Xavier Hachair

Rémy Braive

Gian-Lucca L Lippi

Elvira David

Luc Le Gratiet

Aristide Lemaître

Izo Abram

Isabelle Sagnes

Isabelle Robert-Philip

Alexios Beveratos
email: alexios.beveratos@lpn.cnrs.fr

D Elvira

A Lemaitre

Definition of the stimulated emission threshold in high-β nanoscale lasers through phase-space reconstruction

Keywords: numbers: 42.55.Tv, 42.55.Sa, 42.55.Ah

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Recent progress in the fabrication of nanoscale lasers has opened the possibility of developing extremely efficient devices in which even the spontaneously emitted photons are preferentially funneled into the lasing mode. In these devices, the fraction β of spontaneous emission going into the laser mode becomes a significant fraction of unity, in contrast to conventional lasers, in which β ≃ 10 -5 . High-β nanolasers may be obtained in high-Q small-volume cavities, such as micropillars [1,2] or photonic crystal nanocavities [3][4][5], where the funneling is due to the directional enhancement of spontaneous emission through the Purcell effect, nanowires [6], in which the high β results from the very small number of modes they sustain [7], or surface plasmon lasers (SPASERs) [8][9][10], where the plasmon field strongly enhances spontaneous emission. High-β nanolasers can also be obtained in the strong coupling regime, using polariton condensation [11].

When β approaches unity, the transition from spontaneous to stimulated emission becomes smooth, so that the traditional definition of the threshold as an abrupt change in the Light-in Light-out (L -L) curve of the device is no longer applicable [12]. In fact, for β = 1 the L -L curve is a perfect straight line, with no break, so that from the viewpoint of energy efficiency the β = 1 laser can be considered "thresholdless" [13] as it exhibits the same efficiency for both spontaneous and stimulated emission. However, from the viewpoint of dynamical response it is important to be able to distinguish between the spontaneous and stimulated regimes, as the largesignal direct modulation (gain-switching) bandwidth of the laser is different in these two regimes. In addition, in high-β lasers, thanks to the deterministic channelling of spontaneous photons into the lasing mode, stimulated emission is initiated very rapidly giving rise to a broad direct modulation bandwidth even for large amplitude modulation (On-Off Keying -OOK) , a feature desirable for high-speed optical data transmission.

In order to differentiate the spontaneous from the stimulated regime, it was proposed to monitor the threshold through the evolution of coherence, in particular through the increase in the first order coherence, as dictated by the Schawlow-Townes relation [3,14] (which, however, can only be measured for stabilized continuous-wave operation) or through the decrease of the second order autocorrelation function, from g (2) (0) = 2 below threshold (corresponding to chaotic light) to g (2) (0) = 1 (corresponding to coherent emission) [1,2,4,5,11]. However both techniques have fundamental limitations. In nanoscale lasers, thermal or refractive index fluctuations lead to a deviation from the Schawlow-Townes expectations, while the transition from g (2) (0) = 2 to unity is not abrupt [1] and thus does not permit a more precise determination of the threshold than the L -L curve. At the same time, the threshold obtainable from the evolution of coherence is not related in a simple way to the stimulated emission threshold and does not provide direct information on the response dynamics of the laser. In addition, a definition of the threshold based on g (2) and the underlying amplitude fluctuations would be inapplicable to lasers displaying non-classical photon statistics, such as amplitude squeezed lasers [15].

In this Letter, we propose a phase-space reconstruction technique that permits an unequivocal identification of stimulated emission and provides an unambiguous definition of the threshold in nanolasers that can also be used for nanolasers with β = 1. We illustrate this approach through experiments identifying spontaneous and stim-ulated emission in the output of a high-β (β ≈ 0.25) nanolaser.

Our analysis is based on the traditional laser rate equations which, for a simple laser, are [16]:

dN (t) dt = P f (t) -γ N (t) -βγ N (t)n(t) (1) dn(t) dt = -Γ c n(t) + βγ N (t) +βγ N (t)n(t) (2)
where N (t) is the number of excited dipoles, n(t) the number of photons in the cavity, β the ratio of spontaneous photons in the laser mode over the total number of spontaneously emitted photons, γ -1 the dipole lifetime, Γ c the cavity loss rate and P f (t) describes the pumping, with the integral of f (t) over time, equal to 1. It should be noted that, because of the short round-trip time of light in the nanoscale cavity, the cavity decay is fast, so that nanolasers are generally characterized by Γ c ≥ γ (Class-B lasers). Following [START_REF] Loudon | The Quantum Theory of Light[END_REF], we can define N c = Γc βγ the critical number of dipoles necessary to attain the threshold. More complicated rate equations, taking into account various effects of the gain material, such as absorption of ground-state dipoles or non-radiative losses, would not alter the general discussion of our approach. Our analysis requires that the system be unexcited before and after the experiment, in other words that the pumping last only for a finite time, so that the number of photons and excited dipoles return to zero after the pumping is over. To visualize the evolution of the laser under Eqs. ( 1) and ( 2), we represent its response in a two-dimensional phase-space defined by {dn(t)/dt, n(t)} [START_REF] Packard | [END_REF], in a manner analogous to that for the traditional representation of transient laser dynamics through the parametric curve {N (t), n(t)} in which time is factored out [19]. The advantage of this novel phase-space is that, in contrast to N (t) which cannot be measured directly, dn(t)/dt like n(t) are both accessible experimentally, through the signal intensity measured outside the cavity, S(t) and dS(t)/dt. As the system returns to its initial (ground) state at the end of the excitation process, its trajectory in phase-space is a closed curve. The rate equations (1) and ( 2) are solved numerically, taking f (t) as a Gaussian pulse much shorter than γ -1 . Figure (1a) represents typical phase-space diagrams below (P = 0.1N c ) and above (P = 2N c ) threshold, for a nanolaser with β = 0.1 and Γ c = 10γ . At low pumping energies, when the laser operates in the spontaneous emission regime, the number of photons n(t) and its derivative dn(t)/dt both increase, but attain relatively small values overall. The phase-space trajectory is thus a relatively small loop. On the other hand, at high pumping energies, when the system operates under full laser dynamics [20], the number of photons n(t) and its derivative dn(t)/dt both increase to high values, so that the area enclosed by the trajectory increases dramatically. In addition, for Class-B lasers (Γ c ≥ γ ) the shape of the trajectory loop changes as soon as stimulated emission becomes the dominant dynamical process. Below threshold, after n(t) reaches its maximum, the phase space trajectory returns to the origin through the neg-ative half-plane (dn(t)/dt < 0) quasi-vertically so that the overall curve lies mostly in the right-hand half-plane defined by dn(t)/dt > 0. In that case -Γ c n(t) is slightly bigger than βγ N (t)(1 + n(t)). The number of excited dipoles is still large and decays with a characteristic lifetime γ . On the other hand, above threshold, the number of excited dipoles drops considerably due to stimulated emission and the negative term dominates. Thus the trajectory makes a long excursion in the negative half-plane so that the phase-space curve becomes almost symmetric. Hence, even for β = 1, the effect of stimulated emission is observable in the phase space trajectory and in the area it encloses. The area enclosed within the phase-space trajectory can be calculated through the Gauss-Green formula for a parametric curve {x(t), y(t)}:

A = 1 2 ∞ 0 x(t) dy(t) dt -y(t) dx(t) dt dt, (3)
which in our case reads,

A = 1 2 ∞ 0 dn(t) dt 2 -n(t) d 2 n(t) dt 2 dt. (4)
Equation ( 4) can be integrated by parts, and since at t = 0 and t = ∞ both n(t) and dn(t) dt are zero under pulsed excitation, the area is given by

A = ∞ 0 dn(t) dt 2 dt (5)
which, using Equation ( 2), can be expressed as

A = ∞ 0 Γ 2 c 1 -Ñ (t) 2 n 2 (t) + Ñ 2 (t) -2 1 -Ñ (t) Ñ (t)n(t) dt (6)
where Ñ (t) = N (t)/N c . Below threshold, Ñ (t) < 1 at all times, so that the third term in Eq. ( 6) is negative and is thus subtracted from the overall phase-space area. On the other hand, this term turns positive above threshold and its contribution adds to the total area during the time it remains positive. Thus, the area enclosed by the parametric curve is expected to increase dramatically at the transition between spontaneous and stimulated emission, even in the case β = 1 for which no break in the L -L curve can be observed. Figure 2 shows a log-log plot of the calculated L -L curve as well as the square root of the calculated area enclosed by the phase-space trajectory as a function of the pump energy for β = {1, 0.1, 0.01, 0.001} and Γ c = 10γ . The square root is used to facilitate comparison between the two types of curves, since both the signal and the square root of the area are proportional to P below threshold. The two types of curves coincide at low pumping and display the onset of a nonlinear increase (a "knee") at approximately the same pumping energies. However, while the L -L curves display a level change only for β < 1, the phase-space area curves display such a change even for β = 1, permitting identification of stimulated emission in all cases.

