production of flavours and fragrances [START_REF] Rouillard | Microwave-Assisted Kinetic Resolution of Homochiral (Z)-Cyclooct-5-ene-1,2-diol and (Z)-2-Acetoxycyclooct-4-enyl Acetate Using Lipases[END_REF].Resolution of pentan-2-ol experiments were performed in a continuous solid/gas reactor that allowsthe control and independent variation of thethermodynamic activity of substrates and other components. The solid/gas reactor offers the possibility to modulate, and to study independently, the effect of each component present in the reaction medium or the effect of each reaction parameter, all other things being equal [START_REF] Lamare | Solid/gas bioreactors: powerful tools for fundamental research and efficient technology for industrial applications[END_REF].

Materials and Methods

Materials

Lipases fromCandida antarctica (immobilized on acrylic resin 077K1155, 10 000 PLU - 1 Novozym 435 ® )and fromBurkholderiacepacia (free and immobilized on diatom MKBB3465, 500 PLU -1 )were purchased from Sigma Aldrich (Sigma-Aldrich ChemieS.a.r.l., Saint Quentin Fallavier,France). For pentan-2-ol resolution, free CaLB (Lipozyme  ) was used. It was a gift from NovozymesA/S (Bagsvaerd, Denmark) and was used after purification by gel filtration on PD 10 column (Health Care, Danemark), using ultra-pure water as elutant and then freezedried. For diol resolution, free CaLB was purchased from Sigma Aldrich. All enzymes were stored in a vacuum dessicator containing P 2 O 5 before use.

All other chemicals were purchased from Sigma Aldrich and were used without further purification except in the case of vinyl acetate which was used after fresh distillation.

Preparation of lipase co-dried with imprinting molecules

In order to prepare Novozym435® co-dried with imprinting molecules, 0.5g of immobilized biocatalyst was mixed in 3mL of hexane at room temperature during two hours with (R)-2aminohexane, (S)-2-aminohexane or 1-aminohexane at different concentrations. The aim was to obtain a ratio of 1, 10, 100 or 1000 moles of amine per mole of enzyme for each amine, assuming that the amount of active lipase on the carrier was equal to 1000nmole/g carrier [START_REF] Hedfors | Lipase chemoselectivity towards alcohol and thiol acyl acceptors in a transacylation reaction[END_REF].

General procedure for Sol-Gel Immobilization of Lipase

Sol-gel immobilized CaLB in presence of imprinting molecules or not, was prepared according to previous works [START_REF] Hellner | Novel Sol-Gel Lipases by Designed Bioimprinting for Continuous-Flow Kinetic Resolutions[END_REF][START_REF] Weiser | Disubstituteddialkoxysilane precursors in binary and ternary sol-gel systems for lipase immobilization[END_REF]. The following compounds were mixed in a 5mL glass vial: 390µL Tris-HCl buffer (0.1M, pH 7.5), 200µL propan-2-ol, possibly the imprinting molecule ((R)-2-aminohexane, (S)-2-aminohexane or 1aminohexane at a ratio of 1or 1000 moles of amine per mole of enzyme, assuming a quantity of enzyme of 50mg), 100µL NaF 1M, 200µL polyethylene glycol 1000(4% w/v) at room temperature for 10 min. The silane precursors TEOS(tetraethoxysilane):OcTEOS(octyltriethoxysilane):PhTEOS(phenyltriethoxysilane)(10: 7:3mmol) (in the given molar ratio) and 50mg of purified free CaLB were then added under stirring, submitted to sonication during 15 min and agitation during 12 hours at room temperature in order to obtain complete polymerization. The resulting solid was washed with propan-2-ol (7mL), distilled water (5mL), propan-2-ol (5mL) and hexane (5mL). The residual white powder was dried in a vacuum desiccator.

Instrumentation

IR spectra were performed with a Perkin-Elmer Spectrum 100 IRFT-ATR instrument. 1 H and 13 C NMR were obtained with a JEOL JNM LA400 (400 MHz) spectrometer. Chemical shifts () are given in parts per million (ppm) downfield from internal standard tetramethylsilane (TMS). Coupling constants J are given in Hz. The high resolution mass spectra (HRMS) were obtained with a Varian MAT311 spectrometer in the Centre Régional de Mesures Physiques de l'Ouest (CRMPO, University of Rennes, France).

