Marion Guthmuller

Martin Quinson

System-level State Equality Detection for the Dynamic Verification of Distributed Applications

The poster is about the joint correction and performance study of distributed applications, without any manual rewrite between models, algorithms and real implementations. Our goal is to dynamically verify unmodified legacy distributed applications written with MPI, or a similar API that allows concurrent sequential processes (CSP) to interact through message passing.

Our work aims to improve the dynamic verification tool SimGridMC, which exhaustively verifies distributed applications by executing the real program. For this, the applications can be specified using either one of the interfaces provided by the versatile SimGrid framework, or the classical MPI standard. Prior to this work, SimGridMC was stateless, only checkpointing the initial system's state. When needing to backtrack a previous state to explore another path, this initial state was first restored and then all transitions leading to the desired state were replayed. In the context of HPC applications however, this approach reveals problematic since application's computations are much more expensive than with Peer-to-Peer protocols for which Sim-GridMC was initially intended. It then becomes interesting to save more system states so that they can be restored without re-executing previous computation. Moreover, stateless verification only enables the verification of safety properties while the most interesting properties include vivacity. The verification of liveness properties is based on the detection of cycles during the execution of the application and thus on the detection of system's state equality. Finally, this work can also be used to exhaustively verify cyclic applications which may be infinite-time, such as cyclic protocols that are governed by periodic events (the P2P protocol Chord for example). This poster organized as follows : we first present our motivation -the study and verification of distributed applications. As explained earlier, a preexisting version of a dynamic verification tool, SimGridMC, was developed to verify real implementations of distributed applications. This tool was however limited to the verification of one kind of property and its stateless approach wasn't suitable for the HPC applications or for some cyclic applications. The detection of system state equalities is one of major challenges for the improvement of this tool. We detail in a second part the OSlevel challenges encountered when detecting the system state equality of MPI applications and propose in a third part solutions to mitigate these difficulties.

Since the byte per byte memory comparison leads to the detection of syntactic differences that are not significant, a memory introspection and a semantic comparison is necessary. We identified four major causes of these differences : (1) the memory overprovisioning implemented by many memory allocation libraries, such as the size of memory chunks is a power of two. For example, a memory area of 64 bytes will be used to serve a request of 48 bytes. Unused bytes may thus result in irrelevant differences, as they contain unspecified values ; (2) some padding bytes added by the compiler between the variables to enforce memory alignment constraints. Like with the memory overprovisioning, these bytes must not be considered in the comparison ; (3) some irrelevant differences such as PID, or file descriptors ; (4) the order in memory of malloced blocks may leads to two heaps at two different execution times syntactically different while being equal from a semantic point of view. This situation often occurs in our context because the block ordering stems from the order of malloc requests, which change when the process execution order changes, as in the dynamic verification.

For each of these OS-level issues, we propose several solutions leveraging debugging information and tools. To the best of our knowledge, this is the first heuristic that can be used on programming languages and systems without automatic garbage collection to reconstruct the needed semantic information.

We show in another part the effectiveness of our state equality mechanism through several experiments such as the dynamic verification of a liveness property on a custom MPI application and the verification of some MPICH3 integrated tests. Finally, we briefly present the future work associated to this work.


Samia Ayadi
email: samiaay@yahoo.com

Olivier Haeberlé
email: olivier.haeberle@uha.fr

The Lorenz model for single-mode homogeneously broadened laser: analytical determination of the unpredictable zone Research Article

Keywords: 55.-f, 42.55.Ah, 05.45.Pq Laser instabilities, Lorenz-Haken equations, Self-pulsing, Chaos

. In this work, we extend this method in the aim of obtaining the higher harmonics. We show that this iterative method is indeed limited to the fifth order, and that above, the obtained analytical solution diverges from the numerical direct resolution of the equations.