In steady-state lasers the threshold is conventionally defined as the x-intercept of the high pumping power asymptote in the linear L -L curve, which occurs at P = (1β)N c . Clearly, this definition cannot be applied to the β = 1 laser, as it sets the threshold at P = 0. Alternatively, the threshold is often defined as the inflec-

FIG. 2:

Comparison of the theoretical Light-In Light-Out curves (continuous black lines) and the phase-space area curves (dashed red lines) under pulsed pumping for a nanolaser with Γc = 10γ and β = {1, 0.1, 0.01, 0.001} tion point of the log-log L -L curve, where the number of photons in the cavity per dipole lifetime is 1/ √ β, so that stimulated emission dominates the response of the laser beyond that point. On the other hand, the onset of stimulated emission is associated with the "knee" of the log-log L -L curve, where the number of photons starts exceeding the value of 1. Thus the "'knee"' of the area curve can be defined as the thershold. This definition is also valid for β = 1 laser and identifies the point beyond which the laser dynamics (including its modulation bandwidth) change.

To illustrate the above considerations, we have investigated experimentally the phase-space transition for a photonic crystal nanolaser. The cavity consists of a photonic crystal double heterostructure [21] etched in a 180 nm-thick suspended GaAs membrane [22], incorporating a single layer of self-assembled InAs quantum dots at its vertical center plane. Details of the structure and experimental setup can be found in [23]. The nanolaser is optically pumped by a Ti:Sa laser delivering 3 ps-long pulses at a 81.8 MHz repetition rate and tuned to 840 nm near the energy gap of the wetting layer, in order to reduce heating of the membrane. Time traces are obtained by a Syncroscan streak camera (Hamamatsu), with a temporal resolution of 3 ps. Two representative experimental time traces are represented on Figure 1(b), one in the spontaneous emission regime (P sp = 1.7 µJcm -2 ) and one in the stimulated regime (P st = 8.6 µJcm -2 ). The decay rates of these two traces permit us to deduce γ -1 = 50 ps and Γ -1 c = 13 ps. The phase-space trajectories were obtained by measuring the cavity signal as function of time S(t) and then computing dS(t)/dt after applying a numerical low pass filter to smooth out the noise in the data. Figure 1(c) represents the phasespace trajectories that correspond to the two experimental time-traces of Figure 1(b). The two key features expected upon passage through the threshold, namely the change in symmetry and the dramatic increase in the area are clearly seen. Figure 3 presents a log-log plot of the square root of the integrated phase-space area, appropriately normalized, and compares it with the L -L curve. A spontaneous emission fraction of β ≈ 0.25 can be inferred from the intensity ratio between the spontaneous and stimulated emission levels. The "knee" of the area plot coincides with that of the L -L curve, giving for the threshold P th = 2.1 µJcm -2 .

In summary, the emission properties of nanoscale lasers with large spontaneous emission coupling factors have been studied from a dynamical point of view. In such lasers, the transition from spontaneous to stimulated emission extends over a wide pump power range in the Light in -Light out curve, and the threshold cannot be defined easily, especially when β = 1. We provide a simple and unequivocal identification of laser threshold through a new indicator: the area A enclosed by the phase-space trajectory that corresponds to the impulse response of the device. With this definition a threshold is defined even when none is visible in the L -L curve, as is the case for β ≈ 1 lasers. This definition is directly related to the onset of stimulated emission and thus directly pertains to the dynamical response of the laser, while at the same time it provides a practical tool for quick and easy measurements.

FIG. 1 :

 1 FIG.1: Phase-space trajectories of the nanolaser response: (a) Theoretical trajectories for a nanolaser of β = 0.1 and Γc = 10γ , under pumping below (P = 0.1Nc) and above (P = 2Nc) threshold. (b) Experimental time-traces of the output of a nanolaser under impulse excitation, below (Psp = 1.7µJ cm -2 ) and above (Pst = 8.6µJ cm -2 ) threshold. (c) Phase-space trajectories corresponding to the experimental time-traces above.

FIG. 3 :

 3 FIG. 3: Log-log plot of the Light in -Light out curve (blue open circles) and the square root of the area enclosed by the phase space trajectory (black circles) as a function of pump pulse energy. The dashed lines of slope unity in log-log scale are a guide to the eye

The authors gratefully acknowledge financial support from the Triangle de la Physique under the BIRD project and from the French National Agency (ANR) through the Nanoscience and Nanotechnology Program (project NATIF ANR-09-NANO-P103-36). The authors also thank K. Gauthron and J. Bloch for lending critical equipment and S. Barbay for helpful discussions.


Micha Pfeuty

Lionel Thivard

Sophie Dupont

Claude Adam

Michel Baulac

Séverine Samson
email: severine.samson@univ-lille3.fr

Postoperative recovery of hippocampal contralateral diffusivity in medial temporal lobe epilepsy correlates with memory functions (Running head: Diffusivity and memory)

Keywords:

This study aimed at determining if the recovery of mean diffusivity (MD) in the contralateral non sclerotic hippocampus is correlated with change in memory outcome after surgery in patients with medial temporal lobe epilepsy (MTLE). Verbal and nonverbal memory scores and MD were assessed in 23 patients with MTLE before and after surgical treatment of epilepsy. The recovery of MD in the left hippocampus was correlated with the performance on verbal memory tests and the recovery of MD in all patients was correlated with the performance on non-verbal memory tests. This finding strengthens the hypothesis that reversible diffusion abnormalities in the contralateral hippocampus in MTLE are linked to the active epileptic process that seems to interfere with memory abilities.

INTRODUCTION

Using interictal diffusion tensor imaging (DTI) in medial temporal lobe epilepsy (MTLE) patients with unilateral hippocampal sclerosis, we demonstrated bilateral hippocampal diffusion abnormalities: an increased mean diffusivity (MD) in the epileptic hippocampus and a decreased MD in the contralateral non sclerotic hippocampus. [START_REF] Thivard | Diffusion tensor imaging in medial temporal lobe epilepsy with hippocampal sclerosis[END_REF] Subsequently, we showed that the reduced MD in the contralateral hippocampus recovers after epilepsy surgery. [START_REF] Thivard | Postoperative recovery of hippocampal contralateral diffusivity in medial temporal lobe epilepsy[END_REF] This result supports previous proton magnetic resonance spectroscopy studies showing that spectroscopic abnormalities tend to normalize in the postoperative period. [START_REF] Cendes | Normalization of neuronal metabolic dysfunction after surgery for temporal lobe epilepsy. Evidence from proton MR spectroscopic imaging[END_REF][START_REF] Hugg | Normalization of contralateral metabolic function following temporal lobectomy demonstrated by 1 H magnetic resonance spectroscopic imaging[END_REF][START_REF] Serles | Time course of postoperative recovery of Nacetyl-aspartate in temporal lobe epilepsy[END_REF][START_REF] Vermathen | Temporal lobectomy for epilepsy: recovery of the contralateral hippocampus measured by (1)H MRS[END_REF] These neuroimaging findings suggest that contralateral abnormalities before surgery reflect neuronal dysfunction due to the epileptic process. Although we were not able to demonstrate a relationship between the recovery of MD in the contralateral non sclerotic hippocampus after surgery and the electro-clinical outcome of these patients, we hypothesised that MD recovery should be associated with changes in memory abilities.

It is well known that the hippocampus is involved in memory functions, the left and right mediating verbal and non-verbal memory functions respectively. [START_REF] Milner | Interhemispheric differences in the localization of psychological processes in man[END_REF] Links between MD variations in the hippocampus and neuropsychological testing have already been reported in MTLE patients [START_REF] Lui | Correlation of apparent diffusion coefficient with neuropsychological testing in temporal lobe epilepsy[END_REF] but there has been no study of the relationship between postoperative changes in MD and memory functions to date. The purpose of the present study is to examine correlations between the recovery of MD in the contralateral non sclerotic hippocampus and postoperative changes in memory scores. We predicted that postoperative memory outcome should be better in the patients showing a higher MD increase in the contralateral hippocampus. According to the functional lateralization of hippocampus, [START_REF] Lui | Correlation of apparent diffusion coefficient with neuropsychological testing in temporal lobe epilepsy[END_REF] we hypothesized that postoperative verbal memory outcome should be correlated with MD recovery in the left hippocampus (patients with right MTLE), while postoperative non-verbal memory outcome should be correlated with MD recovery in the right hippocampus (patients with left MTLE).