General procedure for enantioselective acetylation of racemic pentan-2-ol using lipase

For pentan-2-ol resolution reactions, initial rates of reaction were calculated at different temperatures in a solid-gas reactor (Scheme 1). In this type of reactor,the enzyme in the solid state, is packed between two layers of glass wool, to form a packed bed, which is percolated by a carried gas, transporting gaseous substrates. A full description of this type of reactor was given previously [START_REF] Lamare | Solid/gas bioreactors: powerful tools for fundamental research and efficient technology for industrial applications[END_REF]. Thermodynamic activities for ester and alcohol substrates were respectively a ester =0.1 anda alcohol =0.05. Reactions were carried out under anhydrous conditions. Pentan-2-ol commercial solutions was dried byrefluxing with magnesium and then distilled. Methyl esterscommercial solutions were dried with Na 2 CO 3 anddistilled from P 2 O 5. The dried substrates were stored under argon atmosphere and over molecular sieves.The amount of enzyme in the fixed-bed of the reactor was equal to 2.5 mg of Novozym 435 ® or 0.15 mg of free CaLB. The total gaseous flow was equal to 2000 µmol.min -1 .

For experiments with Novozym435 ® co-dried with imprinting molecules, the mass of enzyme used was 1 mg and for sol-gel immobilized CaLBit was equal to 2mg, with a total flow rate equal to 1400µmol.min -1 .In these conditions, conversion level ofsubstrates was comprised between 2 and 5% and allowed initial reaction rates to be calculated.

For the solid-gas system analysis, the vapor phase leaving thebioreactor was injected in a gas chromatograph (Agilent model6890N Series) with the column Chirasil-Dex CB (25m×0.25mmi.d.×0.25µm -cyclodextrin; Chrompack,France), as previously described [START_REF] Létisse | Solid/gas biocatalysis: an appropriate tool to study the influence of organic compounds on kinetics of lipase catalyzed alcoholysis[END_REF].The column temperaturewasprogrammed to hold between 5 and 20 min at 55°C then to increase at different rates from 55 to 85 to 160°C, depending on the ester used as substrate.The external calibration of the two substrates (pentan-2-ol and methyl esters) was carried out by programminga range of their partial pressures in the bioreactor andby analyzing with the gas chromatograph. For the productspent-2-yl esters, an external calibrationwas carried out with pure esters synthesized as described below, by using gas chromatography (Agilent 7890A) equipped with an auto-sampler (7688B), in the same analytical conditions as described above.For accurate determination of E values the vapor phaseleaving the bioreactor was re-condensed if necessary and was then partiallyevaporated in order to enhance pent-2-yl esters detection andquantification.The enantiomeric ratio was calculated using the equationfrom Wescott and Klibanov [START_REF] Wescott | Predicting the solvent dependence of enzymic substrate specificity using semiempirical thermodynamic calculations[END_REF]):E = Vi R / Vi S , Vi R and Vi S are the initial rates of (R)-pent-2-yl ester and (S)-pent-2-yl estersynthesis respectively.The enantiomeric ratios were calculated by taking the average of three different experiments. The associated standard deviation was calculated for each result.

General procedure for the synthesis of pent-2-yl esters (1-5)

To a solution of pentan-2-ol (1eq) in pyridine (0.5 eq), anhydride (1.1eq) was added and the mixture was kept at room temperature for 16 h. The mixture was poured into ice cold water and extracted with ethyl acetate (3×20 mL). The ethyl acetate extract was washed with water (2×10 mL), 1 N HCl (2×10 mL), and finally with water (3×10 mL), dried over MgSO 4 , (Z)-(1S,8R)-9-oxa-bicyclo[6.1.0]non-4-ene (6). This compound(6) was synthesized and analysed by IR and NMR as previously described [START_REF] Rouillard | Microwave-Assisted Kinetic Resolution of Homochiral (Z)-Cyclooct-5-ene-1,2-diol and (Z)-2-Acetoxycyclooct-4-enyl Acetate Using Lipases[END_REF]. Trans-(Z)-cyclooct-5-en-1,2-diol (7)was thenobtainedfrom(6) andanalyzed as previouslydescribed [START_REF] Rouillard | Microwave-Assisted Kinetic Resolution of Homochiral (Z)-Cyclooct-5-ene-1,2-diol and (Z)-2-Acetoxycyclooct-4-enyl Acetate Using Lipases[END_REF].

