Wolfgang Merkel

Holger Mack

Wolfgang P Schleich

Eric Lutz

Gerhard G Paulus

Bertrand Girard

Chirping a two-photon transition in a multi-state ladder

Keywords: numbers: 32.80. †, 42.50.Md, 02.10.De

We consider a two-photon transition in a specific ladder system driven by a chirped laser pulse. In the weak field limit, we find that the excited state probability amplitude arises due to interference of multiple quantum paths which are weighted by quadratic phase factors. The excited state population has the form of a Gauss sum which plays a prominent role in number theory.

I. INTRODUCTION

Fresnel diffraction at a straight edge leads to an intensity pattern determined by the Cornu spiral [START_REF] Born | Principles of Optics[END_REF][START_REF]Here we concentrate on scalar waves. For the more complicated case of electromagnetic radiation corresponding to vector waves, see[END_REF]. The latter results from an integral with a quadratic phase. The Talbot effect [3] is a generalization of this scattering situation. The discreteness of the grating translates the integral of quadratic phase factors into a sum. Recently, the combination of chirped pulses together with an appropriate atomic level scheme created an atomic analog of the Fresnel diffraction of the straight edge [4,5]. In the present paper we propose the atomic analog of the Talbot effect by chirping a two-photon transition in a multi-state ladder.

Chirped pulses are characterized by a non-linear phase dependence and have a vast variety of applications [4,6,8,9,10,11]. Examples include, the realization of a timedomain Fresnel lens with coherent control [12] and quantum state measurement using coherent transients [13]. Of particular relevance in the context of the present paper are interference phenomena in quantum ladder systems induced by two-photon excitation with chirped laser pulses [14,15].

The central idea is summarized in Fig. 1. We consider a two-photon transition from the ground state |g to the excited state |e . In the case of a single intermediate state [16,17] discussed in the top row of this figure the total excitation probability W e ≡ |c e | 2 is due to the interference of two paths: (i) in the direct path two photons of energy ω 0 are absorbed instantaneously and (ii) in the sequential excitation path the total energy of 2 ω 0 * Electronic address: wolfgang.schleich@uni-ulm.de is absorbed in two steps: in the first step the system is excited to the intermediate state |m , whereas the second, delayed step provides the residual energy to reach the excited state |e . This interference effect gives rise to an excitation probability determined by a Fresnel integral. It is experimentally accessible by measuring the population in the excited state through the detection of the fluorescence signal. Population transfer with chirped laser pulses has been experimentally demonstrated in the three-state ladder of Rubidium [6].

In the case of a multi-state ladder with a manifold of intermediate states shown in the second row of Fig. 1 we find a generalization of this two-path interference to a multi-path interference. Here the interference is mainly due to the indirect paths through the individual intermediate states. The direct path does not play a dominant role anymore. The problem of scattering a chirped pulse from D intermediate levels is very much in the spirit of diffraction from a D-slit grating. For an equidistant manifold the phase of the excitation path through the m-th level contributing to the total excitation probability depends quadratically on m in complete analogy to the Talbot effect. We emphasize that the present proposal is different from the temporal Talbot effect suggested in Ref. [18].

Our paper is organized as follows: In Sec. II, we investigate the population transfer in the D-state ladder system shown in Fig. 1 driven by a weak chirped laser pulse. We use second order perturbation theory to derive an analytic expression for the probability amplitude to be in the excited state. This result is of the form of a Gauss sum [START_REF] Maier | Prime Numbers 101: A Primer on Number Theory[END_REF][START_REF] Davenport | Multiplicative Number Theory[END_REF] which plays a prominent role in number theory. We dedicate the following sections to a discussion of the physical origin of this Gauss sum.

We start in Sec. III by first considering the transition probability amplitude c e in the limit of a single intermediate state. In this case c e follows from the complementary error function with a complex-valued argument which depends on the chirp parameter. With the help of appropriate asymptotic expansions of the complementary error function we bring to light the oscillations in the transition probability W e = |c e | 2 shown in Fig. 1 for negative values of the chirp. These oscillations arise due to the interference of two excitation paths as shown in Sec. IV. Moreover, a primitive version of the method of stationary phase allows us to identify the times of the transitions. We show that the accumulated phases depend quadratically on the offset δ m . Since our analysis works in frequency-time phase space our approach is different from Refs. [14,15,[START_REF] Dudovich | [END_REF] which relies on the frequency domain. In the case of many intermediate levels the interference of quadratic phase factors gives rise to a Gauss sum which we express in its canonical form in Sec. V. Here all dimensionless quantities are related to experimental parameters. In Sec. VI we conclude by addressing possible experimental realizations of multi-state systems.

II. EXCITATION PROBABILITY

We consider the multi-state ladder system shown in the bottom of Fig. 1 driven by a weak chirped laser pulse and calculate the probability amplitude of the excited state in second order perturbation theory. We show that this probability amplitude is the interference of many amplitudes which depend quadratically on the offset of the intermediate state. We present a simple phase space argument for the origin of such quadratic phase factors.

A. Model system

The ladder system consists of a ground state |g , an excited state |e separated by an energy 2 ω 0 and intermediate states |m with quantum numbers M ′ ≤ m ≤ M , as depicted in Fig. 1. These intermediate states are displaced by the offset

δ m = δ + m ∆. ( 1
)
with respect to the central frequency ω 0 . The harmonic manifold is characterized by the offset δ of a specific central state |0 and separation ∆ of adjacent states.

In the interaction picture the Hamiltonian describing the interaction between the electric field of the chirped pulse

E(t) = E 0 e -iω0t f (t) + c.c. ( 2
)
of amplitude E 0 , carrier frequency ω 0 and pulse shape function f (t) and our ladder system reads

V (t) = - m Ω mg e i δmt f (t) |m g|+ Ω em e -i δmt f (t) |e m| + c.c. ( 3
)
where we have introduced the Rabi frequencies Ω ij ≡ E 0 ℘ ij / with the dipole matrix element ℘ jm of the transition |j → |m and |j = |g, e .

B. Time evolution

Throughout the paper we consider a situation in which the laser pulse is short (femtosecond-time scale) compared to the characteristic time scale of radiative decay processes. As a consequence we can neglect spontaneous emission and the time-evolution of the state vector

|ψ(t) = k c k (t) |k (4)
is given by the Schrödinger equation

i d dt |ψ(t) = V (t)|ψ(t) . (5)
Since the laser field is assumed to be weak, the probability amplitude c e of the excited state can be simply calculated in second order perturbation theory. We find

c e (t) = - 1 2 2 t -∞ dt ′ t ′ -∞ dt ′′ e|V (t ′ ) V (t ′′ )|g (6)
The first-order term vanishes since we assume that at t 0 = -∞, that is before the interaction, the ladder system is initially prepared in the ground state |g and a two-photon transition is necessary in order to reach the excited level.

When we substitute the explicit expressions for the interaction Hamiltonian, Eq. ( 3), and concentrate on times after the pulse has passed, that is t → ∞, we obtain

c e = - 1 2 m Ω em Ω mg I m (7)
for the probability amplitude c e = c e (t → ∞).

The domain of the integral

I m ≡ ∞ -∞ dt ′ t ′ -∞ dt ′′ e -i δm(t ′ -t ′′ ) f (t ′ )f (t ′′ ) (8)
extends over the area below the diagonal crossing the (t ′ , t ′′ )-plane from the lower left to the upper right. When we introduce the new integration variables t ≡ t ′ + t ′′ and t ≡ t ′ -t ′′ and use the formula

∞ -∞ dt ′ t ′ -∞ dt ′′ • • • = 1 2 ∞ -∞ d t ∞ 0 dt . . . (9)
we eventually arrive at

c e = - 1 4 m Ω em Ω mg × ∞ -∞ d t ∞ 0 dt e -i δmt f t + t 2 f t -t 2 . ( 10
)
This expression is valid for an arbitrary pulse shape f . In the next section we restrict ourselves to a linearly chirped pulse with a Gaussian envelope.

C. Description of the chirped pulse

We now specify the pulse shape

f (t) = f 0 exp - 1 2 (∆ωf 0 ) 2 t 2 (11)
with the complex-valued amplitude

f 0 = 1 + ia 1 + a 2 , (12)
and the dimensionless parameter a denotes the second order dispersion

a = ∆ω 2 φ ′′ . ( 13
)
Here ∆ω is the bandwidth of the pulse and φ ′′ = d 2 φ(ω)/dω 2 is a measure for the quadratic frequency dependence of the phase of the laser pulse.