METHOD

Patients

The participants included 23 patients (15 women and 8 men, mean age at surgery 31.4±9.0 years, ranging from 19 to 52) with MTLE who had undergone a presurgical evaluation at the Epilepsy Unit (La Salpêtrière Hospital). Unilateral left (n = 12) or right (n = 11) hippocampal sclerosis (HS) was preoperatively diagnosed from qualitative MRI assessment. All patients, except one, were right-handed. Left cerebral lateralization of language was confirmed by Wada tests or fMRI in most of the patients (n = 15) and one patient with left HS had bilateral language representation. With the exception of four patients who underwent a selective amygdalo-hippocampectomy, they underwent anteromedial temporal lobe (ATL) resection. Systematic pathological examination of the operated tissue confirmed HS in all patients. The number of antiepileptic medications varied from one to four (mean = 3) before surgery and from zero to four (mean = 2) after surgery. The number of antiepileptic drugs taken before and after surgery was identical for 16 patients, 6 had one medication withdrawn and one had one medication added.

Based on Engel's classification, 18 patients were seizure free (class Ia), and 5 had only rare persistent auras (class Ib) after surgery. The local ethics committee approved the study and informed written consent was obtained from all the patients.

Imaging protocol

Conventional MRI and DTI data were acquired on a 1.5 T scanner (GE, Milwaukee, USA). All the patients reported that they were free of seizure in the 24 hours preceding the DTI scan. The mean interval between the pre-and postoperative MRI was 451±158 days, ranging from 252 to 790 days. The mean delay between surgery and the postoperative MRI was 243±100 days, ranging from 69 to 629 days. Details on the image acquisition and the definition of regions of interest are reported in Thivard and collaborators (2007). 2

Memory evaluation of patients

Span and supra-span learning tasks were used to assess verbal and non-verbal memory abilities. The auditory verbal span test was performed by all patients. It contains an increasing number of digits to be immediately recalled. The score corresponded to the maximum number of digits recalled in the correct order. The abstract word learning test [START_REF] Jones-Gotman | Learning and retention of words and designs following excision from medial and lateral temporal-lobe structures[END_REF] was performed by 10 left and 11 right temporal lobe patients. It contains 13 abstract visual words that have to be recalled across four successive learning trials. The score was calculated as the total number of words recalled over all trials (score max = 52). The visual spatial span test was performed by all but one left temporal lobe patient. It comprises an increasing number of blocks displayed on a board to be immediately recalled by pointing. The score corresponded to the maximum number of blocks recalled in the correct order. The Aggie figure learning test 10 was performed by 5 left and 6 right temporal lobe patients. It consists of 15 non-verbal simple line drawings that have to be recalled across five successive learning trials. The score was calculated as the total number of figures recalled over all trials (score max = 75). The mean delay between the pre-and postoperative memory evaluation of patients was 548±213 days, ranging from 256 to 984 days.

Statistical analysis

Two by two ANOVAs were performed on the contralateral hippocampal MD values and on the memory scores obtained by the two patient groups (right vs. left) before and after surgery. Furthermore, we calculated the number of patients whose performance increased after surgery in the two verbal memory scores or in the two non-verbal memory scores.

Chi-square was used to compare the number of participants with postoperative improvement in the two groups of patients (right vs. left).

A percentage of postoperative recovery of contralateral MD was calculated for each patient as follows: MD recov = 100 × ((MD post -MD pre ) / MD pre ), in which MD recov is the percentage of postoperative recovery of contralateral MD whereas MD pre and MD post correspond to values of MD before and after surgery respectively. For each patient and each test performed, we subtracted the score before surgery from the score after surgery to obtain a score post-pre . To allow comparison between tests, each score post-pre was converted to a z-score post-pre using the mean and standard deviation obtained by all the patients for each test. For patients who performed the two verbal memory tests, zscores post-pre were averaged to obtain a global verbal memory z-score post-pre . For patients who performed the two non-verbal memory tests, z-scores post-pre were averaged to obtain a global non-verbal memory z-score post-pre .

Pearson's correlation coefficients were first calculated to assess potential associations between postoperative recovery of hippocampal contralateral diffusivity and postoperative memory outcome. However, the change in memory scores were likely to be related to the delay between the pre-and postoperative memory evaluation of patients (delay post-pre ) and to the memory scores before surgery (z-scores pre ) as observed in a recent study. [START_REF] Baxendale | Improvements in memory function following anterior temporal lobe resection for epilepsy[END_REF] Thus, partial Pearson's correlation coefficients were used to adjust for delay post-pre and z-scores pre . Furthermore, Mann Whitney U tests were performed to determine the effect of the type of surgery (selective amygdalo-hippocampectomy vs. anterior temporal lobectomy) on MD recov and the different memory z-scores post-pre .

RESULTS

Postoperative recovery of the hippocampal contralateral diffusivity

Contralateral hippocampal MD value increased after surgery (Mean (SD), preoperative: 895 (25) × 10 6 mm 2 /s, postoperative: 914 (28) × 10 -6 mm 2 /s; F(1, 21)=22.81, p<0.0001).

There was no significant effect of lesion laterality (F<1) or interaction between factors (F<1).

Postoperative outcome of memory scores

The score in the abstract word learning test decreased after surgery (Mean (SD), preoperative: 32.4 (7.5), postoperative: 28.8 (8.6); F(1, 19)=8.32, p=0.009) and was higher for right than for left ATL patients (right ATL patients: 34.7(6.0), left ATL patients: 26.1(6.4); F(1, 19)=10.35, p=0.005). There was no significant interaction between factors (F(1, 19)=1.81, p=0.194). The other memory scores did not significantly change after surgery and there were no significant effect of lesion laterality or interaction between factors.

The number of patients whose performance increased after surgery in the two verbal memory tests was significantly more important in the right ATL (3 out of 11) than in the left ATL (0 out of 10) group (chi-2=10.71, df=1, p=0.001). No patients demonstrated postoperative improvement in the two non-verbal memory tests.

Correlations between recovery of the hippocampal contralateral diffusivity and memory scores

Figure 1 presents scatter plots of the different memory z-scores post-pre against the postoperative recovery of MD in the contralateral hippocampus for the whole group of patients and for the left and right ATL patients separately. Pearson's correlation coefficients are shown in each plot. Furthermore, partial correlation coefficients adjusting for delay post-pre and z-scores pre , are shown in Table 1. No significant effect of the type of surgery was found on the MD recov and on the different verbal and non-verbal memory zscores post-pre .

Verbal memory

There was no significant correlation between the postoperative recovery of MD in the contralateral hippocampus and the different verbal memory z-scores post-pre in all patients.

When the left ATL patients were separated from the right ATL patients, the recovery of MD in the contralateral hippocampus was positively correlated with the auditory verbal span z-score post-pre for the right ATL patients (p=0.023). Furthermore, an almost significant positive correlation was observed with the global verbal memory score (p=0.053). For the left ATL patients, an almost significant negative correlation was observed with the auditory verbal span z-score post-pre (p=0.071).

In a post-hoc analysis, right ATL patients were separated in two groups: G 1 made up of the 3 patients whose performance increased after surgery in the two verbal memory tests and G 2 made up of the 8 other patients. MD recov value was significantly higher for G 1 patients (Mean(SD), G 1 : 4.9(1.4) %, G 2 : 1.7(1.8) %, p=0.025, Mann-Whitney U-test).

Non-verbal memory

The postoperative recovery of MD in the contralateral hippocampus was positively correlated with the global non-verbal memory z-score post-pre (p=0.005) and with the Aggie figure learning z-score post-pre (p=0.007) in all patients. When the left ATL patients were separated from the right ATL patients, the recovery of MD in the contralateral hippocampus was positively correlated with the Aggie figure learning z-score post-pre for the left ATL patients (p=0.032), but the correlation was not significant after adjustment for delay post-pre and z-scores pre . However, the number of patients within each group was very small (<7) for the Aggie figure learning z-score post-pre and the global non-verbal memory score post-pre .

DISCUSSION

The results of the present study demonstrate that postoperative recovery of MD in the non sclerotic hippocampus contralateral to the resection is linked to memory outcome in patients who have undergone surgery for intractable MTLE epilepsy.