General procedure for enantioselective acetylation of racemic trans-(Z)-cyclooct-5-en-1,2-diol (7) using lipase

To a solution of racemic diol (7) (0.2 g, 1.41 mmol) solubilized in 2.5 mL of THF and under inert atmosphere, the corresponding vinyl ester (ethanoate, propanoate, butanoate, 14 mmol)

and free or immobilized lipase were added. The mixture was stirred at room temperature and then filtrated, extracted and purified as previously described [START_REF] Rouillard | Microwave-Assisted Kinetic Resolution of Homochiral (Z)-Cyclooct-5-ene-1,2-diol and (Z)-2-Acetoxycyclooct-4-enyl Acetate Using Lipases[END_REF] Compounds(8a) and (8b)were analysed by IR and NMR as previously described [START_REF] Rouillard | Microwave-Assisted Kinetic Resolution of Homochiral (Z)-Cyclooct-5-ene-1,2-diol and (Z)-2-Acetoxycyclooct-4-enyl Acetate Using Lipases[END_REF].

(Z)-1-hydroxy-cyclooct-4-enyl propanoate(9a and 9b)

Compounds(9a and 9b) were obtained as colorless oil.  max (cm For the determination of enantiomeric excess (ee) of monoesters gas chromatography (Agilent 7890A) was used as previously described [START_REF] Rouillard | Microwave-Assisted Kinetic Resolution of Homochiral (Z)-Cyclooct-5-ene-1,2-diol and (Z)-2-Acetoxycyclooct-4-enyl Acetate Using Lipases[END_REF]. The enantiomeric excesses and yields were calculated by taking the average of three different experiments. The associated standard deviation was calculated for each result.

For the determination of enantiomeric excess of remaining diols, high performance liquid chromatography was carried out in a Waters 600s combined with an auto-sampler (Waters 717 plus) as previously described [START_REF] Rouillard | Microwave-Assisted Kinetic Resolution of Homochiral (Z)-Cyclooct-5-ene-1,2-diol and (Z)-2-Acetoxycyclooct-4-enyl Acetate Using Lipases[END_REF].

Results and Discussion

Enantioselective acylation of pentan-2-ol using free andimmobilized CaLB

Novozym 435 (CaLB immobilized on acrylic resin) was used here as a biocatalyst. The effect of acyl donor chain length on resolution was studied by measuring the enantiomeric ratio (E) in the kinetic resolution of racemic pentan-2-ol through acylation with different methyl esters (Figure 1), at different temperatures ranging from 50 to 100°C, in a continuous solid-gas reactor.

When CaLB is immobilized on acrylic resin, the highest E values are obtained with methyl propanoate as an acyl donor, at each temperature tested. E values decrease when the chain length of the acyl donor increases from C3 to C4 and to C6, at each temperature tested. E value is very low when methyl isobutanoateis used as an acyl donor (Figure 2A). The same trend was obtained for enzyme activity at 60°C, but the valueswere divided by 3 and 5 going from a C3 acyl chainto C4 andC6 acyl chains, respectively, and divided by 10 when methyl isobutanoateis used as an acyl donor.