It is useful to decompose the argument

1 2 (∆ωf 0 ) 2 ≡ α r + i α i (14)
of the Gaussian pulse shape given by Eq. ( 11) into the real and imaginary parts

α r ≡ ∆ω 2 2 1 1 + a 2 and α i ≡ ∆ω 2 2 a 1 + a 2 (15)
resulting in

f (t) = f 0 exp -α r t 2 exp -i α i t 2 . ( 16
)
This representation brings out most clearly that the electric field Eq. ( 2) of the chirped laser pulse features a linear variation of the instantaneous frequency

ω(t) = ω 0 + 2α i t. (17)

D. Excitation probability amplitude

When we now substitute the pulse shape, Eq. ( 11), into the expression Eq. ( 10) for the excitation probability amplitude we obtain

c e = - 1 4 m Ω em Ω mg I 1 I 2 (18)
with

I 1 ≡ f 0 ∞ -∞ d t e -1 4 (∆ωf0) 2 t2 = 2 √ π ∆ω ( 19
)
and

I 2 ≡ f 0 ∞ 0 dt e -iδmt e -1 4 (∆ωf0) 2 t 2 . ( 20
)
The substitution

z ≡ ∆ω 2 f 0 t + i δ m ∆ωf 0 (21)
casts the integral I 2 into the form

I 2 = √ π ∆ω exp - η m f 0 2 erfc i η m f 0 (22)
where we have introduced the dimensionless offset

η m ≡ δ m ∆ω . ( 23
)
and have recalled [START_REF] Abramowitz | Handbook of mathematical functions[END_REF] the definition

erfc(ζ) ≡ 2 √ π C(ζ) dz e -z 2 (24)
of the complementary error function.

The integration in Eq. ( 20) over t from 0 to ∞ translates into a path C(ζ m ) in the complex plane which according to Eq. ( 21) starts at the point ζ m ≡ i η m /f 0 and follows a straight line to infinity. This path encloses an angle arg(f 0 ) with respect to the positive real axis.

When we introduce the abbreviation

d m ≡ π Ω em Ω mg ∆ω 2 (25)
the definition Eq. ( 12) of f 0 yields the expression

c e = m w m exp i η 2 m a (26)
for the transition probability amplitude Eq. ( 18). Here we have introduced the weight factors

w m ≡ - d m 2 erfc (ζ m (a)) exp -η 2 m ( 27
)
with

ζ m (a) = η m i √ 1 -i a. (28)
Equation ( 26) is the central equation of the present paper. It demonstrates that the probability amplitude c e to be in the excited state is a sum of weighted phase factors in which the summation index m enters through the dimensionless offset η m . Since according to Eq. ( 1) the offset δ m is linear in m and the phase is quadratic in η m the phase factors are quadratic in m. Such sums are called Gauss sums [START_REF] Maier | Prime Numbers 101: A Primer on Number Theory[END_REF][START_REF] Davenport | Multiplicative Number Theory[END_REF].

III. SINGLE INTERMEDIATE STATE

In order to gain some insight into the dependence of the transition probability amplitude c e on the dimensionless chirp parameter a we first concentrate on a single intermediate state |m . In this case the sum in Eq. ( 26) reduces to a single term and the transition probability

W e = |c e | 2 = 1 4 d 2 m e -2η 2 m |erfc (ζ m (a))| 2 (29)
is solely determined by the weight factor w m . As a consequence the quadratic phase η 2 m a in Eq. ( 26) has no chance to contribute. Nevertheless, under appropriate conditions [6,7,14,15] we find interference due to the rather intricate properties of the complementary error function. In order to bring this subtle phenomenon to light we now analyze the asymptotic behavior of erfc.

ζ m ≡ η m (x + i y) (31)
and take the real and imaginary parts of

x 2 -y 2 + 2i xy = i a -1, (32)
we find the trajectory

y = 1 + x 2 (33)
and the translation

a = 2xy = 2x 1 + x 2 (34)
between x and a.

For large positive or large negative values of x the trajectory, Eq. ( 33), of ζ m approaches the diagonals of the first or second quadrant, respectively. The straight path C of integration defined in Eq. ( 21) has a steepness determined by the coefficient (35) in front of t. Hence, for large positive values of a the straight path encloses an angle with respect to the positive real axis which is slightly smaller than π/4. For

1 2 ∆ωf 0 = ∆ω 2 √ 1 -i a = ∆ω 2 (1 + a 2 ) -1/4 e i 2 arctan(a)

B. Asymptotic expressions

The top of Fig. 1 and the bottom left of Fig. 2 demonstrate that the value a = 0 of the chirp parameter separates two distinct domains of the transition probability: (i) For positive values of a W e decays as a increases, and (ii) for negative values of a the probability oscillates. These qualitatively different behaviors of W e originate from the different slopes of the paths of integration as we now show.

Decaying regime

We start our discussion by analyzing the behavior of erfc(ζ m (a)) for large positive values of a. For this purpose we integrate the integral in the definition Eq. ( 24) by parts which yields

erfc(ζ m ) = 1 √ π    - e -z 2 z C+ + C+ dz e -z 2 z 2    . (36)
Since the path C + shown in Fig. 3 reaches the domain where e -z 2 decays the boundary term originating from the end point of the path C + , that is from infinity, vanishes. As a consequence only the starting point of

C + contributes leading to erfc(ζ m ) = 1 √ π e -ζ 2 m ζ m + C+ dz e -z 2 z 2 . ( 37
)
We can now repeat this procedure of partial integration to obtain a power series of erfc in 1/ζ m . In lowest order we arrive at the asymptotic expression

erfc(ζ m ) ∼ = 1 √ π e -ζ 2 m ζ m (38)
valid for 1 ≪ a.

Oscillatory regime

We now turn to the limit of large negative values of a. It is tempting to apply again the technique of integration by parts. However, this idea is bound to fail since the partial integration creates inverse powers of z in the integrand. As shown in Fig. 2 for large negative values of a the path of integration gets very close to the origin of complex space. As a result the series cannot converge in this case. This heuristic argument clearly indicates that the behavior for negative values of a is substantially different from the one arising for 0 < a.

In order to evaluate erfc(ζ m (a)) for large negative values of a we first recall that e -z 2 is an analytic function. Therefore, the Cauchy theorem applied to the closed path C c depicted in Fig. 3

yields the identity

Cc dz e -z 2 = 0.

(

) 39
Since in infinity the integrand vanishes the contributions from the paths C ∞ and C -∞ are zero and we find immediately 3: Paths C+ and C-of integration in the complex plane corresponding to a positive and a negative value of the chirp parameter a . We depict the domains in the complex plane where the integrand e -z 2 of the complementary error function increases by shaded areas. For 0 < a the starting point ζm of the integration is located in the first quadrant and the straight path C+ is inclined with a steepness slightly smaller than the first diagonal. Thus C+ proceeds almost completely in the domain of the complex plane where the integrand exp(-z 2 ) of erfc decays. For a < 0 the path C-of integration starts in the second quadrant and proceeds slightly above the second diagonal into the decaying region of e -z 2 . In order to evaluate the resulting integral we close the path as indicated by the dashed line consisting of the paths C-, C∞, Cr, C-∞ and C+.

C- dz e -z 2 = √ π - C+ dz e -z 2 . ( 40
) C+ C- C+ C∞ C-∞ Cr ζm(a > 0) ζm(a < 0) z r z i FIG.
The path C+ is the mirror image of the path C + and avoids the origin. Therefore, we can use the technique of the preceding section to approximate the remaining integral. When we traverse C+ in the opposite direction we arrive at

erfc(ζ m ) ∼ = 2 + 1 √ π e -ζ 2 m ζ m . (41)
In this approximation the complementary error function is a sum of two complex-valued contributions: a constant term and a phase factor. As a result the interference between these two terms manifests itself in oscillations of the probability W e as a function of a.

IV. INTERFERENCE OF EXCITATION PATHS

The preceding section has shown that the two paths C+ and C r of integration in complex space correspond to two interfering contributions in the total probability amplitude c e . We now show that they represent two different paths of excitation in the atom. Indeed, the term e -ζ 2 m /ζ m results from a direct two-photon transition whereas the term 2 results from a sequential excitation.

A. Sequential excitation

So far our analysis was based on the integral I 2 . However, in order to identify the sequential character of the excitation process it is more convenient to use the representation Eq. ( 7) for the transition probability amplitude c e with the integral I m . The latter is the product of two integrals which reflects the sequential character of the excitation process: in the first step, the atom is promoted to the intermediate state |m , while the second step provides the residual energy to reach the excited state.