The number of patients with postoperative improvement in verbal memory was more important in the right ATL than in the left ATL group, in agreement with a recent study. [START_REF] Baxendale | Improvements in memory function following anterior temporal lobe resection for epilepsy[END_REF] Furthermore, postoperative verbal memory outcome was positively correlated with MD recovery of the contralateral hippocampus only for the right ATL patients. It has been previously reported that patients with right MTLE are impaired in verbal memory before surgery and this impairment is associated with a global decrease in left hemisphere functional activity. [START_REF] Dupont | Bilateral hemispheric alteration of memory processes in right medial temporal lobe epilepsy[END_REF] Furthermore, the MD in the left hippocampus of rightsided MTLE patients is reduced before surgery and returns to normal after surgery. [START_REF] Thivard | Diffusion tensor imaging in medial temporal lobe epilepsy with hippocampal sclerosis[END_REF][START_REF] Thivard | Postoperative recovery of hippocampal contralateral diffusivity in medial temporal lobe epilepsy[END_REF] Although several explanatory hypothesises have been previously proposed, [START_REF] Thivard | Postoperative recovery of hippocampal contralateral diffusivity in medial temporal lobe epilepsy[END_REF] the functional mechanisms leading to the recovery of MD remain speculative. The present results reveal that this return to normal is associated with better postoperative outcome in verbal memory and more specifically in the auditory verbal span score. This latter finding would be related with the role of the left hippocampus in verbal encoding. [START_REF] Milner | Interhemispheric differences in the localization of psychological processes in man[END_REF] This is consistent with a recent study which shows that cognitive rehabilitation has stronger effect on verbal memory after right than after left temporal lobe surgery. [START_REF] Helmstaedter | The effects of cognitive rehabilitation on memory outcome after temporal lobe epilepsy surgery[END_REF] Cognitive rehabilitation could take effect by favouring return to normal of the contralateral non sclerotic hippocampus, although only return to normal of the left hippocampus would be associated with better postoperative verbal memory outcome.

Even though not significant, increasing diffusion in the right hippocampus of the left ATL patients is associated with worse postoperative verbal memory outcome. A hypothesis is that MD recovery of the right hippocampus would improve functions of the right hemisphere at the expense of the left, which would amplify their deficit in verbal memory.

Postoperative non-verbal memory outcome was positively correlated with MD recovery of the contralateral hippocampus only when right and left ATL patients were pooled together. This is consistent with the idea that non verbal and specifically visual spatial memory depends on the integrity of both right and left hippocampi. [START_REF] Van Asselen | 7)=0.26 p=0.502 Rp(1)=-0.76 p=0.449 df =0 Right ATL patients Rp(7)=0[END_REF] However, the absence of significant correlation in separated groups could be due to a lack of statistical power related to the large proportion of missing data.

In conclusion, these results strengthen the hypothesis that reversible diffusion abnormalities in the hippocampus contralateral to the seizure focus in MTLE are functional and linked to the active epileptic process. Furthermore, diffusion abnormalities appear to be associated with changes in memory functions, suggesting a correlation between cognitive outcome and neuronal changes measured by diffusion tensor imaging in MTLE patients who have undergone unilateral antero-mesial temporal lobe resection.

Figure 1 .

 1 Figure 1. Scatter plots of the different z-scores post-pre against the postoperative recovery

Figure 1 Table 1 .

 11 Figure 1

ACKNOWLEDGMENTS

The authors are grateful to Amee Baird for her helpful comments on previous version of the manuscript. This work was supported by the IFR 49, by a grant from the college des Enseignants de Neurologie (L.T.) and was part of a post-doctoral project at the Epilepsy Unit -La Salpêtrière Hospital (M.P). The authors confirm that they have read the Journal's position on issues involved in ethical publication and affirm that this report is consistent with those guidelines.

DISCLOSURE OF CONFLICTS OF INTEREST

The corresponding author certifies that all co-authors have no financial or other conflict of interest to disclose.


M.D Luana Benedetti
email: luanabenedetti@libero.it

Chiara Briani

Diego Franciotta

Raffaella Fazio

Ilaria Paolasso

Cristoforo Comi

Marco Luigetti

Mario Sabatelli

Fabio Giannini

Giovanni Luigi Mancardi

Angelo Schenone

Eduardo Nobile-Orazio

Dario Cocito

Benedetti

Rituximab in patients with CIDP: A report of 13 cases and review of the literature

Keywords: Rituximab, CIDP, polyradiculoneuropathy

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Introduction

Chronic inflammatory demyelinating polyradiculoneuropathy (CIDP) is a relapsingremitting, or chronic-progressive disease of the peripheral nervous system [START_REF] Dyck | Chronic inflammatory polyradiculoneuropathy[END_REF]. The etiology of CIDP is unknown, but immunologic mechanisms, which involve both B and T cellmediated responses, presumably play a central role in the pathogenesis of the damage to peripheral myelin. Said mechanisms offer a rationale for immunoactive therapies, such as intravenous immunoglobulins (IVIg), corticosteroids, and plasma exchange (PE), but inadequate responses to these treatments lead to the use of immunosuppressant drugs. [START_REF] Brannagan | Current treatments of chronic immune-mediated demyelinating polyneuropathies[END_REF] Rituximab, a monoclonal anti-CD20 antibody, has recently been used for the treatment of anti-MAG antibody-associated polyneuropathy [START_REF] Benedetti | Predictors of response to Rituximab in patients with neuropathy and anti-MAG IgM[END_REF][START_REF] Dalakas | A placebo-controlled trial of rituximab in IgM anti-myelin-associated glycoprotein antibody demyelinating neuropathy[END_REF], another demyelinating neuropathy with immune-mediated pathogenesis. In CIDP, the drug proved to be efficacious in single cases [START_REF] Kilidireas | Rituximab therapy in monoclonal IgM-related neuropathies[END_REF][START_REF] Knecht | Sustained remission of CIDP associated with Evans syndrome[END_REF][START_REF] Műnch | Rituximab in chronic inflammatory demyelinating polyneuropathy associated with diabetes mellitus[END_REF][START_REF] Kasamon | EBV-associated lymphoma and chronic inflammatory demyelinating polyneuropathy in an adult without overt immunodeficiency[END_REF][START_REF] Briani | Rituximab-responsive CIDP[END_REF][START_REF] Benedetti | Rituximab efficacy in CIDP associated with idiopathic thrombocytopenic purpura[END_REF], whereas in a prospective pilot trial (2 cases) it afforded no reduction in IVIg dosage [START_REF] Gorson | Rituximab treatment in patients with IVIgdependent immune polyneuropathy: a prospective pilot trial[END_REF].

We present 13 CIDP patients who were treated with rituximab in eight Italian centres after unsatisfactory responses to standard therapies.

Methods

We reviewed nationally registered [START_REF] Cocito | Register of CIDP patient non responders to traditional therapy (abstract)[END_REF] CIDP patients diagnosed in accordance with the EFNS/PNS criteria [START_REF] Hughes | European Federation of Neurological Societies/Peripheral Nerve Society guideline on management of chronic inflammatory demyelinating polyradiculoneuropathy: report of a joint task force of the European Federation of Neurological Societies and the Peripheral Nerve society[END_REF]. Two of these cases had been previously reported [START_REF] Briani | Rituximab-responsive CIDP[END_REF][START_REF] Benedetti | Rituximab efficacy in CIDP associated with idiopathic thrombocytopenic purpura[END_REF]. Table 1 summarizes the demographic and clinical characteristics of the selected 13 patients (8 men, 5 women; age range, 30-72 years, mean age, 55 years; mean neuropathy duration, 7 years).

Routine laboratory tests, including glycaemia, serum thyroid hormones, B 12 vitamin, tumor markers, and hepatitis C serology excluded other causes of neuropathy. Anti-MAG and antisulfatide antibody tests were negative. Clinical impairment and disability prior to rituximab were highly variable: mean value of 48 (range, 36-56) on the Medical Research Council (MRC) sumscore [START_REF] Kleyweg | Interobserver agreement in the assessment of muscle strenght and functional abilities in Guillain-Barrè syndrome[END_REF], and of 5 (range, 3-8) on the Inflammatory Neuropathy Cause and Treatment (INCAT) disability score [START_REF] Hughes | the Inflammatory Neuropathy Cause and Treatment (INCAT) Group. Randomized Controlled Trial of Intravenous Immunoglobulin Versus Oral Prednilosone in Chronic Inflammatory Demyelinating Polyradiculoneuropathy[END_REF]. Disease course was relapsing-remitting in 9 Benedetti 4 patients, and chronic-progressive in 4. In 8 cases, CIDP co-occurred with haematological diseases. Rituximab was proposed for the following reasons: 7 patients were refractory to conventional immunosuppressive therapies, and quality of life was compromised to 6 patients by frequent recourse to PE/IVIg cycles (mean between-cycle interval, 44 days; range, 14-120); additionally, haematological diseases worsened in 4 cases. All patients gave written informed consent to rituximab treatment.

Rituximab dosage was 375 mg/sqm IV, weekly for 4 consecutive weeks, for 12 patients; one patient, who had severe Waldenstrom macroglobulinemia, received 1000 mg IV every six months for 2 years (4 infusions).

Neurological assessments were performed at baseline and every month for the first 6 months, then every 3 months (MRC sumscore for muscle strength, and INCAT scales for arm and leg disability), with ≥ 1 year follow-up. CD19+ B cells, serum immunoglobulin levels, and platelet counts were checked regularly.