The enantiomeric ratiosand enzyme activitieswere then measured with free CaLB instead of immobilized lipase, under the same conditions and using a longer acyl donor(C10 instead of C6). The highest enzymatic activities were obtained with the shortest acyl chain tested (C3), both at 60 and 80°C. Three fold higher values of enantiomeric ratio Ewereobtained compared to immobilized enzyme on acrylic resin (Figure 2A and 2B).Interestingly,in previous studies, it was shown by Jacobson et al. that free and immobilized CaLBon acrylic resin presented the same enantioselectivity toward other secondary alcohols such as 1-phenoxy-propan-2-ol or 3chloro-1-phenoxy-propan-2-ol, through acylation with vinyl butanoate in hexane [START_REF] Jacobsen | Immobilization does not influence the enantioselectivity of CAL-B catalyzed kinetic resolution of secondary alcohols[END_REF].

The effect of acyl donor lengthonthe resolution of pentan-2-ol with free CaLBdemonstrates that the highest E valueisobtained at 50°C with aC3 acyl chain, and at 80°C with a C10 acyl chain (Figure 2B).CaLB has a deep narrow active site into whichthe substrate ester binds in a hairpin structure [START_REF] Uppenberg | Crystallographic and molecular-modeling studies of lipase B from Candida antarctica reveal a stereospecificity pocket for secondary alcohols[END_REF].The acyl and pentan-2-ol moieties arethus broughtspatially close together during catalysis;this could explainlipase sensitivity in relation to the length of the acyl chain. Each different acyl-enzyme intermediate formed leads to different active site structures, with different enantioselectivity toward the secondary alcohol.

Furthermore CaLB exhibits a funnel-like scissile fatty acid binding site, inside which fatty acids with chain lengths as long as 13 carbons can bind completely [START_REF] Pleiss | Anatomy of lipase binding sites: the scissilefatty acid binding site[END_REF];this means that amino acid residues in the acyl binding site that are at some distance from the point of chirality also affectenantioselectivity [START_REF] Haeffner | Molecular Modeling of the Enantioselectivity in Lipase-Catalyzed Transesterification Reactions[END_REF]. Finally, it is known that immobilization of a protein may produce distortion of the active site, therefore, it is not surprising that enantioselectivityof CaLBvaries both with the acyl chain length of the donor and with immobilization. Recently two different conformations, open and closed, were observed for crystal structure of CaLB at very high resolution [START_REF] Stauch | Open and closed states of Candida antarctica lipase B: protonation and the mechanism of interfacial activation[END_REF]. The shift from the open form to the closed form involves the unfolding of the -helix 5, that allows the formation of a salt bridge between Asp145 and Lys290, thus closing the catalytic cavity.

Consequently, changes in both dielectric constant and ionic strength of the medium are able to affect interfacial activation of CaLB [START_REF] Stauch | Open and closed states of Candida antarctica lipase B: protonation and the mechanism of interfacial activation[END_REF]. These recent findings give new insights into the possible effects of substrates, media and immobilization on CaLB fold and activity. These features make it extremely difficult to predict a priori and explain

CaLBenantioselectivity.

The resolution of 3-methyl-2-butanol through acylation with vinyl esterswas studied by

Ottosson and Hultand it was found thatlong alkyl chains like vinyl octanoate afforded the highest resolution.A decrease in resolution was obtained with shorterchain lengths for vinyl hexanoate and vinyl butanoate.Shortening the acyl chain to vinyl propanoategave a slightincreasein resolution [START_REF] Ottosson | Influence of acyl chain length on the enantioselectivity of Candida antarctica lipase B and its thermodynamic components in kinetic resolution of secalcohols[END_REF]. There is no general rule concerning the effect of acyl chain length on CaLB-catalyzed resolution of secondary alcohols as it differs from one combination of acyl donor/alcohol to another. In conclusion, the choice of an achiralacyl donor strongly influences the efficiency of the resolution ofa chiral alcohol in acylation reactions with CaLB.

We then wished to ascertain the conditions necessary to obtain a high enantioselectivityin acylation reactions of pentan-2-ol usingimmobilized lipase, compared with free CaLB.

Methyl propanoate was chosen as acyl donor as it gives the best resolution compared to other acyl donors and also has the highest enzyme activity. 70°C was chosen as reaction temperature as it gives the best compromise between high enzyme activity and long-term stability.