Times of excitation

In order to bring out this feature most clearly and to identify the times when the transitions occur we now substitute the expression Eq. ( 11) for the shape f (t) of the pulse into the definition Eq. ( 8) of I m which leads us to the integral

I m = f 2 0 e i δ 2 m 4α i ∞ -∞ dt ′ e -αrt ′2 e -i αi t ′ + δm 2α i 2 ×e i δ 2 m 4α i t ′ -∞ dt ′′ e -αr t ′′2 e -i αi t ′′ -δm 2α i 2 . (42)
So far the calculation is exact. We now recognize that for large values of the chirp parameter a the definition Eq. ( 15) of α r and α i provides us with the inequality

α r α i = 1 |a| ≪ 1. (43)
Consequently, the real-valued Gaussians in the integral Eq. ( 42) are slowly varying compared to the oscillatory terms resulting from the quadratic phases. The main contributions to the integral emerge from the oscillatory terms and in particular from the neighborhood of the times

t ′′ s ≡ δ m 2α i = η m ∆ω a 1 + 1 a 2 ∼ = η m ∆ω a (44)
and

t ′ s = - δ m 2α i = -t ′′ s ( 45
)
when the phase factors are slowly varying. Therefore, the transition from the level |g to the level |m appears at the time t ′′ = t ′′ s followed by the transition from |m to |e at the time t ′ = t ′ s . In order to ensure this time ordering enforced by the limits t ′′ < t ′ of the integral I m defined in Eq. ( 42) we need to have

t ′′ s < t ′ s = -t ′′ s . ( 46
)
This inequality is only satisfied provided t ′′ s is negative. From Eq. ( 44) we recognize that it is the product η m a which determines the sign of t ′′ s . Consequently we only find points of slowly varying phase for η m a < 0. In the case 0 < η m a no such points exist and the integral I m is decreasing rapidly.

Origin of quadratic phases

Moreover, this analysis of the integral I m brings out most clearly that each transition is associated with a phase which is quadratic in the dimensionless offset η m . The phases of both transitions are identical. The total acquired phase in the two-photon transition is twice of that of the individual one-photon transition.

We can now evaluate the integrals in an approximate way by replacing the variables t ′ and t ′′ in the real-valued Gaussians by t ′ s and t ′′ s and factoring them out of the integral which yields

I m ∼ = Θ(-η m a)f 2 0 ∞ -∞ du e -i αiu 2 2 × exp - α r α i δ 2 m 2α i exp i δ 2 m 2α i . ( 47
)
Here we have also extended the integration over t ′′ to +∞.

With the help of the integral relation [START_REF] Abramowitz | Handbook of mathematical functions[END_REF] ∞ -∞ du e -i γu 2 = π i γ (48) we find

I m ≈ Θ(-η m a)f 2 0 π i α i exp - α r α i δ 2 m 2α i exp i δ 2 m 2α i .
(49) When we now take the limit of 1 ≪ |a| in the definitions Eq. ( 12) and Eq. ( 15) of f 0 and α i which yields f 2 0 ∼ = i/a and α i = ∆ω 2 /(2a) we obtain together with α r /α i = 1/a the final approximate expression

I m ≈ 2π ∆ω 2 Θ(-η m a) exp -η 2 m exp i η 2 m a . ( 50
)
We substitute this formula into Eq. ( 7) for the probability amplitude c e and arrive at Eq. ( 26) with the approximate weight factor

w m = -d m Θ(-η m a) exp -η 2 m . (51)
This expression also results from the exact formula Eq. ( 27) with the help of the relation

√ 1 -ia ∼ = (-i) 1/2 a together with the primitive asymptotic expansion erfc(ζ m ) ≈ 2Θ(-η m a) ( 52
)
of the complementary error function following from Eqs. (38) and (41).

Interference in time-frequency phase space

This analysis of the transition probability amplitude in second order perturbation theory translates itself into an

t |m |g ω(t) E/h ω 0 2ω 0 |e 0 ω 0 -δ m ω 0 + δ m t ′′ s t ′ s A 1 A 2
FIG. 4: Origin of Gauss sum from phase space interpretation of two-photon excitation induced by a chirped pulse. In the phase space spanned by frequency ν and time t we indicate the ground state |g , the intermediate level |m and the excited state |e of frequencies 0, ω0 + δm and 2ω0, respectively, by thick horizontal lines. We also denote the onephoton resonance of energy ω0 and the virtual level of frequency ω0 -δm by dotted horizontal lines. The instantaneous frequency ω(t) = ω0 + 2αi t is depicted for one negative value of the chirp by the tilted thin line. A stationary phase analysis of the double integral Eq. ( 8) determining the two-photon transition probability amplitude brings out most clearly that the sequential excitation appears at the times t ′′ s ≡ δm/(2αi) and t ′ s . Indeed, at t ′′ s the instantaneous frequency is resonant with the transition from |g → |m whereas at t ′ s it is resonant with the transition from |m to |e as indicated by the dotted and the thick arrow. In this phase space representation the times when the transitions occur are determined by the crossings of the tilted thin line with the thick and the lower dotted horizontal lines corresponding to the frequencies ω0 + δm and ω0 -δm. The enclosed shaded area A1 + A2 corresponds to the phase acquired during the transition. Each triangle has the area δmt ′′ s /2 = δ 2 m /(4αi). As a consequence, the phase associated with this transition is quadratic in the offset δm.

In the case of a multi-state ladder with a harmonic manifold we have the interference of all excitation paths, that is the weighted sum of phase factors with quadratic phases giving rise to a Gauss sum. elementary geometrical representation in a phase space spanned by frequency ν and time t. In Fig. 4 we consider a single sequential excitation path in this space. We indicate the frequencies 0, ω 0 + δ m and 2ω 0 of the ground state, the m-th intermediate state and the excited state by thick horizontal lines. We also denote the frequency ω 0 of the two-photon resonance and the virtual level of frequency ω 0 -δ m by dotted horizontal lines. The instantaneous frequency ω(t) = ω 0 + α i t is depicted for one negative value of the chirp by the tilted thin line.

The transitions from |g to |m and from |m to |e occur at times t ′′ s and t ′ s , respectively. These times are determined geometrically by the crossings of the representations of the instantaneous frequency ω(t) and the two frequencies ω 0 + δ m and ω 0 -δ m , that is by the crossings of the tilted thin line with the thick and the lower dotted horizontal lines. Each crossing contributes to the total transition probability amplitude with a phase determined by the enclosed triangular shaded area A 1 + A 2 corresponds to the phase acquired during the transition. Each triangle has the area t ′′ s δ m /2 = δ 2 m /(2α i ). As a consequence, the phase associated with this transition is quadratic in the offset δ m .

B. Direct excitation

We conclude by briefly discussing the direct twophoton excitation. In the analysis of the time ordered integrals Eq. ( 42) we have implicitly assumed that the moments t ′ s and t ′′ s of the transitions are appropriately separated in time. However, since we deal with a double integral the instant t ′ = t ′′ can make a significant contribution. This fact stands out most clearly in the re-formulation, Eq. ( 9), of the double integral where the line t ′ = t ′′ translates into the condition t = 0. When we recall the definition Eq. ( 21) of the path C(ζ) of integration the moment t = 0 translates into the starting point ζ m of the path. As a consequence the main contribution to the integral I 2 arises from the lower limit. The partial integration that we have used in Sec. III B to derive the asymptotic expansion of the complementary error function took advantage of this fact.

V. MANIFOLD OF INTERMEDIATE STATES

So far we have concentrated on a single intermediate state. We now briefly address the problem of a manifold of intermediate states. In the case of a multi-state ladder with a harmonic manifold we have the interference of all excitation paths, that is the weighted sum of phase factors with quadratic phases giving rise to a Gauss sum. In the case of a positive chirp parameter a these paths consist solely of direct two-photon excitations. In the limit of negative values of a each interfering path through the m-th level consists of an interference of a sequential and a direct excitation. However, the direct excitation is less pronounced especially for appropriately negative values of the chirp. In this case the probability amplitude is predominantly determined by the interference of the quadratic phases arising from the sequential excitations.

A. Gauss sum made explicit

In order to bring out most clearly the connection to the Gauss sums, we make the dependence of the weight factors and the phase on m more explicit. For this purpose we use the definitions, Eq. ( 1) and Eq. ( 13) for the offset δ m of the harmonic manifold and the second order dispersion a to verify the quadratic nature of the phase

η 2 m a = δ m ∆ω 2 a = π 2 N ξ + 2π m + m 2 N ξ (53)
in the probability amplitude c e . Here we have introduced the quantity

N ≡ 2δ ∆ ( 54
)
and the dimensionless chirp

ξ ≡ δ∆ π a ∆ω 2 = δ∆ π φ ′′ . (55)
Moreover, in terms of N the offset δ m defined by Eq. ( 1) reads

δ m = ∆ m + δ ∆ = ∆ m + N 2 . ( 56
)
Hence, the final expression for the excited state probability amplitude takes the canonical form

c e = exp i π 2 N ξ M m=-M ′ w m exp 2πi m + m 2 N ξ
(57) of a Gauss sum. Here we have numbered the intermediate levels from -M ′ to M .