Before-and after-treatment electrophysiological data were available for 6 patients. Our analysis was based on ulnar nerve motor conduction velocity (MCV), because peroneal nerve MCV was not recordable in 3 patients. We considered ≥ 10% changes in MCV values as indicators of therapeutic efficacy.

Responders were defined as those patients who improved by at least 2 points on each of the two clinical scales, or who maintained the degrees of improvement obtained with IVIg/PE cycles, without further cycles of IVIg/PE over the follow-up.

Our review of the literature included patients with pure CIDP, and excluded CIDP associated with anti-MAG/-sulfatide antibodies.

Benedetti 5

Results

Table 1 shows the clinical data. Nine out of 13 (69%) patients responded to rituximab.

Clinical improvements occurred at a median time of 2.0 months after the end of rituximab cycles, and lasted for up to one year. Among the 7 patients who were refractory to conventional therapies, one worsened under rituximab, whereas the other 6 improved by at least 4 points in the MRC score, and by 2 points in the INCAT scales, with a reduction of disability in quality-of-life-related daily activities, such as deambulation, handling knife and fork, washing hair, and doing-undoing zips. Of the 6 patients for whom rituximab replaced IVIg/PE cycles, 3 were responders: two patients ceased IVIg/PE and maintained the improvement achieved with these treatments prior to rituximab (benefits started at a median time of 2.0 months after rituximab, and lasted for ≥ 1 year); one patient stopped IVIg, and clinically improved [START_REF] Benedetti | Rituximab efficacy in CIDP associated with idiopathic thrombocytopenic purpura[END_REF]. The 3 non-responders recorded worsened clinical symptoms after a 25% reduction in IVIg dosage (Table 1). No responders required any re-treatment throughout the follow-up period. We tried to correlate the response to rituximab with clinical variables, namely disease duration, course of disease, and association with haematological diseases.

Mean disease durations in responders (4.3 years) were significantly shorter than in nonresponders (12.7 years; p=0.025, Mann-Whitney U test). Clinical improvement and course of the disease did not correlate. Seven of the 9 responders, and one of the 4 non-responders had haematological diseases, but the between-group difference was not significant (p=0.12, Fisher exact test). Blood CD19+ B cells were undetectable in all 13 patients at month 1 post-rituximab cycles; they reappeared at month 6, and returned to pre-treatment values at months 9-12.

Serum IgM levels were about 50% lower at 1-3 months post-treatment, and were still lower at month 12 than at baseline in patients with IgM monoclonal gammopathy of undetermined significance (MGUS) or with Waldenstrom macroglobulinemia. In contrast, serum IgA levels Benedetti 6 were unchanged in the IgA MGUS patient. Platelet counts improved in the patient with idiopathic thrombocytopenic purpura (ITP) [START_REF] Benedetti | Rituximab efficacy in CIDP associated with idiopathic thrombocytopenic purpura[END_REF]. Rituximab therapy respectively circumvented autologous peripheral blood stem cell transplantation and splenectomy in Waldenstrom macroglobulinemia and ITP patients.

Electrophysiological data showed ≥ 10% ulnar nerve MCV improvement in 5 responders, and worsening in one non-responder (data not shown).

No major adverse effects were recorded. During the first rituximab infusion, one patient complained of flu-like symptoms, and another showed a mild skin allergy, which responded to corticosteroids.

Discussion

B cell-depleting antibody rituximab has current FDA approval for the treatment of non-Hodgkin lymphoma and rheumatoid arthritis, but its off-label use extends to several autoimmune diseases. Since rituximab could target the immunological mechanisms deemed to be involved in CIDP pathogenesis, the drug qualifies for CIDP patients who do not respond to standard therapies. The few relevant reports show controversial results in terms of drug efficacy. One study on 2 CIDP cases reports no benefit in the primary endpoint, namely a 25% reduction in IVIg dosage 1 year after rituximab therapy vs IVIg dosage in the year prior to rituximab [START_REF] Gorson | Rituximab treatment in patients with IVIgdependent immune polyneuropathy: a prospective pilot trial[END_REF]. In contrast, four case reports describe clinical improvement in four rituximab-treated CIDP patients (Table 2) (5-8). Specifically, rituximab induced: the complete disappearance of a gastric lymphoma, and long-lasting improvements in neurological symptoms in a CIDP case [START_REF] Kilidireas | Rituximab therapy in monoclonal IgM-related neuropathies[END_REF]; the remission of both neurological symptoms and haematological abnormalities for 17 months in a patient with CIDP and Evans syndrome (6); the stabilization of CIDP course for 11 months in a patient with diabetes mellitus [START_REF] Műnch | Rituximab in chronic inflammatory demyelinating polyneuropathy associated with diabetes mellitus[END_REF].

Combined with chemotherapy, rituximab proved successful in a case of CIDP associated with non-Hodgkin lymphoma [START_REF] Kasamon | EBV-associated lymphoma and chronic inflammatory demyelinating polyneuropathy in an adult without overt immunodeficiency[END_REF]. Benedetti 7 To date, our cohort is the largest to be used to evaluate rituximab in CIDP. Our data show that rituximab induced a sustained remission of neurological symptoms in about 70% of CIDP patients. Although we do not advocate a syncretic approach, we note that this percentage is similar to that obtainable by pooling the 6 CIDP cases previously reported in the literature. Comparably with what was reported for anti-MAG neuropathy [START_REF] Benedetti | Long-term effect of Rituximab in anti-MAG polyneuropathy[END_REF], rituximab efficacy lasted one year, or longer, both in our series and in those previously reported in the literature. As a result, patients either improved clinically, or were spared periodic IVIg/PE cycles. Regarding the responders, 3 of the 6 previously reported cases, and 7 of our 13 patients, had associated haematological diseases, and rituximab was effective for both pathologies. In all the given series, the drug was less effective for idiopathic CIDP than for CIDP and haematological diseases. These findings suggest that, when associated with haematological diseases, CIDP may have different pathogenic mechanisms from those underlying isolated CIDP. Although CIDP is considered an autoimmune disease, the target antigen and the precise roles of humoral and cell-mediated immunity remain unknown [START_REF] Hughes | Chronic inflammatory demyelinating polyradiculoneuropathy[END_REF].

Rituximab greater efficacy in CIDP associated with haematological diseases could imply that B cells, whether as antibody, or cytokine-producing cells, play a predominant role in the pathogenesis of these CIDP types. Responses to rituximab in demyelinating lymphomaassociated neuropathies [START_REF] Briani | Improvement of peripheral nervous system manifestations of B-cell non-Hodgkin's Lymphoma after Rituximab therapy[END_REF] could support this view.

As a whole, the percentages of responders reported in the literature and in our series are similar to those reported for anti-MAG polyneuropathy [START_REF] Benedetti | Predictors of response to Rituximab in patients with neuropathy and anti-MAG IgM[END_REF][START_REF] Dalakas | A placebo-controlled trial of rituximab in IgM anti-myelin-associated glycoprotein antibody demyelinating neuropathy[END_REF], but the intervals between rituximab cessation and clinical improvement differed: 2-3 months in CIDP versus 6-8 months in anti-MAG polyneuropathy [START_REF] Benedetti | Predictors of response to Rituximab in patients with neuropathy and anti-MAG IgM[END_REF][START_REF] Dalakas | A placebo-controlled trial of rituximab in IgM anti-myelin-associated glycoprotein antibody demyelinating neuropathy[END_REF]. It is likely that the longer intervals in anti-MAG polyneuropathy depend on the fact that rituximab does not target plasma cells, namely the cells directly involved in disease pathogenesis, and that benefits thus occur only after a sustained depletion of their precursors, the CD20+ B cells. Benedetti 8 Both for our CIDP patients and for those previously reported, mean disease duration in responders was shorter than in non-responders. The phenomenon might derive from cumulative axonal damage in patients with greater disease durations, and from the possible down-regulation of CD20 molecule as a result of aggressive and prolonged pre-rituximab immunosuppressive therapies; such down-regulation could deny targets, and hence efficacy to rituximab.

In conclusion, although obtained in a relatively small number of patients, our data suggest that rituximab therapy for refractory CIDP may be promising and free from major adverse events, particularly if the disease has short duration, and is associated to haematological diseases that are potentially responsive to the drug.