The effect ofco-drying CaLBNovozym 435 with imprinting molecules in hexane, before performing the acylation reaction on immobilized enzyme activity and enantioselectivity was first studied.

The different amines ((R)-, (S)-2-aminohexane and 1-aminohexane)have a similar effect on enantioselectivity: Edoubles for 1 mole of amine per mole of enzymebut the increase of E is lower at higher concentrations of amines (Figure 3A).Enzymatic activity is increased 1.4

times to 1.8 times for 1 mole of amine per mole of enzyme and by lower amounts at higher concentrations of amines (1.5 to 1.7 times for 10 molesof amine per mole of enzyme, 1.3 to 1.6 times for 100 moles of amine per mole of enzyme and 1.15 to 1.5 times for 1000 moles of amine per mole of enzyme; Figure 3B). In all cases no amides were detectable in the reaction product. This is in agreement with the fact that alcohols deacylate the acyl-enzyme intermediate faster than amines (Öhrner et al., 1996).

The analysis of these different results is complex. The fact that the amines have a significant effect on enantioselectivity at a ratio 1/1 with the enzyme supports the hypothesis of an imprinting effect, but this is called into question by the fact that the three amines, with different structures, have an almost similar effect. The decrease in both enantioselectivity and activity using 100 or 1000 moles of amine per mole of enzyme instead of 1 or 10 moles could arise from the formation of non-reactive enzyme-amine complexes, as was hypothezised by Garcia-Urdialeset al., when studying the role of amines in the CaLB-catalyzed alcohol resolution (Garcia-Urdialeset al., 2001).

The effect of encapsulation prior to the acylation reaction of free CaLB, in the absence or presence of imprinting molecules, on enzyme activity and enantioselectivity toward pentan-2ol was then studied.

Encapsulation of free CaLBleadsto an increase of E from 109.0 ± 2.7 for free enzyme (lipozyme) to 154 ± 1 for CaLBencapsulated in sol/gel matrix (Figure 4).The value corresponding to the activity of the encapsulated CaLBis multiplied by a factor of 1.7 compared to free enzyme.A substantially improved enantioselectivityand activity of sol-gel entrapped CaLBcompared to non-immobilized lipase was also obtained by Ursoiuet al.in the acylation reaction of octan-2-ol [START_REF] Ursoiu | Sol-gel Entrapped Candida antarcticalipase B -A Biocatalyst with Excellent Stability for Kinetic Resolution of Secondary Alcohols[END_REF].

Enantioselectivityremainsalmost identical with or without addition of amines in the sol/gel matrix, at 1 or 1000 moles per mole of enzyme and with all aminestested.Enzymatic activity remained constant with or without amines added in sol/gel matrix for all the amines tested.

The enantioselectivity of CaLB toward pentan-2-ol varies substantially between the free form (E=109.0 ± 2.7),Novozym 435 (E=51.2 ± 1.0) and the encapsulated enzyme (E=154 ± 1).It has already been shown that Novozym 435, that corresponds to CaLB immobilizedon Lewatit VP OC 1600, a macroporous hydrophobic acrylic polymer resin, displays very different selectivity compared to free CaLB or CaLBimmobilized on other supports [START_REF] Cabrera | Novozym 435 displays very different selectivity compared to lipase from Candida antarctica B adsorbed on other hydrophobic supports[END_REF][START_REF] Mateo | Improvement of enzyme activity, stability and selectivityvia immobilization techniques[END_REF]. Recently, it was shown by Zisiset al. by combining experiment and simulation, that CalB is an interfacially activated enzyme and that the hydrophobicity of the support used for its immobilization influences both enzyme conformation and substrate specificity [START_REF] Zisis | Interfacial Activation of Candida antarctica Lipase B: Combined Evidence from Experiment and Simulation[END_REF].