When we express δ m with the help of Eq. ( 56) the weight factors

w m ≡ - d m 2 erfc (ζ m ) exp - m + N/2 ∆m 2 (58)
as well as the argument

ζ m ≡ m + N/2 ∆m i √ 1 -ia (59)
of the complementary error function contain the width ∆m ≡ ∆ω/∆. The transition probability amplitude consists of a sum of weighted phase factors with phases that depend linearly and quadratically on the summation index m. Closely related sums appear in the context of fractional revivals [START_REF] Leichtle | [END_REF]24], the Talbot effect [25] and curlicues [26,27] and Josephson junctions [28].

B. Asymptotic expansion of weight factors

With the help of the asymptotic expansion Eq. ( 51) the approximate real-valued weight factors

w m ≈ -d m Θ -a m + N 2 exp - m + N/2 ∆m 2 (60)
are given by half a Gaussian centered around m ≈ -N/2. The sign of the chirp determines which half. Indeed, for negative chirps that is a < 0 we find only contributions for m > -N/2. In contrast for positive chirps, that is for 0 < a only levels with m < -N/2 contribute. Moreover, by controlling the width ∆m we can ensure that all quantum paths enter the sum with approximately the same weight. In Fig. 5 we compare and contrast the exact (black dots) and the approximate (gray dots) expressions, Eqs. ( 58) and (60), for the ratio |w m /d m | of the weight factors w m and the matrix element d m . To be specific, we have used N = 15 and a negative chirp. The figure clearly shows that in this case the weight factors are nonvanishing only on the right half of the Gaussian, that is for -N/2 = -7.5 < m. Moreover, there is an excellent agreement between the exact result and the approximation.

We conclude by noting that there are small oscillations in the exact weight factor distribution which are not contained in the approximate expression. They are a remnant of the interference between the direct and the sequential path as discussed in Sec. IV. Here we consider the case N = 15 with ∆m = ∆ω/∆ = 50.8 for the negative dimensionless chirp a = -10824 < 0.

VI. CONCLUSION AND OUTLOOK

In the present paper we have analyzed a two-photon transition in a ladder system driven by a chirped laser pulse. We have shown that the resulting excitation probability amplitude is of the form of a Gauss sum. However, one task remains: Find a quantum system which displays such an equidistant ladder system.

Before we address this task it is helpful to recall the origin of the Gauss sum -the interference of paths with quadratic phases. Fig. 4 brings out most clearly that these phases arise already from the first step of the excitation, that is from transition into the equidistant manifold. Therefore, a one-photon transition from a ground state to an equidistant manifold of excited states driven by a chirped pulse also leads to a Gauss sum. In this simplified system the fluorescence signal is acquired from the states in the harmonic manifold. Interference from the individual levels occurs during spontaneous emission of the states.

At least five different quantum systems offer themselves as candidates for such one-photon excitations: (i) Rydberg atoms in electric or magnetic fields, (ii) vibrational energy levels in a molecule, (iii) atoms in traps, (iv) laser driven one-photon transitions, and (v) quantum dots. However, these systems have to be subjected to additional requirements: (i) the Rabi-frequencies for the interfering excitation paths should ideally be of the same order of magnitude in order to guarantee that they contribute in a democratic way, that is with the same weight, (ii) since anharmonicities in the target manifold change the character of the resulting probability amplitude, only perfectly equidistant states result in a Gauss sum. We now briefly discuss these quantum systems and address these requirements Rydberg atoms [29] are highly sensitive to electric or magnetic fields. In an electric field the energy levels of a Rydberg atom split into a manifold of states [30]. The so-called Stark maps which show the dependence of the splitted spectral lines as a function of the electric field have been measured for many alkali atoms. Whereas for a modest field strength the separation between neighboring splitted energy levels is constant, for larger values of the field also higher order corrections to the energy have to be taken into account. A closer analysis of the Stark map reveals that the dipole moments vary for each member of the manifold and the individual paths do not contribute equally. Due to the Zeeman effect a homogeneous magnetic field also creates a manifold of equidistant states |m . However, in this case the problem are the selection rules which eliminate most of the transitions. A change of the orientation of the magnetic field might offer a possibility to overcome this problem.

Diatomic molecules may also serve as a candidate system for identifying a harmonic manifold of (almost) equidistant states [31]. The diagonalization of the rovibrational Hamiltonian yields that the inter-nuclear potential in a diatomic molecule can be indeed approximated as a harmonic oscillator with equidistant spectrum. However, anharmonicities have again to be taken into account and would destroy the regularity of the Gauss sum.

Another promising realization of an equidistant spectrum relies on cold trapped atoms or ions [32,33]. The central idea of this approach is that the trapping in magnetic microtraps relies on the magnetic dipole of the atom which depends on its internal state. Therefore, the center-of-mass motion of an atom depends on its internal state. Our suggestion for the realization of an equidistant spectrum is reminiscent of the quantum tweezer [34,35]. Whereas the atom initially in the ground state is weakly bound to a shallow trap, the atom in its excited state feels a steep harmonic potential. The vibrational states in the excited electronic state constitute the desired harmonic manifold.

Yet another method to obtain an equidistant ladder system is to apply two electromagnetic fields to a twolevel atom which has a permanent dipole moment in the excited state. A strong cw-modulation field generates an equidistant Floquet ladder in the excited state. The second field, that is the weak chirped pulse, leads to the excitation probability amplitude in the form of a Gauss sum.

Quantum dots [36] may be the perfect choice in the end. Indeed, they allow for designing the discrete energy spectrum [37] of a trapped electron and can thus be viewed as an artificial atom. Quantum dots have already been discussed as a candidate system for implementing quantum logic [38].

In conclusion, we have shown that a two-photon transition in a ladder system driven by a chirped laser pulse leads to an excitation probability amplitude in the form of a Gauss sum. We have discussed several physical systems in the context of realizing such an excitation scheme and have also addressed the experimental challenges. We emphasize that this example also establishes an interesting connection between number theory and wavepacket dynamics.

  A. Path of integration induced by chirpAccording to Eq. (29) the transition probability W e follows from the absolute value squared of the complementary error function evaluated at the argument ζ m (a). The function erfc(ζ m ) is an integral of the analytic function e -z 2 = e -(z 2 r -z 2 i ) e -i 2 zr zi (30) along the path C(ζ m ). For |z i | < |z r | the integrand exp(-z 2 ) decreases rapidly for increasing |z r |. The decaying domains for positive and negative real parts are connected via a saddle at the origin. Indeed, in the sector |z r | < |z i | we deal with a rapidly growing integrand as |z i | increases. As a function of the chirp parameter a the starting point ζ m (a) of the integration traverses a path in complex plane. When we square the representation

  FIG.2: Complementary error function in its dependence on the chirp parameter a (left bottom) as obtained by integrating along a straight path in complex space (top). The starting point ζm(a) of this path traverses as a function of a a trajectory in complex space which for large positive and large negative values of a approaches the diagonals of the first and the second quadrants, respectively. The path is always a straight line which is tangent to this trajectory and tends towards infinity as shown on the top. The gray curve below translates between a and Re ζm.

FIG. 5 :

 5 FIG.5: Comparison between the exact (black dots) and the approximate (gray dots) expression for the scaled weight factors |wm/dm| given by Eqs. (58) and (60), respectively. Here we consider the case N = 15 with ∆m = ∆ω/∆ = 50.8 for the negative dimensionless chirp a = -10824 < 0.