Competing interests: None

The Corresponding Author has the right to grant on behalf of all authors and does grant on behalf of all authors, an exclusive licence (or non exclusive for government employees) on a worldwide basis to the BMJ Publishing Group Ltd and its licencees, to permit this article (if accepted) to be published in JNNP and any other BMJ Group products and to exploit all subsidiary rights, as set out in our licence (http://jnnp.bmjjournals.com//ifora/licence.pdf)

Table 1 :

 1 Demographic and clinical characteristics of the CIDP patientsTreatment arm and leg disability scores; * from the onset of neurological symptoms; § the first number refers to values at baseline, and the second number to value at improvement; † the degree of improvement obtained after PE and IVIg cycles and maintained after rituximab; ^ which corresponded to duration of improvemen

	Patient	Disease	Neuropathy	Hematological	Pre-rituximab	Rationale	MRC § INCAT §	Months	Duration of
	(sex, age)	course	duration*	Disease	therapy	for			before	follow-up ^
			(years)			rituximab			improvement	(years)
	1 (M, 72) 9	RR	3	Low-grade small B cell lymphoma	Ster, IVIG, AZA, Cyclosp	Refractory CIDP	36-48	5-3	6	5
	2 (F, 56)	RR	2	Ig M MGUS	Ster, IVIg, PE	To spare PE 47-53 †	6-3	6	4
	3 (M, 35)	RR	10	-	Ster, PE, IVIg, AZA, MM	To spare IVIg	55-55	5-5	-	1
	4 (M, 63)	RR	0	-	Ster, PE, IVIg, AZA	Refractory CIDP	50-56	5-2	2	4
	5 (F, 65)	CP	4	Ig A MGUS	Ster, PE, IVIg, AZA	Refractory CIDP	56-47	3-4	-	1
	6 (F, 67)	CP	1	Non-Hodgkin Lynphoma	Ster, PE, IVIg	Refractory CIDP	46-54	5-3	1	2
	7 (M, 59) 10	RR	1	ITP	Ster, IVIg	To spare IVIg, Ster	47-60	5-0	1	2
	8 (M, 50)	RR	7	Waldenstrom Macroglobulinemia	Ster, IVIg	To spare IVIg	47-51 †	5-3	1	2
	9 (F, 30)	RR	5	-	Ster, IVIg, AZA	Refractory CIDP	52-56	3-1	1	1
	10 (M, 65)	RR	4	Ig M MGUS	Ster, IVIg, PE	Refractory CIDP	48-52	6-4	2	1
	11 (F, 59)	CP	6	Ig M MGUS	Ster, IVIg, AZA	Refractory CIDP	40-50	8-4	4	1
	12 (M, 53)	RR	18	-	Ster, IVIg, AZA, Cyclopho	To spare IVIg	50-50	5-5	-	1
	13 (M, 45)	CP	19	-	Ster, IVIg, AZA, MM, Cyclosp	To spare IVIg, Ster	51-51	4-4	-	1

Table 2 :

 2 Results of published studies on rituximab therapy in patients with CIDP

	Author	N° of	Neuropathy	Pre-rituximab	Comorbidity	Clinical	Months	Duration of
		patients	duration	therapy		response	before	improvement
			(months)				improvement	(years)^
	Gorson et al.		60	IVIg, AZA, MM,
		2			No	No	-	-
	2007		(mean)	Ster, PE, Cyclopho
	Kilidireas et al.				Gastric
		1	10	No		Yes	2	> 5
	2006				lymphoma
	Knecht et al.			Ster, PE, AZA,	Evans
		1	17			Yes	6	1.5
	2004			Cyclopho	Syndrome
	Kasamon et al.				Non-Hodgkin
		1	< 12	Ster		Yes	2	n.a.
	2002				lymphoma
	Műnch et al.
		1	20	IVIg	DM	Yes	1	1
	2007
	IVIg: intravenous immunoglobulin; AZA: azathioprine; Cyclopho: Cyclophosphamide; Ster: steroids; PE: plasma exchange; MM: Mycophenolate
	mofetil; DM diabetes mellitus; n.a.: not available; ^ which corresponded to duration of follow-up


Michel Planat

Patrick Solé

Sami Omar

Alf Van Der Poorten

RIEMANN HYPOTHESIS AND QUANTUM MECHANICS

Keywords: Bost-Connes model, Riemann Hypothesis, Primorial numbers, Quantum Information. MSC codes: 46L05, 11M26, 11A25, 81P68

published or not. The documents may come

Introduction

The Riemann Hypothesis (RH), that describes the non trivial zeroes of Riemann ζ function, is a Holy Grail of Mathematics [START_REF] Borwein | The Riemann hypothesis: a resource for the afficionado and virtuoso alike[END_REF][START_REF] Lachaud | L'hypothèse de Riemann : le Graal des mathématiciens[END_REF]. Many formulations of RH may be found in the literature [START_REF] Conrey | The Riemann hypothesis Notices[END_REF]. In this paper, we mainly refer to the Nicolas inequality [START_REF] Nicolas | Petites valeurs de la fonction d[END_REF] (1)

N k ϕ(N k ) > e γ log log N k ,
where N k = k i=1 p i is the primorial of order k, ϕ is the Euler totient function and γ ≈ 0.577 is the Euler-Mascheroni constant.

The inequality [START_REF] Borwein | The Riemann hypothesis: a resource for the afficionado and virtuoso alike[END_REF] holds for all k ≥ 1 if RH is true [START_REF] Nicolas | Petites valeurs de la fonction d[END_REF]Th. 2 (a)]. Conversely, if RH is false, the inequality holds for infinitely many k, and is violated for infinitely many k [14, Th. 2 (b)].

The Bost and Connes (BC) system is a quantum dynamical model whose partition function is the Riemann zeta function ζ(β) (with β = 1 kT the inverse temperature) and that exhibits a spontaneous symmetry breaking at the pole β = 1. The behaviour of the system reminds that of a ferromagnet. At a temperature larger than the critical temperature T c ≈ 10 3 K, the disorder of the spins dominates and the thermal equilibrium state is unique while for T < T C the individual spins tend to align to each other, which in the three-dimensional space R 3 of the ferromagnet yields a set of equilibrium thermal states parametrized by the symmetry group SO(3) of rotations in R 3 . In Bost and Connes approach, the Hamiltonian operator is given by the logarithm of the number operator N as H 0 = log N. This contrasts with the ordinary Hamiltonian H 0 = N + 1 2 for the harmonic oscillator. It may be that the logarithmic Hamiltonian is realized in the field of quantum physiology, where the perception states are proportional to the logarithm of the excitation instead of being proportional to it.

For a first account of this process, see [START_REF] Planat | On the cyclotomic quantum algebra of time perception[END_REF][START_REF] Planat | Riemann and Galois: phase-locking[END_REF]. See also [START_REF] Mack | Riemann zeta function from wave-packet dynamics[END_REF] for a model of a wave-packet dynamics based on a logarithmic energy spectrum.

The purpose of the paper is two-fold. First, we rewrite the abstract mathematical language of BC theory with, as much as possible, concepts of quantum information processing (QIP).

Second, we uncover a relationship between the statistical phase states (KMS states) of BC theory and Riemann hypothesis (RH). The Riemann hypothesis is shown to be equivalent to an inequality satisfied by the KMS states of the BC theory at a selected low temperature β > 2. A special case (at [START_REF] Solé | Extreme values of the Dedekind ψ function[END_REF]. It plays a leading role for counting the number of maximal commuting sets of qudits [START_REF] Planat | Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function?[END_REF]. Higher temperatures KMS states (1.5 < β < 2) require RH to be analyzed with conventional tools such as the prime counting function π(x). Very high temperatures KMS states (0 < β < 1.5) are left as an open problem.

β = 3) of this inequality is ψ(N k )/(N k log log N k ) > e γ /ζ(2) for k > 3, where ψ(n) = ψ 2 (n) is the Dedekind ψ function

On modular exponentiation: classical and quantum

Let us start the parallel between the BC theory and QIP with the problem of finding the multiplicative order r in the modular equation ( 2)

a x = 1 (mod q).
Given the integer a and a positive integer q with (a, q) = 1, the order r = ord q (a) is the smallest positive integer x satisfying [START_REF] Bost | Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory[END_REF]. It is straigthforward to establish the set of inequalities (3) ord q (a) ≤ λ(q) ≤ ϕ(q) ≤ q -1, where ϕ(q) is the Euler totient function [it counts the number of irreducible fractions p q with p < q] and λ(q) is the Carmichael lambda function [it is the smallest positive integer x satisfying (2) for every integer a < q with (a, q) = 1] 1 . Both ϕ(q) and λ(q) are universal exponents that satisfy (2) for any integer a coprime to q.

Order-finding is a task that can in principle be performed more efficiently on a quantum computer than on a classical computer. It is a building block of the Shor algorithm for factoring the integers [START_REF] Nielsen | Quantum computation and quantum information[END_REF].