For enzyme encapsulated in sol/gel matrix, neither the enantioselectivitynor the activity wereimproved by the addition of chiral aminesor by the primary amine.On the contrary, these parameters were significantly improved with the same amines when co-driedin hexane with the enzyme immobilized on acrylic resin.In a recent study, docking experiments into CaLBof various additives with imprinting effects, were performed, using the two different enzyme conformations(open and closed) observed by [START_REF] Stauch | Open and closed states of Candida antarctica lipase B: protonation and the mechanism of interfacial activation[END_REF][START_REF] Weiser | Bioimprinted lipases in PVA nanofibers as efficient immobilized biocatalysts[END_REF]. These studies revealed synergistic effects includingthe stabilization of the active conformation of CaLB and provision of extra space for the mobile active site-covering lids after theirremoval.

These types of effects were caused by large additives only, whereas smaller additives were unable to significantly stabilizethe active conformation of the active site [START_REF] Weiser | Bioimprinted lipases in PVA nanofibers as efficient immobilized biocatalysts[END_REF].

On that basis, it is probable that the "imprinting effect" for the investigated amines is rather based on allosteric effects.Further studies are required to fully explain these data.

Enantioselective acylation of diols using CaLB and BC lipases

Trans-(Z)-cyclooct-5-en-1,2-diol (7)was usedas a modelracemicsubstrate to study the ability of lipase to catalyze the resolution of these difunctionalized compounds through enantioselective acylation.The comparison of acyl donors and enzyme, together with the effects of immobilization of the enzyme, were carried out as outlined below.

Theracemicdiol( 7)was synthesizedfromthe commercial cycloocta-1,2-diene in two steps : epoxidation followed by ring opening in acidic conditions(Figure 5), as previously described [START_REF] Rouillard | Microwave-Assisted Kinetic Resolution of Homochiral (Z)-Cyclooct-5-ene-1,2-diol and (Z)-2-Acetoxycyclooct-4-enyl Acetate Using Lipases[END_REF].

The preparation of an optically enriched monoester was first performed using free or immobilized CaLB(Novozym 435) to desymmetrizemeso-symmetric diol (7)usingvinyl ethanoate, vinyl propanoate and vinyl butanoate as the acylatingagents in THF solvent,at room temperature (Figure 6A and Table 1).

When vinyl ethanoate is used as the acyl donor, immobilized CaLBgives a higher conversion than free CaLB (16% compared to 8%) and also demonstrates a higher enantioselectivity for the (1R,2R) diol. It increased the enantiomeric excess (ee) of (1R,2R)-monoethanoate(8a)from 50%to 55% and the ee of (1S,2S)-diol (7b)increasedfrom 4% to 9%. The immobilized CaLB was then chosen for reactions with three acyl donorscontaining different chain lengths.

The conversion and the enantiomeric excess of(1R,2R)-monoacetatewere enhanced when the chain length increased from C2 to C4.

Conversion increased from 16% using vinyl ethanoate to 33% using vinyl propanoate and 48%, using vinyl butanoate (Table 1).The enantiomeric excess of the monoester (8a)increased from 55%with a vinyl ethanoate donor to 73% or 96% using vinyl propanoate or vinyl butanoate, respectively. In all cases, the (1R,2R) diol was the preferred substrate for monoacetylation.The enantiomeric excess of the remaining (1S,2S)-diol (7b)increased from 9% using a vinyl ethanoate donor to 55% using vinyl propanoateand more than95% using vinyl butanoate.

The same reaction wasthen performed using BC Lipase (Figure 6B and Table 2).

When vinyl ethanoate was used as an acyl donor, BC lipase resulted in a higher conversion than CaLB lipase, and interestingly,under the same conditionsshowed an inversion ofenantioselectivity, with a preference for (1S,2S)-diol, and the formation of monoester (8b).

Enzyme immobilization slightly enhanced conversion (26% compared to 17%) but not selectivity (Table 2).No reaction was observedwhen an acyl donor with longer chainlength was used with immobilized BC Lipase.