  6 fs 2 )

	| 2
	|c e
	FIG. 1: Chirping a two-photon excitation through a single
	intermediate state (top) and through an equidistant manifold
	(bottom). The left and right columns show the level system
	and the resulting excitation probabilities as a function of the
	chirp parameter φ ′′ , respectively. In both systems the ground
	state |g is connected by a two-photon transition to the ex-
	cited state |e . In the case of a single intermediate state |m
	with the offset δ > 0 with respect to the central frequency
	ω0 discussed in the top row the population |ce| 2 of the ex-
	cited state displays oscillations for negative chirps, whereas
	for positive values of the chirp it is decreasing rapidly. The
	oscillations result from an interference of the direct and the
	indirect excitation. The direct path is a direct two-photon
	transition through the virtual state of frequency ω0. The in-
	direct path consists of first an excitation to the level |m and
	then a transition to |e . Here we have used the parameters
	δ = 0.0225 fs -1 , ∆ω = 0.1525 fs -1 , ΩemΩmg = 1 as suggested
	by the experiment[14]. In the bottom row we discuss the
	case of a manifold of M ′ + M + 1 equidistant intermediate
	states |m with -M ′ ≤ m ≤ M which are shifted by the
	offset δm = δ + m ∆ with respect to the central frequency
	ω0. The offset of the central state is δ whereas the states in
	the harmonic manifold are separated by ∆. The population
	|ce| 2 of the excited state is due to interference of compet-
	ing quantum paths via the intermediate states and displays
	strong modulations as a function of the chirp. Parameters
	are M ′ = 7, M = 7, δ = 0.0225 fs -1 , ∆ = 0.003 fs -1 , ∆ω =
	0.1525 fs -1 , ΩemΩmg = 1

Acknowledgments

We thank I. Sh. Averbukh, M. Bienert, P. Braun, P. T. Greenland, F. Haug, P. Knight, T. Pfau, F. Schmidt-Kaler, S. Wallentowitz and A. Wolf for stimulating discussions. W. M., E. L. and W. P. S. acknowledge financial support by the Landesstiftung Baden-Württemberg. In addition, this work was supported by a grant from the Ministry of Science, Research and the Arts of Baden-Württemberg in the framework of the Center of Quantum Engineering. Moreover, W. P. S. also would like to thank the Alexander von Humboldt Stiftung and the Max-Planck-Gesellschaft for receiving the Max-Planck-Forschungspreis.


Daniel Carrié

Hassan Srour

Paul Le Maux

Gérard Simonneaux

Daniel Carrie

Gerard Simonneaux
email: gerard.simonneaux@univ-rennes1.fr

Synthesis and chemical reactivity of new zinc porphyrin diazoacetates catalyzed by ruthenium and iron porphyrins

Keywords: Meso-phenyl-diazo-porphyrin ester, ruthenium porphyrin, iron porphyrin, catalytic carbene transfer

come

preparing different types of diazocarbonyl compounds and there are now well-established procedures. 1,2 Important examples of these processes include the functionalization of porphyrin macrocycles using diazo compounds which has been recently reviewed. 3 Thus porphyrin derivatives such as chlorins with a diazo function in the macrocycle core were first reported by Cavaleiro and coworkers in 1997. 4 These chlorin diazoketones can be used as versatile intermediates for the synthesis of new compounds. 5 In 2006, Zaleski's group reported an alternative strategy to obtain 2-diazo-3-oxochlorins. 6 In all these cases, the diazo group is directly inserted in the core yielding chlorin derivatives and to our knowledge, there is no diazo group directly appended to the periphery of the porphyrin ring. We previously reported the use of metalloporphyrins for catalytic insertion of diazo compounds in N-H bonds, 7,8 including insulin protein. 9 Asymmetric cyclopropanation was also developed using chiral iron and ruthenium porphyrins as catalysts. 10,11 In this paper, we report the first synthesis of zinc porphyrin diazoacetates and some of their reactivity using metalloporphyrins as catalysts (Figure 1). A convenient synthetic method for the preparation of a variety of diazoacetates from the corresponding bromoacetates by treatment with N,N'-ditosylhydrazine has been recently reported. 12 Herein, we successfully used this recent method with hydroxymethyl-porphyrins that were prepared as previously reported by Tamiaki. 13 After Zn metalation, reaction with bromoacetate and then with N,N'-ditosylhydrazine yielded the expected diazo derivatives. To demonstrate the presence of possible metallocarbene intermediate, first its decomposition to form olefinic products at room temperature in the presence of tetraphenylporphyrin ruthenium carbon monoxide (Figure 1) as catalyst was examined.

Starting from 1a, this coupling reaction catalyzed by ruthenium porphyrin to form olefins, proceeds slowly, possibly through a metallocarbene intermediate, yielding the cis isomer 2 in overall 91% yield (Scheme 2). In our hands, the trans isomer was not detected. 14 However the reaction needs 24 hours to be completed. In comparison, a similar ruthenium(II) porphyrin species has been reported to catalyze rapidly the decomposition of ethyl diazoacetate to form diethyl maleate and diethyl fumarate with a large cis/trans ratio of 15. 15 The stereochemical contraints that result from the steric interaction of the axial porphyrin carbene atoms with atoms of the porphyrinato core 16 may explain the decrease of the rate. The generally accepted reaction pathway for transition metal catalyzed carbene dimerization of diazo derivatives involves initial attack of the diazo compound at the metal center to generate a metal carbene. 17,18 Nucleophilic attack on the metal carbene by diazo compound, followed by dissociation of the olefin from the metal complex, completes the catalytic cycle. The origin of the cis selectivity during the coupling of -diazo compounds has been previously interpreted on the basis of steric and electronic effects. [19][20][21] To evaluate the reactivity of diazo porphyrin ester, its catalytic decomposition was also examined in the presence of styrene in toluene at room temperature using tetraphenylporphyrin ruthenium carbon monoxide (Figure 1) as catalyst. The cyclopropane 3 was formed with 79% yield and complete trans diastereoselectivity with a concomitant formation of the dimer (15%), resulting from coupling of two carbene precursors. The formation of the trans isomer, which was confirmed by NOESY spectroscopy, is similar to that observed when ethyl diazoacetate is used as reactant. 22 The insertion of electrophilic carbenes in the N-H bonds of protected -amino esters or amides is a powerful method for N-alkylating this class of compounds. 23 Since we 7,8 and others [24][25][26][27] previously reported N-H insertion of diazoesters catalyzed by metalloporphyrins (Fe, Ru), N-H insertion with a diazo porphyrin ester was also investigated (Scheme 2). The results presented in Scheme 2 show that the iron complex is a good catalyst for the transformation of tryptophan methyl ester into the expected N-H inserted products with 69% yield. It should be noted that only the dimerization was observed with the ruthenium complex.

Scheme 2. Reactivity of diazoporphyrin esters.

Since peptidyl diazomethyl ketones appeared initially to be specific inactivators of cysteine proteinases, 28 insertion of diazoporphyrin ester into S-H bonds, catalyzed by metalloporphyrins, was assayed (Scheme 2). Treatment of thiophenol with diazoporphyrin ester catalyzed by complex 1a gave insertion of the diazo derivative into the S-H bond with 85 % yield. The mechanisms of the cyclopropanation reaction, N-H insertion and S-H insertion, catalyzed by metalloporphyrins, have been previously discussed by us 8 and others 25- 27 several times, and will not be presented herein. All the new derivatives were identified by NMR and mass spectral analysis. 29 In summary, the synthesis of new zinc diazoporphyrin esters occurs with good yields offering for the first time a general access to original porphyrins. Studies on the application of the insertion reaction into N-H and S-H bonds for bioconjugation of these potentially fluorescent probes to proteins are in progress.

29.

All new compounds reported here gave spectral data consistent with the assigned structures. Reaction conditions and selected data: For 1. To a distilled CH 2 Cl 2 solution (5 mL) of Zinc 5-[4-(hydroxymethyl)phenyl]-10,15,20-triphenylporphyrin 13 (100 mg, 0.14 mmol), 3 eq of DBU (0.42 mmol) were first added under argon at 0°C and then 1.5 eq of bromo acetyl bromide (0.21 mmol). The reaction mixture was stirred for 10 min at room temperature. After cooling the solution again to 0°C, a THF solution of ditosylhydrazine (2 eq) and DBU (5 eq) was added and the mixture was stirred for 30 min at room temperature. The solution was then evaporated, dissolved in CH 2 Cl 2 and purified through a silica gel column (CH 2 Cl 2 ). Yield = 72%.

Figure 1 .

 1 Figure 1. Structure of the catalysts.

From 5 -

 5 Scheme 1. General pathway for the synthesis of Zn diazoporphyrin ester derivatives.

  The mixture was concentrated by vacuum evaporation and purified by silica gel chromatography. Yield = 85%). 1 H NMR (CD 2 Cl 2 , ppm, 400 MHz): δ 9.10-8.90 (m, 8H, pyrrole), 8.35-8.15 (m, 8H, phe), 7.95-7.75 (m, 9H, phe), 7.75-7.68 (m, 2H, phe), 7.56-7.50 (m, 2H, phe), 7.43-7.35 (m, 1H, phe), 7.33-7.25 (m, 2H, phe) 5.52 (s, 2H, CH 2 O), 3.88 (s, 2H, CH 2 S). 13 C NMR (CD 2 Cl 2 , ppm, 100 MHz): 169.7 (CO), 150.3, 142.9, 127.0, 134.5, 132.0, 130, 129.1, 127.5, 127.1, 126.6, 126.3, 121.2, 120.4, 67.36 (CH 2 O), 36.6 (CH 2 S). HRMS (ESI): calcd for C 53 H 36 N 4 O 2 SZn (M+H) + : 856.18449, found: 856.1841. UV-vis (CH 2 Cl 2 ): max , nm ( cm -1 mM -1 ) : 422 (251.76), 550 (8.93), 592 (1.32).