Introduction

The simplest laser model is a single mode unidirectional ring laser containing a homogeneously-broadened, twolevel medium, commonly designated as the Lorenz-Haken model. In 1975, Haken [1] showed equivalence between the Lorenz model that describes fluid turbulence [2], already known for leading to deterministic chaos, and the equations of a homogeneously-broadened, single-mode laser. In this context, such a laser can be viewed as a system, which becomes unstable under suitable conditions related to the respective values of the decay rate (bad cavity condition) and of the level of excitation (second laser threshold). The numerical integration of the Lorenz-Haken model has indicated that the system undergoes a transition from a stable continuous wave output to a regular pulsing state. However, it also sometimes develops irregular pulsations (chaotic solutions). The nature of such irregular solutions was explained by Haken [1]. For more than thirty years, the approach towards solving Haken-Lorenz equations was dominated by the general line of thought that "the pulsing solutions of the singlemode laser equations must be found with numerical integration". Bougoffa and Bouggouffa [3][START_REF] Bougouffa | AIP Conf. Proc[END_REF][START_REF] Bougouffa | [END_REF] presented an analytical approach based on Adomian decomposition to find the solutions of the Lorenz system. Their method was applied to find the stationary states. In a previous work [6,7], it was shown that a simple harmonic expansion method permits to obtain analytical solutions for the laser equations, for physical situations where the long-term signal consists of regular pulse trains (periodic solutions). The corresponding laser field oscillates around a zero mean-value. In particular, we have shown that the inclusion of the third-order harmonic term in the field expansion allows for the prediction of the pulsing frequencies. The analytical expression of pulsing frequencies excellently matches their numerical counterparts, when the solution consists of regular period-one pulse trains. We have also derived a natural frequency, typical of the permanent pulsing-regime of operation. The aim of the present work is first to extract analytical information about the zone of period doubling and about the chaotic region, from the third-order expansion analysis. Then, the validity of the analytical development is discussed. We demonstrate in particular that the analytical expression of the periodic solutions diverges from the numerical one if one extends the development above the fifth order.

Haken-Lorenz equations: period-one oscillations

The model we start from is based on the Maxwell-Bloch equations in single mode approximation, considering a unidirectional ring laser containing a homogeneously broadened medium. The equations of motion are derived using a semi-classical approach, considering the resonant field inside the laser cavity as a macroscopic variable interacting with a two-level system. Assuming exact resonance between the atomic line and the cavity mode, and after adequate approximations, one obtains three coupled non-linear differential equations for the field, polarization, and population inversion of the medium, the so-called Lorenz-Haken model [START_REF] Haken | Light[END_REF][START_REF] Narducci | [END_REF][START_REF] Harrison | [END_REF]:

d E (t) dt = -κ {E (t) + 2 C P (t)} (1a) d P (t) dt = -P (t) + E (t) D (t) (1b) d D (t) dt = -℘ {D (t) + 1 + P (t) E (t)} (1c)
where E(t) represents the electric field in the laser cavity having a decay constant k, P (t) is the polarization of this field, D(t) is the population difference having a decay constant ℘. Both κ and ℘ are scaled with respect to the polarization relaxation rate, and 2C is the pump rate required for obtaining the lasing effect. To obtain the steady-state solution of system (1), all derivatives with respect to time are set to zero. Under suitable conditions, the steady-state solution becomes unstable. We can delimit the boundary regime where Eqs. (1) involve unstable solutions by linear stability analysis (LSA) [START_REF] Haken | Light[END_REF][START_REF] Ya | Fundamental of Laser Dynamics[END_REF]. This analysis leads to the following results: the loss of stability simultaneously requires a bad cavity (i.e., κ being sufficiently larger than ℘ + 1) and a pumping parameter 2C larger than 2C 2th (the instability threshold condition). This threshold 2C 2th corresponds to the onset of instability and is given by:

2C th = 1 + (κ + 1) (κ + 1 + ℘) (κ -1 -℘) (2)
At this critical value of the excitation parameter, the solution undergoes a subcritical Hopf bifurcation [START_REF] Sparrow | The Lorenz Equation: Bifurcation, Chaos and Strange Attractors[END_REF] and loses stability, leading to a large-amplitude, pulsing solution. We have solved numerically Eqs. (1) using a Runge-Kutta method with an adaptive integration step, and with the following parameters κ = 3, ℘ = 0.1 and 2C = 10 > 2C th (2C 2th = 9.63). The field and polarization oscillate around a zero mean-value, while the population inversion oscillates with a dc component. The corresponding frequency spectra obtained by Fast Fourier Transform (FFT) exhibit odd components at ∆, 3∆, 5∆,... for the field and polarization. On the contrary, the population inversion spectrum exhibits even components at 2∆, 4∆, 6∆,.... These properties are at the basis of the strong-harmonic expansion method [6,7,[START_REF] Meziane | Atomic, Molecular and Optical Physics[END_REF] that we use to construct analytical solutions. We now give a brief outline of the main steps of the adapted strong-harmonic expansion method we use. This yields analytical expressions for the angular frequency of the periodic solutions and for the first harmonics of the corresponding analytical solutions.