Let q be an m-bit integer and a be an integer smaller than q with (a, q) = 1. The quantum algorithm for order-finding is based on the following unitary operator acting on m-bits and such that for any n < q

(4) |n → U a |n = |an mod q .
Since a is coprime to q, it has an inverse modulo q and U a U ⋆ a = 1 (U a is unitary). Operator U a is also multiplicative:

U a U b = U b U a , ∀a, b < q.
The eigenvalues of the operator (4) are in the form exp(2iπ k r ) and the corresponding eigenvectors are given as a quantum Fourier transform ( 5)

|u k = 1 r 1/2 r-1 j=0 exp( -2iπkj r ) a j mod q .
The efficient implementation of the quantum Fourier transform on a quantum computer allows the efficient estimation of the period r of the function f (x) = a x mod q in (2), and subsequently the efficient factorization of an integer (in polynomial time instead of the exponential time required on a classical computer).

1 The recursive calculation of λ(n) is as follows. For a positive integer

k ≥ 3, λ(2 k ) = 2 k-2 . For p an odd prime ≥ 3 or 0 < k ≤ 2, λ(p k ) = ϕ(p k ) = (p -1)p k-1 . For q = n i=1 p ki i a product of primes, λ(q) = lcm[λ(p ki i )] i
, where lcm denotes the least common multiple of the numbers in the bracket.

On the KMS-states of Bost-Connes model

Bost and Connes theory is a skilfull balance of concepts related to QIP and to quantum statistical physics [START_REF] Bost | Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory[END_REF]. The Bost-Connes C ⋆ algebra is the cross product [START_REF] Laca | Semigroup of *-endomorphisms, Dirichlet series and phase transitions[END_REF] (6)

C Q = C ⋆ (Q/Z) ⋊ N ⋆
of the C ⋆ -algebra C ⋆ (Q/Z) of additive/phase operators e δ acting on qudits |n as [START_REF] Hardy | An introduction to the theory of numbers[END_REF] |n → e δ |n = exp(2iπ

p q ) |n , δ = p q ∈ Q/Z, by the semigroup N ⋆ of multiplicative operators µ a (8) |n → µ a |n = |an mod q , a ∈ N ⋆ .
The operator µ a is closely related to U a but acts on qudits |n .

The BC system is the pair (C Q , σ t ) with entries the algebra C Q and a quantum statistical flow σ t .

In standard quantum statistical mechanics, given an observable Hermitian operator M and a Hamiltonian H 0 , one has the evolution σ t (M) versus time t as For the more general case of an algebra of observables A, the Gibbs state is replaced by the so-called Kubo-Martin-Schwinger (or KMS β state). One introduces a 1-parameter group σ t of automorphisms of A so that for any t ∈ R and x ∈ A

(11) σ t (x) = exp(itH 0 )x exp(-itH 0 ).
For the algebra A, the equilibrium state remains unique only if some conditions regarding the evolution of the operator x ∈ A are satisfied: the KMS states φ β on A have to fulfill the KMS β condition with respect to σ t , namely that for any x, y ∈ A, there exists a bounded

holomorphic function F x,y (z), 0 ≤ ℑz ≤ β such that (12) F x,y (t) = φ β (xσ t (y)) and F x,y (t + iβ) = φ β (σ t (y)x), ∀t ∈ R.
The KMS flow of BC system. The Hamiltonian on the BC system is defined from its action on the qudits |n by Applying [START_REF] Planat | On the cyclotomic quantum algebra of time perception[END_REF] to the Hamiltonian (13) one obtains [START_REF] Koch | Sur la distribution des nombres premiers[END_REF] σ t (µ a ) = a it µ a and σ t (e δ ) = e δ for any a ∈ N ⋆ , δ ∈ Q/Z and t ∈ R.

It can be shown [START_REF] Bost | Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory[END_REF][START_REF] Laca | Semigroup of *-endomorphisms, Dirichlet series and phase transitions[END_REF] that in the high temperature regime 0 < β ≤ 1 there is a unique KMS β state of the quantum statistical system (C Q , σ t ). It corresponds to the partition function [START_REF] Nicolas | Petites valeurs de la fonction d[END_REF]. In the low temperature regime β > 1, the KMS β is no longer unique. The symmetry group G of the quantum statistical system is a subgroup of the automorphisms of (C Q , σ t ) which commutes with σ t , i.e. ( 16)

w • σ t = σ t • w for any w ∈ G and t ∈ R,
where • is the composition law. The symmetry group G permutes a family of extremal KMS β states generating the possible states of the system after phase transition. The symmetry group of the algebra (µ a , e δ ) endowed with the Hamiltonian H 0 in (13) is the Galois group

G = Gal(Q cycl /Q) of the cyclotomic extension Q cycl of Q. The field Q cycl is generated in
Q by all the roots of unity. The Galois group G is aut(Q cycl ). It is isomorphic to Z ⋆ q . At any temperature β < ∞, the restriction of the KMS β state to C ⋆ (Q/Z) is given by [START_REF] Mack | Riemann zeta function from wave-packet dynamics[END_REF] φ β (q) = q -β p∈P,p|q

1 -p β-1 1 -p -1 and for β = ∞ it is (18) φ ∞ (q) = µ(q)/ϕ(q),
where µ is the Möbius function.

The KMS state (17) may be easily rewritten as φ β (q) = N β-1 q-1 (q)ψ β-1 (q), where N q = q k=1 p k is a primorial number and ψ b (q) is defined in the abstract.

The Bost-Connes KMS states and Riemann hypothesis

In section(4.1) one formulates Riemann hypothesis as a property of the KMS states [START_REF] Mack | Riemann zeta function from wave-packet dynamics[END_REF] with the inequality

(19) ǫ β (q) = N q |φ β (N q )| log log N q - e γ ζ(β -1) > 0 when β > 2.
The special case β = 3 corresponds to our recent paper [START_REF] Solé | Extreme values of the Dedekind ψ function[END_REF]. In section (4. to obtain

ψ b (N n ) ∼ e γ ζ(b) log(p n ),
Now the Prime Number Theorem [7, Th. 6, Th. 420] shows that x ∼ θ(x) for x large where θ(x) stands for Chebyshev's first summatory function:

θ(x) = p≤x log p.
This implies that, upon taking x = p n we have p n ∼ θ(p n ) = log(N n ). The result follows.

Proposition 2. Pick a real number b > 1. For n ≥ 2 we have

n 2 > ϕ(n)ψ b (n) ≥ n 2 ζ(b) .
Proof: The first inequality follows at once upon writing

ϕ(n)ψ b (n) n 2 = p|n (1 -1/p b ),
a product of finitely many terms < 1. Notice for the second inequality that

ϕ(n)ψ b (n) n 2 = p|n (1 -1/p b ) ≥ p (1 -1/p b ),
an infinite product that is the inverse of the Eulerian product for ζ(b).

The next result is a far reaching consequence of Nicolas study of the small values of Euler ϕ function.

Theorem 1. Under RH the ratio R b (N n ) is > e γ ζ(b) for n ≥ 3.
If RH is false, this is still true for infinitely many n.

Proof:

Follows by Proposition 2, combined with [14, Theorem 2].

In view of this result and of numerical experiments the natural conjecture is

Conjecture 1. Fix a real number b > 1. For all n ≥ 3 we have R b (N n ) > e γ
ζ(b) . The main result of this note is the following. Theorem 2. Conjecture 1 is equivalent to RH.

Proof: If RH is true we refer to the first statement of Theorem 1. If RH is false we consider the function

g(x) := e γ ζ(b) log θ(x) p≤x (1 -1/p) p≤x (1 -1/p b ) . Observing that log θ(p n ) = log N n , we see that g(p n ) < 1 is equivalent to R b (N n ) > e γ ζ(b)
. We need to check that for x large enough g(x) can be > 1 or equivalently log g(x) > 0. Using [6, Lemma 6.4], we obtain, upon writing

-log ζ(b) = p≤x log(1 -1/p b ) + p>x log(1 -1/p b ),
the bound log g(x) ≥ log f (x) -b/x, where f is the function introduced in [14, Theorem 3], that is

f (x) := e γ log θ(x) p≤x (1 -1/p).
We know by [START_REF] Nicolas | Petites valeurs de la fonction d[END_REF]Theorem 3 (c)] that, if RH is false, there is a constant 0 < c < 1 such that lim sup x -c f (x) > 0.

Since for x large b/x << x -c , the result follows. Recall the Prime Number Theorem with rest from 1896 [START_REF] Hardy | An introduction to the theory of numbers[END_REF],

π(x) = Li(x) + O(x exp(-c log x)),
for some constant c > 0, as well as the Prime Number Theorem with rest under RH ( [START_REF] Koch | Sur la distribution des nombres premiers[END_REF]).