	(ESI): calcd for C 55 H 38 N 4 O 2 Zn (M+H) + : 850.22807, found: 850.2276. UV-vis
	1 H NMR (CD 2 Cl 2 , ppm, 400 MHz): 8.99 (m, 8H, pyrrole), 8.33-8.20 (m, 8H, phe), 7.90-7.72 (m, 11H, phe), 5.55 (s, 2H, CH 2 O), 5.00 (s (broad), 1H, CHN 2 ). 13 C NMR (CD 2 Cl 2 , ppm, 125 MHz): 166.8(CO), 150.1, 142.5, 135.5, 134.5, 132.0, 127.5, 126.5, 126.2, 121.1, 66.4 (CH 2 O), 30.6 (CHN 2 ). HRMS (ESI): calcd for C 47 H 30 N 6 O 2 Zn (M+H) then evaporated and purified by chromatography on silica gel column (CH 2 Cl 2 ). Yield (CH 2 Cl 2 ): max , nm ( cm -1 mM -1 ) : 421 (357.50), 550 (10.43), 592 (1.08). For 4. To a distilled CH 2 Cl 2 solution (1mL) containing 2.3 l of triethylamine (1.5 eq), 0.6 mg of TPPFeCl and 2.8 mg (0.01 mmol) of L-tryptophan methyl ester hydrochloride, 8.6 mg (0.01 mmol) of 1 dissolved in 2 ml of CH 2 Cl 2 was added under argon. The mixture was stirred for 15 min. The solution was then evaporated and purified by chromatography on silica gel column (CH 2 Cl 2 /CH 3 OH: 98/2).Yield = 69 %. 1 H NMR (CD 2 Cl 2 , ppm, 400 MHz): 8.97 (m, 8H, pyrrole), 8.36-8.24 (m, 8H, Phe), 7.93 (s, 1H, NH), 7.81-7.60 (m, 11H, Phe), 7.27 (d, 1H, tryp), 7.11 (t, 1H, tryp), 6.94 (t, 1H, tryp), 6.84 (d, 1H, tryp), 5.87 (s, 1H, tryp), 4.93 (s, 2H, CH 2 O), 3.05 (s, 3H, CH 3 ), 1.4 and 0.5 (m, 2H, CH 2 ), 0.4 (m, 2H, CH 2 N), 0.39 (m, 1H, CH*). 13 C NMR (CD 2 Cl 2 , ppm, 100 MHz): 169.5 (CO), 168.0 (CO), 149.1, 141.5, 135.0, 134.5, 132.0, 127.0, 126.5, 126.0, 121.1, 110.0, 67 (CH 2 O), 57.5 (C*), 43.0 (CH 2 N), 52.0 (CH 3 ), 24.8 NH of tryptophan ester by porphyrin diazo esters Cyclopropanation of styrene by porphyrin diazo esters Preparation of porphyrin dimers by diazo coupling reaction (CH 2 ). HRMS (ESI): calcd for C 59 H 45 N 6 O 4 Zn (M+H) First synthesis of Zinc porphyrin diazoacetates SH insertion of thiophenol by porphyrin diazo esters
	= 91 %. 1 H NMR (CD 2 Cl 2 , ppm, 400MHz):	8.85 (m, 16H, pyrrole), 8.3-8.1 (m,
	18H, Phe), 7.90-7.72 (m, 20H, Phe), 6.62 (s, 2H, CH 2 ), 5.67 (s, 4H, CH 2 O). 13 C NMR
	(CD 2 Cl 2 , ppm, 100 MHz): 165.2 (CO), 151.0, 143.0, 135.1, 134.9, 132.5, 130.1
	(CH cis ), 127.5, 126.4, 126.2, 67.2(CH 2 O). HRMS (ESI): calcd for C 94 H 60 N 8 O 4 Zn 2
	(M) +. : 1492.33149, found: 1492.3311. UV-vis (CH 2 Cl 2 ): max , nm ( cm -1 mM -1 ) : 421
	(485.59), 550 (20.58), 594 (5.15). For 3. To a distilled CH 2 Cl 2 solution (1mL)
	containing 25 l of styrene and 0.9 mg of TPPRuCO under argon, 15.8 mg of 1
	dissolved in 2 ml of CH 2 Cl 2 was added progressively (1ml/h). The mixture was stirred
	overnight. The solution was then evaporated and purified by chromatography on silica
	gel column. Yield = 79 %. 1 H NMR (CD 2 Cl 2 , ppm, 400 MHz): 8.99 (m, 8H,
	pyrrole), 8.31-8.21 (m, 8H, phe), 7.90-7.72 (m, 11H, phe), 7.38-7.30 (m, 2H, phe),
	7.28-7.23 (m, 1H, phe), 7.22-7.16 (m, 2H, phe), 5.53 (s, 2H, CH 2 ), 2.64-2.57 (m, 1H,
	cyclo), 2.12-2.05 (m, cyclo), 1.72-1.64 (m,1H, cyclo), 1.49-1.41 (m,1H, cyclo). 13 C
	NMR (CD 2 Cl 2 , ppm, 100 MHz): 173.6 (CO), 150.1, 142.8, 142.7, 140.2, 135.6,
	134.5, 132.0, 128.4, 127.5, 126.5, 126.2, 121.6, 121.2, 66.4, 26.2, 24.1, 17.1. HRMS

+ : 774.17162, found: 774.1719. UV-vis (CH 2 Cl 2 ): max , nm ( cm -1 mM -1 ) : 421 (365.90), 549 (14.39), 592 (3.47). For 2. To a distilled CH 2 Cl 2 solution (1 mL) containing 0.9 mg of TPPRuCO, 8.6 mg (0.01 mmol) of 1 dissolved in 2 ml of CH 2 Cl 2 was added under argon. The mixture was stirred for 24h at room temperature. The solution was + : 965.27882, found: 965.2790. UV-vis (CH 2 Cl 2 ): max , nm ( cm -1 mM -1 ) : 421 (272.90), 551 (11.47), 592 (1.90). For 5. To a distilled toluene solution (1mL) o2.86 mg (0.026 mmol) of thiophenol and 0.6 mg of TPPRuCO under argon, 2.9 mg (0.02 mmol) of 1 dissolved in 0.5 ml of toluene was added over a period of 30 min. The reaction was then stirred for 4 hours at room temperature.


Brahim Belaoucha

Maureen Clerc

Théodore Papadopoulo

Cortical Surface Parcellation via dMRI Using Mutual Nearest Neighbor Condition

Keywords: Parcellation, dMRI, Mutual Nearest Neighbor

In this paper, we present a method that aims at parcellating the cortical surface from individual anatomy. The parcellation is obtained using the Mutual Nearest Neighbor (MNN) criterion to obtain regions with similar structural connectivity. The structural connectivity is obtained by applying a probabilistic tractography on the diffusion MRI (dMRI), a non-invasive modality allowing access to the structural information of the white matter. The results are compared to some of the atlases that can be found in the literature. We show that these atlases have lower similarity of structural connectivity than the proposed algorithm implying that the regions of the atlases may have lower functional homogeneity.

INTRODUCTION

Understanding the human cortex organization is a very challenging research field. One of the challenges is to find a relation between anatomical and functional regions. Physiologically, the brain is organized in functional areas, and structural connectivity of the cortical region is the primary indicator of its function [START_REF] Sporns | Organization, development and function of complex brain networks[END_REF]. In [START_REF] Rakic | Neurocreationism-making new cortical maps[END_REF], the author described cortical maps suggesting a specific organization of the brain functions in relation to the cortical anatomy. Diffusion Magnetic Resonance Imaging (dMRI) is a non-invasive imaging modality that allows the understanding and exploration of the underlying tissue structure of the human brain.

There have been several methods to label the cortical surface [START_REF] Fischl | Automatically parcellating the human cerebral cortex[END_REF][START_REF] Desikan | An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest[END_REF][START_REF] Klein | Mindboggle: a scatterbrained approach to automate brain labeling[END_REF][START_REF] Moreno-Dominguez | Whole-brain cortical parcellation: A hierarchical method based on dMRI tractography[END_REF][START_REF] Roca | Parcellisation du manteau cortical à partir du réseau de connectivité anatomique cartographié par imagerie de diffusion[END_REF] and usually researchers register their subjects to some atlases that can be found in the literature; Destrieux (DX) [START_REF] Fischl | Automatically parcellating the human cerebral cortex[END_REF], Desikin-Killiany (DK) [START_REF] Desikan | An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest[END_REF], and Mindboggle (ME) [START_REF] Klein | Mindboggle: a scatterbrained approach to automate brain labeling[END_REF], see Fig. 1. The registration can be misleading due to variations in the cortical surface from one subject to another. Previous methods for dMRI based parcellation use atlases as a pre-parcellation step to reduce the computational cost and to obtain sub-regions that have similar structural connectivity [START_REF] Anwander | Connectivity-based parcellation of Broca's area[END_REF][START_REF] Philippe | A nested cortex parcellation combining analysis of MEG forward problem and diffusion MRI tractography[END_REF]. The method presented in this paper uses the underlying white matter pathways to separate regions. It is similar to the one presented in [START_REF] Bellec | Identification of large-scale networks in the brain using fMRI[END_REF], but diffusion MRI is used instead of functional MRI information. This allows us to parcellate the whole cortical surface and not only an active region of the brain. We also compare connectivity profiles of the resulting regions to both atlases and subdivided atlases using k-means.