Conclusion

In this study, it was shown that for pentan-2-ol resolution in a continuous solid/gas reactor enantioselectivityand temperature dependence of CaLB vary and depend on the conditions of immobilization of the enzyme. This indicates that the enzyme may take on different conformations induced by interactions with the immobilization supports. The acyl donor has a pronounced effect on CaLB-catalyzed resolution through acylation, due to the fact that the acyl and alcohol moieties are brought spatially close together during catalysis. A significant enhancement of resolution of the trans-(Z)-cyclooct-5-en-1,2-diolusing CaLBNovozym 435 and vinyl butanoate,rather thanvinyl ethanoate, as acyl donor, was also noticed. The data presented here contribute to the researchintoimproving the stereochemical purity of

  to give the esters:(1R,2R)-(Z)-1-hydroxy-cyclooct-4-enylethanoate(8a), (1S,2S)-(Z)-1-hydroxycyclooct-4-enyl ethanoate(8b), (1R,2R)-(Z)-1-hydroxy-cyclooct-4-enyl propanoate(9a), (1S,2S)-(Z)-1-hydroxy-cyclooct-4-enyl propanoate(9b), (1R,2R)-(Z)-1-hydroxy-cyclooct-4enyl butanoate(10a) and(1S,2S)-(Z)-1-hydroxy-cyclooct-4-enyl butanoate(10b).

  = 8.7, 3.8 Hz, CH-CO), 3.91 (1H, td, J = 8.5, 3.9 Hz, CH-COH),2.30-2.48(4H, m, 2xCH 2 )2.03-2.22 (5H, m 4xCH 2, C-OH), 1.60 -1.83(m, 4H, 2xCH 2 ), 0.96 (3H, t, J = 8.0 Hz, CH 3 ).

	-1 ): 3413, 2942, 1718, 1356,
	1272, 1183, 1028. H NMR (400 MHz, CDCl 3 ):5.54 -5.81 (2H, m, CH=CH), 4.96 (1H, td, J =
	8.7, 3.8 Hz, CH-CO), 3.90 (1H, td, J = 8.4, 3.8 Hz, CH-COH), 2.33 -2.46 (4H, m, 2xCH 2 ),
	2.01 -2.26 (5H, m, 2xCH 2 , C-OH), 1.66 -1.79 (2H, m, CH 2 ), 1.34 -1.03 (3H, m, CH 3 ).
	(Z)-1-hydroxy-cyclooct-4-enyl butanoate (10a and 10b)

Compounds(10a and 10b) were obtained as colorless oil. max (cm -1 ):

3414, 2962, 1718, 1255, 1177, 1042

. H NMR (400 MHz, CDCl 3 ): 5.

m, CH=CH), 4.97 (1H, td, J

Acknowledgments

The authors would like to thank ANR EXPENANTIO, la "LigueNationaleContre le Cancer" for financial supports and the "Ministère de la Recherche et de l'EnseignementSupérieur" for a PhD grant to Hervé Rouillard.The manuscript was revised and the English corrected by traduction@lefevere-laoide.net

compounds obtained by biocatalysisas well asother possibilities such as the optimisation of solvent, temperature, water activity, choice of enzyme and enzyme immobilization methods.

Author

Conflict of Interests

The authors declare that there is no conflict of interestregarding the publication of this paper.


Yann Brouet
email: yann.brouet@obspm.fr

Jérémie Lasue

Anny-Chantal Levasseur-Regourd

Pierre Encrenaz

Atmospheric entry of fluffy cometary particles under early Earth's conditions

INTRODUCTION

At the epoch of the Late Heavy Bombardment (LHB), the delivery of complex organic compounds on Earth, necessary for the apparition of life, was likely favored by the fluffy structure of the interplanetary dust of cometary origin (e.g. [START_REF] Lasue | Inferring the interplanetary dust properties from remote observations and simulations[END_REF]. We have used Direct Simulation Monte Carlo (DSMC) methods [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF] to simulate the entry of compact spherical particles and fractal aggregates of spherical grains (representative of fluffy cometary particles found in the interplanetary dust cloud), with a fractal dimension of 2.31, under early Earth's conditions. We performed 2-D calculations at the onset of particle ablation. Particle size varies from 2 mm to 100 µm. Several entry speed have been tested: 12 km.s -1 (prograde orbit), 30 km.s -1 and 71.5 km.s -1 (retrograde orbit). Results show significant differences between compact and irregular particles in terms of temperature distribution.