π(x) = Li(x) + O( √ x log x).
Plugging these estimates into the definition of J b we obtain Lemma 2. Let 0 < b < 1. We have, unconditionally,

J b (x) = x 2 Li(t)dt t 1+b + O( x 2 exp(-c √ log t)dt t b
).

and, under RH

J b (x) = x 2 Li(t)dt t 1+b + O( x 2
log tdt t b+0.5 ).

The preceding result shows that the assumption b > 1/2 is necessary to have asymptotic results to the precision O(1), since we need the integral in the error term to converge. where

I b (x) = x 2
Li(t)dt t 1+b .

Proof: Combine Lemma 1, Koch's paper [START_REF] Koch | Sur la distribution des nombres premiers[END_REF] and Lemma 2. Theorem 4. Under RH and for 0.5 < b < 1, there is constant K b > 0, such that, for large n, we have

ψ b (N n ) ∼ K b log p n exp(-B b (p n )).
Proof: Note that the definition of A b (x) entails

A b (x) = C b + o(1) - p≤x 1/p b .
Combine Theorem 1 with the above and with Mertens formula

p≤x (1 -1/p) -1 ∼ e γ log x.
The result follows upon putting x = p n .

We now give a lower bound on ψ b (N n ).

Theorem 5. Under RH and for 0.5 < b < 1, for every ǫ > 0, there is an N such that n ≥ N yields

ψ b (n) > K b e γ (1 -ǫ)N n log 2 2 N n exp(-B b (p n )). Proof: First, note that ϕ(N n ) N n ψ b (N n ) = p≤pn (1 -1/p b ),
a divergent product bounded below by

K b (1 -ǫ) log 2 N n exp(-B b (p n )),
for n large enough, by Theorem 4.

Under RH Nicolas inequality

N n ϕ(N n ) > e γ N n log 2 N n ,
holds for all n. The result follows.

Conclusion and open problems

In the present paper, we established the new criterion (19) (theorem 2) for Riemann hypothesis from the low temperature KMS states of the Bost-Connes model. Our vision is related to but distinct from Connes vision of the zeroes of ζ as an absorption spectrum [START_REF] Connes | Trace formula in noncommutative geometry and the zeros of the Riemann zeta function[END_REF]. In our approach the zeroes lurk in the backstage, and partial Eulerian products appear on the forefront. While the low temperatures KMS states (β > 2) lend themselves to an elegant analysis leading to infinitely many equivalence criterion with RH, high temperatures

( 9 )

 9 σ t (M) = exp(itH 0 )M exp(-itH 0 ), and the expectation value of M at the inverse temperature β = 1 kT is the unique Gibbs state (10) Gibbs(M) = Trace(M exp(-βH 0 ))/Trace(exp(-βH 0 ).

( 13 )

 13 H 0 |n = ln n |n . Using the relations exp(-βH 0 ) |n = exp(-β ln n) |n = n -β |n , it follows that the partition function of the model at the inverse temperature β is (14) Trace(exp(-βH 0 )) = ∞ n=1 n -β = ζ(β), ℜ(β) > 1.

4. 1 .Proposition 1 .

 11 Low temperature KMS states. To simplify the notation, we define the ratio R b (n) := ψ b (n) n log log n . The motivation for this strange ratio will become apparent in the following application of Mertens formula [7, Th. 429]. Pick a real number b > 1. For n going to ∞, we have lim R b (N n ) = e γ ζ(b) . Proof: By the definition of ψ b (n) we can combine the Eulerian product for ζ(b) with the Mertens formula p≤x (1 -1/p) -1 ∼ e γ log(x).

4. 2 . 1 . 1 . 2 π

 2112 High temperature KMS states. To investigate high temperature KMS states at large dimensions n, we will need the following lemmas. Recall π(x) the prime counting functionπ(x) = p≤x Lemma Let 0 < b < 1. We have p≤x 1 p b = π(x) x b + bJ b (x),where J b (x) =x (t)dt t 1+b . Proof: Write the above sum as a Stieltjes integral, and integrate by parts.

Lemma 3 .

 3 Let 0.5 < b < 1. We have, under RH p≤x 1 p b = Li(x) x b + bI b (x) + O(1),

Theorem 3 .

 3 Let 0.5 < b < 1. We have, under RH, Use lemma 3 and integrate I b by parts. If 0.5 < b < 1, denote by C b the sum of the convergent series C b = p (log(1 -1/p b ) + 1/p b ). Denote by A b (x) the partial sum A b (x) = p≤x log(1 -1/p b ). Denote by B b (x) the Bertrand integral B b (x) = x 2 dt t b log t .

( 2 5 .

 25 > β > 3/2) forced us to assume RH to establish the asymptotic formulas in theorems 4 and Future work should focus on the physical understanding of the lower bound on KMS states (equivalent to RH) we established. Another direction of inquiry is the study of such a lower bound attached to a quantum statistical system whose partition function is the Dedekind zeta function[START_REF] Cohen | Quantum statistical mechanics and number theory[END_REF]. As a finer point, we mention very high temperatures KMS states ( β < 3/2), which are challenging to analyze even under RH.

Table 1 .

 1 N q 6.4 × 10 10 4.7 × 10 219 6.7 × 10 3392 9.1 × 10 45336 A check of KMS inequality (19).

	2), new conditional

Acknowledgement: The second and third author are grateful to the Middle East Center of Algebra and its Applications (MECAA) at King Abdulaziz University for its warm and fruitful hospitality.

Institut FEMTO-ST, CNRS, 32 Avenue de l'Observatoire, F-25044 Besanc ¸on, France E-mail address: michel.planat@femto-st.fr Telecom ParisTech, 46 rue Barrault, 75634 Paris Cedex 13, France. E-mail address: sole@enst.fr King Khalid University, Faculty of Sciences, Dept of Math, 9004 Abha, Saudi Arabia also: Faculty of Sciences of Tunis, Dept of Math, 2092 El Manar. Tunis. Tunisia E-mail address: Sami.Omar@fst.rnu.tn


Franck Lefèvre

François Eric Hébrard

Chemistry of Methane and Related Hydrocarbons in the atmosphere of Mars

CHEMISTRY OF METHANE AND RELATED HYDROCARBONS IN THE ATMOSPHERE OF MARS.

F. Lefèvre, LATMOS, Paris, France (franck.lefevre@latmos.ipsl.fr), E. Hébrard, Laboratoire d'Astrophysique de Bordeaux, Bordeaux, France, F. Forget, Laboratoire de Météorologie Dynamique, Paris, France.

Objectives:

The possible detection of methane on Mars [1,2,3,4] has revived the possibility of past or extant life on this planet, despite the fact that an abiogenic origin is thought to be equally plausible. An intriguing aspect of the recent observations of methane on Mars is that methane concentrations appear to be locally enhanced and change with the seasons [3,4,5]. This contradicts the logic that a gas of lifetime much longer (~330 years) than the time required for global mixing should have a constant and spatially uniform distribution [6]. We will review in this paper our current understanding of the methane photochemistry. Assuming a methane source exists at the surface of Mars, we will present its expected distribution in the atmosphere, and investigate the possible processes that could lead to methane variability. These mechanisms could for instance involve atmospheric chlorine, interactions of methane with the surface, or electrochemical processes. Finally, we will examine the chemistry of the hydrocarbons produced by the degradation chain of methane, such as formaldehyde (CH 2 O), methanol (CH 3 OH), or ethane (C 2 H 6 ). These species are often cited as measurable markers of a possible Martian hydrocarbon chemistry, and are among the key targets of the next Exomars Trace Gas Orbiter mission. Using three-dimensional simulations, we will determine what quantities of these products can be expected from the amounts of methane currently detected on the planet.

Method:

We have employed an updated version of the LMD Global Climate Model with interactive photochemistry, extensively described in [7]. Hydrocarbon and chlorine chemistries have recently been implemented in the model, as well as other various improvements that will be detailed in the presentation. This new version of the GCM provides a comprehensive description of the chemistry of CO 2 , CO, oxygen, hydrogen, hydrocarbons, and chlorine by means of 31 species and 107 chemical or photolytic reactions.

References: [1] Krasnopolsky et al., Icarus 172, 537 (2004). [2] Formisano et al., Science 306, 1758(2004).

  [3] Mumma et al., Science 323, 1041 (2009). [4] Fonti et al., Astron. Astrophys 512, A51 (2010). [5] Geminale et al., Planet. Space Sci., in press. [6] Lefèvre and Forget, Nature 460, 720 (2009). [7] Lefèvre et al., J. Geophys. Res. 109, (2004).

Figure 1 .

 1 Figure 1. Zonally-averaged distributions of methane (top), formaldehyde (middle), and methanol (bottom) calculated by the GCM for L s = 0-5°. The model includes a local source of methane at the surface which maintains an equilibrium value of 10 ppbv.