To the best of our knowledge, no one has yet investigated the connectivity profiles of the atlases' regions.

METHODOLOGY

Image acquisition

Structural and diffusion MRI data were taken from 6 healthy subjects [START_REF] Wakeman | A multi-subject, multi-modal human neuroimaging dataset[END_REF], 4 males and 2 females with an average age of 26.83 years. The T1 weighted images of size 256×256×192 were acquired by Siemens 3T Trio with GRAPPA 3D MPRA-GE sequence (TR = 2250 ms; TE = 2.99 ms; flip-angle = 9 • ; acceleration factor = 2) at 1 mm isotropic resolution. The diffusion weighted images of size 96×96×68 were collected by the same scanner at 2 mm isotropic resolution (64 gradient directions and b-value =1000 s/mm 2 ), with one b0 image.

Image processing

The white/gray matter interfaces (cortical surface) were extracted using Freesurfer [START_REF] Dale | Cortical surface-based analysis I: Segmentation and surface reconstruction[END_REF] from T1 images and remeshed to 10 4 vertices, then projected from the anatomical to the diffusion space. The projection was obtained by registering linearly the brain between the two spaces using FSL [START_REF] Smith | Advances in functional and structural MR image analysis and implementation as FSL[END_REF].

Cortex parcellation

Connectivity profile of a seed: Each node in the cortical mesh is considered as a seed. For each seed, we compute the connectivity profile, i.e. its connectivity with every voxel in the diffusion image space. Connectivity profiles are obtained by running probabilistic tractography for all seeds. We assume a model of multiple fiber orientations for each voxel developed in FSL [START_REF] Behrens | Probabilistic diffusion tractography with multiple fibre orientations: What can we gain?[END_REF] with 10 4 samples. Clustering the cortical surface: The goal of parcellation algorithm is to divide the cortical surface into a set of non overlapping and connected regions that have similar connectivity profiles. In the first step, regions are singletons of seeds and are possible candidates for merging. The algorithm is explained in Algorithm 1. CP represents the matrix of the seeds' connectivity profiles and M C is the mesh connectivity Initialize:

s ← N S T , Label=1:N S 3: while K > 0 do 4:
for Each region do 5:

N r ← Neighbors(M C, r) 6: B(r)← arg max r k ∈Nr SM (r, r k ) 7:
end for 8:

for Each region do end for return Label 20: end procedure For every iteration, we obtain the set of neighboring regions N r to the region r (line 5), and compute the similarity measure, SM , between region r and N r (line 6). The vector B of merging regions candidates contains the labels of the possible candidates: B(r) is the neighbor of region r that has the highest SM value. We define the SM between two regions R and S as;

SM (R, S) = 1 |R||S| (v,w)∈(R×S) corr(CP v , CP w ) (1)
where corr is the P earson s correlation and |R| is the number of seeds in R. Region r and B(r) are merged if they satisfy the Mutual Nearest Neighbor (MNN) condition:

B(r) = arg max r k ∈Nr SM (r k , r) r = arg max r k ∈N B(r) SM (B(r), r k )

RESULTS AND DISCUSSION

The mean of the similarity values (MSV) between all the connectivity profile pairs inside the regions is used to measure the homogeneity of the parcellation's regions. High MSV means that the region has close connectivity profiles. Fig. 2 (a) shows the MSV of the different atlases and subjects. We applied the cortical surface parcellation algorithm explained in Section 2.3 to the dMRI data [START_REF] Wakeman | A multi-subject, multi-modal human neuroimaging dataset[END_REF]. Fig. 2 (b) shows the MSV of the resulting parcellation for all subjects and T ∈ {1000, 800, 600, 400, 200}. To compare the MSV of the atlases, Fig. 2 (a), and the ones of the proposed parcellation, we chose T ∈ {148, 62}, Fig. 2 (c). These values give parcellations with close number of regions to the ones of the atlases (see Fig. 3). Fig. 2 (d) shows the MSV and variance of the similarity measure values of the regions obtained by subdividing the atlas' regions using k-means.

Several papers used the atlases as a pre-parcellation step to reduce the computation time [START_REF] Anwander | Connectivity-based parcellation of Broca's area[END_REF][START_REF] Philippe | A nested cortex parcellation combining analysis of MEG forward problem and diffusion MRI tractography[END_REF]. Each atlas' region is parcellated using k-means. This approach is assumed to group seeds that have similar CP inside the atlas' region. We use the eigenvalues of the CP s' cross-correlation to identify the number of clusters in each region. Based on the eigengap (difference between consecutive eigenvalues), we sort the eigenvalues, α i for i ∈ {1, 2, ..., |r|}, of the cross-correlation matrix in decreasing order and pick the number of clusters, k, as: k = argmax i (α i -α i+1 ). The resulting number of regions for the six subjects is shown in Table 1.

As shown in Fig. 2 (a), the MSV of the DX atlas is higher than the others and is followed by DK and ME atlases for all subjects. This follows the decreasing order of the number of regions of the atlases, see Fig. 1.

In average and as expected, the MSV is proportional to T due to including more dissimilar connectivity profiles for low T values. We observe also that the order is homogeneous across subjects, see Fig. 2 (b). For all subjects in Fig. 2 S1 (c), the parcellations of the proposed method provided regions with higher MSV when compared to the atlases (Fig. 2 (a)). This is due to merging only regions with the highest similarity measure due to the MNN condition. In Fig. 3, which represents the result of the parcellations for T = 62, we can distinguish several classical regions like Fusiform gyrus, Superior Frontal, Occipital and others.

Even if the MSV is increased after applying the k-means on the atlases, Fig. 2 (d), but it still less than the MSV of the proposed parcellation with bigger regions (T ∈ {148, 62}), Fig. 2 (c). This implies that the atlas' regions have low homogeneity of the structural connectivity because they do not include the structural information, which implies that they have lower functional homogeneity.

Table 2 shows the mean and the standard deviation of the Dice coefficients vector (D). The Dice coefficient measures the extent of spatial overlap between regions. Let A i be a region in the parcellation A (n regions) and B j be a region in the parcellation B (m regions). D is a vector of n elements that compares the regions of A to B:

D(i) = max 2(Ai∩Bj )
|Ai|+|Bj | for j = 1, .., m. The mean of D between the atlases and the proposed parcellation is low for all subjects, see the first four rows in Table 2. Atlases and parcellations based on MNN condition do not only differ in the MSV but also have low spatial overlapping. The MSV for ME and DK are close, Fig. 2 (a), this can be explained by their high spatial overlapping (90 %), last two rows in Table 2. This is why we do not compare the Dice coefficient between the parcellation with T = 68 and the DK [START_REF] Anwander | Connectivity-based parcellation of Broca's area[END_REF]. Other similarity measures like T animoto measure can be used and should be tested in this framework to see the effect of the similarity measure on the resulting parcellation. The implementation of the algorithm will be available for comparison studies.

S1 S2 S3 S4 S5 S6

CONCLUSION

We have presented algorithm to subdivide the human cerebral cortex into neuro-anatomical parcels that have similar connectivity profiles. The results show that the algorithm provides regions with higher similarity between the CP with compared to atlases even after subdividing them. These atlases have low structural homogeneity because they do not include any structural information. The algorithm may provide valuable tool for research studies involving group and functional studies. Even though and as expected the parcellation using MNN gives regions with higher MSV, it needs to be validated using functional data.

Fig. 1 :Algorithm 1

 11 Fig. 1: The four views of the atlases obtained from FreeSurfer for Subject 1. From left to right; DX (148 regions), DK (68 regions) and ME (62 regions). The Thalamus is not included in the labeling.

9 :

 9 if |r| < s and MNN(r, B(r)) then

Fig. 2 :

 2 Fig. 2: The MSV and variance of the cross-correlation of the CP over regions of (a) different atlases and subjects, (b) the resulting parcellation for different subjects and T values, (c) parcellations with T ∈ {148, 62} and (d) the regions obtained by subdividing the atlases using k-means.