Methods

We performed our simulations with the Direct Simulation Monte Carlo (DSMC) method implemented by [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF]. This method is based on the random generation of representative atmospheric molecules locations and velocities as initial conditions to simulate the evolution of dilute gas flows. Considering Earth atmosphere as a dilute gas flow at the altitude where the onset of ablation is taking place, the code takes into account the effect of external forces, collisions and interactions of molecules with the particle surface to calculate their subsequent motion. The onset altitude of particle ablation is defined when the molecular mean free path is of the order of the particle size (Öpik, 1958). The simulations were thus done without integrating the physical and chemical modifications that happen during the ablation of the particles (e.g. fusion, ablation, chemical reaction with atmospheric molecules).

Entry parameters :

• Entry velocity: 12, 30 or 71.5 km.s -1 .

• Composition of primitive Earth atmosphere: 100 % CO 2 (e.g. [START_REF] Zahnle | Earth's Earliest Atmospheres[END_REF]).

• Particle size: Spheres with radius of 1, 0.5, 0.25, 0.1, and 0.05 mm; aggregates of 8 or 13 spherical grains (in 2-D) with radius of 0.05 mm each.

• Atmospheric density: 7.22×10 20 mol/m 3 .

• Altitude: 60 km (onset of particle ablation).

CONCLUSION and PERSPECTIVES

Structure of the particles and entry velocity are key parameters to the temperature in the immediate vicinity of the particles. Lower temperatures, which may avoid the destruction of fragile organic compounds, are more easy to obtain with fluffy aggregates than with spheres.

Taking into account the huge amount of dust particles, originating in comets and rich in organic compounds, that have been reaching the early Earth atmosphere, the delivery on Earth's surface of a significant fraction of this organic material is likely to have been favored by the structure of cometary dust (see [START_REF] Levasseur-Regourd | Oral presentation at this conference[END_REF].

Simulations for different sizes and fractal dimensions of dust aggregates, leading to even lower temperatures during atmospheric entry, are still to be done. 12.0 30.0

• It may be noticed that gas temperature close to the particle surface is systematically smaller for an aggregate than for a sphere. For an entry velocity of 12 km.s -1 , gas temperature close to the surface particle is about 6 000 K for a sphere and about 3 000 K for an aggregate of 13 grains (see figure 3).

• With a different orientation of the particle, temperature close to the surface appears to be similar (see figure 4).

• Typically, for an aggregate of 8 grains, temperature close to the particle surface varies between about 200 and 1000 K (see figure 4). At theses temperatures some organic materials, expected to be present in cometary dust, can survive : for HCN polymers degradation occurs for a temperature higher than 450 K [START_REF] Fray | Experimental study of the degradation of polymers : Application to the origin of extended sources in cometary atmospheres[END_REF]) ; for amorphous carbon and carbon nitrile, it occurs between 600 and 800 K [START_REF] Zhang | Thermal decomposition kinetics of amorphous carbon nitride and carbon films[END_REF].

• Both aggregates presented here have the same fractal dimension but not the same number of grains. Aggregate of 8 grains is more compact. This shape particularity seems to be more efficient to protect organic material. • For a given velocity, for a rather large field of view, the thermal environments are about similar for a sphere and an aggregate, whose gyro-radius is similar than of the sphere. It is not necessary the case in the immediate vicinity of the particle (see figure 3).

z 12.0 (for all).

500 µm 500 µm 500 µm 500 µm 500 µm

Figure 1 :

 1 Figure 1: Simulation evolution until steady state reached.

Figure 3 :

 3 Figure 3: Comparison between atmospheric entry of a 1 mm diameter sphere and a fractal aggregate of 13 grains of 0.05 mm each (zoom on the particle)

Figure 2 :•

 2 Figure 2: Comparison between atmospheric entry of a 1 mm diameter sphere and a fractal aggregate of 13 grains of 0.05 mm each (large field of view).

Figure 4 :

 4 Figure 4: Comparison between the atmospheric entry (at 12 km/s) of two fractal aggregates of 13 grains, with orientations at 90°from one another, and a fractal aggregate of 8 grains.