Fig. 3 :

 3 Fig.3: The Resulting parcellation of six subjects on the inflated cortical surfaces with T = 62. Each panel correspond to four views of one subject. . The Thalamus is not included in the parcellation. We obtain number of regions for the different subjects between 60 and 66.

Table 1 :

 1 Number of parcels for different subjects and atlases obtained from k-means parcellation on the atlases.

	Atlas	S1	S2	S3	S4	S5	S6
	Mindboggle	138 157 168 152 190 168
	Desikan-Killiany 155 174 170 167 203 222
	Destrieux	380 392 379 406 409 412

Table 2 :

 2 The Dice coefficient (mean/standard deviation in %) between the different parcellations (T ∈ {148, 62}) and atlases.atlas. DX has low spatial overlapping with DK and ME, see row five to eight in Table2, this is because DX has more regions than DK and ME.We used P earson's correlation in Equation.1 because it was used successfully in different papers like in Anwander et al.


Grasping versus Knitting: a Geometric Perspective Comment on Hand synergies: Integration

Grasping an object is a matter of first moving a prehensile organ at some position in the world, and then managing the contact relationship between the prehensile organ and the object. Once the contact relationship has been established and made stable, the object is part of the body and it can move in the world. As any action, the action of grasping is ontologically anchored in the physical space while the correlative movement originates in the space of the body. Robots-as any living system-access the physical space only indirectly through sensors and motors. Sensors and motors constitute the space of the body where homeostasis takes place. Physical space and both sensor space and motor space constitute a triangulation, which is the locus of the action embodiment, i.e. the locus of operations allowing the fundamental inversion between world-centered and body-centered frames. Referring to these three fundamental spaces, geometry appears as the best abstraction to capture the nature of action-driven movements. Indeed, a particular geometry is captured by a particular group of transformations of the points of a space such that every point or every direction in space can be transformed by an element of the group to every other point or direction within the group. Quoting mathematician Poincaré, the issue is not find the truest geometry but the most practical one to account for the complexity of the world1 . Geometry is then the language fostering the dialog between neurophysiology and engineering about natural and artificial movement science and technology. Evolution has found amazing solutions that allow organisms to rapidly and efficiently manage the relationship between their body and the world2 . It is then natural that roboticists consider taking inspiration of these natural solutions, while contributing to better understand their origin.

The recent European project The Hand Embodied is a remarkable application of this multidisciplinary research paradigm 3 . The human hand is certainly the most sophisticated organ evolution has provided to allow a living system to act on the world. The hand is recognized as a fundamental component of intelligence. Its richness comes from its extraordinary capacity to perform a large range of dexterous manipulation tasks ranging from hammering to knitting. Hammering required maintaining a stable grasp between the handle and the moving arm. How to configure all the degrees of freedom of the hand around the handle, and what configurations obey the better the physical constraints of hammering (e.g. non sliding and force resistant contacts)? They are challenging questions even for a simple hammering task. Knitting is a task much more complicated that hammering. Knitting requires mobile dexterity. Finger movements have to be coordinated in order to steer the thread along the needles while tuning its tension. With respect to hammering, knitting adds another level of challenge. Complexity arises both from the dimension of the hand control space and from the dimension of the task as defined in the physical space.

Grasping an object goes back to establish and maintain a fixed relation between an arbitrary object frame and an arbitrary hand frame. Grasping task and its physical constraints is then described in the physical space by the space of hand placements, i.e. a space of dimension 6. Considering the many possible postures of the hand (i.e. the many placement of the fingers around the object), the question of grasping is to select the ones that fitful the constraints. The link to be settled is between a six dimensional space and the hand high dimensional configuration space. Many hand postures might be admissible and many movements reaching an admissible posture might be feasible. The hand is said to be redundant with respect to grasping task. Redundancy requires methods for posture and movement selection. Posture and movement spaces are highly dimensional spaces. Their dimensions give a measure of the computational complexity of the task. All current researches in life science as well as in engineering explore how living and artificial systems face such a complexity. Both understanding the computation foundations of actions performed by a human hand, and devising a human-like hand impose overcoming the famous curse of dimensionality. This dual perspective is supported by the notion of synergy. Synergies are a way to reduce the dimension of spaces to be explored. At first glance, we may consider two types of synergies. Postural synergies reduce the dimension of the system configuration space. They are the consequence of physical links as cams, or they are derived from holonomic differential links that impose integrable constraints between the velocities of some body parts. Motor synergies tend to reduce the scope of all possible movements by orchestrating all elementary motor controls from a basis whose dimension is less than the dimension of entire motor control space. What makes the outstanding quality of the Pisa/IIT SoftHand is exactly the clever combination of postural and motor synergies.

It is natural that roboticists consider taking inspiration of evolution. It is not mandatory. Evolution did not invent the wheel, and aircrafts do not flap wings, at least at this moment! To face the world complexity, engineering has developed its own concepts without reference to any living system. For instance, the concept of minimalism refers to the methodological approach to design the least complex solutions for a given class of tasks, by, e.g., using the minimal number of actuators or control variables, or the simplest set of sensors. Applied to grasping tasks, this perspective gave rise to clever mechanisms such as a gripper based on the jamming of granular material4 or a generalpurpose hand made of only three rigid fingers5 . Such general principles percolate with the concept of synergies or with the so-called morphological computation 6 .

We have seen that actions are defined in the physics space while their very origin takes place in the control space. The relationship between "action in the real world" and "movement generation" in the motor control space is defined in terms of geometry. It particularly derives from differential geometry, linear algebra and optimality principles [START_REF] Todorov | Optimality principles in sensorimotor control[END_REF][START_REF] Laumond | Optimization as Motion Selection Principle in Robot Action[END_REF] . Optimal control is based on well-established mathematical machinery ranging from the analytical approaches initiated by Pontryagin [START_REF] Pontryagin | The Mathematical Theory of Optimal Processes[END_REF] to the recent developments in numerical analysis [START_REF] Bonnans | Numerical Optimization: Theoretical and Practical Aspects[END_REF] . It allows action-driven movement generation. Such mathematical machinery can operate also in a reverse perspective to elucidate the laws of natural movements and to reveal motor synergies. For instance, inverse optimal control is a way to model the human motion in terms of controlled systems: given an underlying hypothesis of a system, as well as a set of observed natural actions recorded from an experimental protocol performed on several participants, the question is to find the cost function the system is optimizing. From a mathematical point view, the inverse problem is much more challenging than the direct one. There are few (recent) results in this direction of research. They include numerical analysis [START_REF]Fast Motions in Biomechanics and Robotics[END_REF] , statistical analysis [START_REF] Inamura | Embodied Symbol Emergence based on Mimesis Theory[END_REF] and also part of the active area of machine learning [START_REF] Mitchell | Machine Learning[END_REF][START_REF] Kober | Reinforcement learning in robotics: A survey[END_REF] .

As final comment, let us consider the following issue. Imagine a humanoid robot whose arms are equipped with two Pisa/IIT SoftHands as prehensile organs. Is such a robot capable of knitting? The answer is no. The bottleneck comes from synergies. We have seen that the advantage of synergies is to allow space dimensionality reduction, and then complexity reduction. In return, not all possible movements can be generated. Knitting would require facing against the synergies used for grasping. Unraveling the current synergy orchestration would be the first step of a solution. The second step would be to learn new synergies dedicated to knitting. And that is out of the scope of current know-how from robotics and life science. Now, another question is: what is the interest for a robot to learn knitting whereas Joseph Marie Jacquard invented the mechanical loom in 1801? As many human know-how since the beginning of the technology, [great-]grandmother knitting know-how is disappearing.

H. Poincaré, L'espace et la géométrie, Revue de métaphysique et de morale, 1895, vol. III, p. 631-646.

A. Berthoz, Simplexity: Simplifying Principles for a Complex World, Yale Univ. Press, 2012.

M. Santello et al. Hand synergies: Integration of robotics and neuroscience for understanding the control of biological and artificial hands, Physics of Life Reviews, 2016.

E. Brown, N. Rodenberg, J. Amend, A. Mozeika, E. Steltz, M. Zakin, H. Lipson and H. Jaeger, Universal robotic gripper based on the jamming of granular material, Proceedings of the National Academy of Sciences, Vol. 107, No. 44, 2010.

M. T. Mason, A. Rodriguez, S. Srinivasa, A. S. Vazquez, Autonomous Manipulation with a General-Purpose Simple Hand, The International Journal of Robotics Research (IJRR), Vol. 31, No. 5, April, 2012, pp.

688-703. 6 R. Pfeifer, J. Bongard, How the body shapes the way we think: a new view of intelligence, The MIT Press, 2007.