Iterative harmonic expansion

We here briefly summarize the analytical procedure, which we use to determine the long-term frequencies and amplitudes of the first harmonics of the field E(t) [6,7,[START_REF] Meziane | Atomic, Molecular and Optical Physics[END_REF]. We write the interacting variables as the following expansions:

E (t) = n≥0 E2n+1 cos ((2n + 1) ∆ t) (3a)
P (t) = n≥0 P2n+1 cos ((2n + 1) ∆ t) + P2n+2 sin ((2n + 1) ∆ t) (3b)
D (t) = D0 + n≥0 D2n+1 cos ((2n + 2 )∆ t) + D2n+2 sin ((2n + 2 ) ∆ t) (3c)
Limiting these expansions to the third order for the field and polarization, and to the second order for the population inversion in Eqs. (3), we obtain the expression of the angular frequency of the pulsating solution [6,7]:

∆ 2 = (2 C -1) κ ℘ (2 + ℘) -3 (κ + 1) ℘ 2 8 (κ + 1) -℘ (2 κ + ℘ + 4) (4)
The details of these calculations are given in section 2 and appendix A of Ref. [7]. This formula shows dependence to the pumping parameter 2C and indicates that the frequency of the signal increases with the excitation level.

We have previously [6] proved that the analytical frequency given by Eq. ( 4) perfectly matches the frequency numerically derived from Eqs. (1). Another expression for the operating long-term frequency [7] is obtained:

∆p = 3 ℘ + 2κ (1 + 2℘) 24 + 6κ + 9℘ (5)
This expression shows no dependence to the excitation level 2C. It constitutes an expression of the natural frequency that characterizes a given set of κ and ℘ values that allow for periodic solutions. This frequency will be used to delimit the domain where the laser exhibits regular oscillations.

Chaos via a period doubling sequence

A. Narduccsi et al. [START_REF] Narducci | [END_REF][START_REF] Narducci | Laser physics and laser instabilities[END_REF] have shown that parameter ℘ plays an important role in defining the kind of dynamics predicted by the Lorenz-Haken model. Numerical simulations [START_REF] Meziane | Atomic, Molecular and Optical Physics[END_REF] with κ = 3 and ℘ = 0.1 (Fig. 1) have shown that large-amplitude periodic solutions dominate over the range 10 < 2C < 32. The behaviour of these periodic solutions for increasing gain 2C is extremely variable. 1e,1f)). This feature corresponds to the period-doubling bifurcations.

These results illustrate that period-doubling bifurcations can only emerge from asymmetrical solutions, as shown by Swift and Weisenfeld [18]. As the excitation level further increases, more and more complicated patterns emerge, which eventually lead to chaotic behaviour. As shown in Figs. (1g,1h) and Figs. (1i,1j), the temporal trace of the field exhibits asymmetric solution with period four when 2C = 29.7, and finally becomes chaotic for We now attempt to delimit the zone where the laser can exhibit periodic pulsations using the analytical method.

2C = 32. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)
At first, we derive the analytical expression of the pumping rate 2C that leads the laser to oscillate with period one, period two and over. To get this expression, we assume that the pulsation frequency Eq. ( 4) is proportional to the eigenfrequency Eq. ( 5):

∆ = α ∆p (6)
α being a rational number. Then, 2Cα takes the following form:

2 Cα = (2k + 3℘ + 4κ℘) α 2 [8(κ + 1) -℘(2κ + ℘ + 4)] + 3℘(8 + 2κ + 3℘) 3κ℘(2 + ℘)(8 + 2κ + 3℘) (7)
For different values of α, we get the pumping level 2Cα that we have used for numerically solving Eqs. (1). These numerical solutions for the chosen parameters κ = 3 and ℘ = 0.1 can be categorized into four classes, depending on the values of α:

Symmetric and period-one solutions:

1 ≤ α < 1.5 (9.63 ≤ 2C ≤ 18.4)
Asymmetric solutions with period one:

α ≤ 1.75 (18.4 < 2C ≤ 27.8)
Asymmetric solutions with period two and over:

1.75 < α < 2 (27.9 ≤ 2C < 32) Chaotic solutions: α ≥ 2 (2C > 32)
These results are illustrated in Fig. (1), and the route to chaos via period doubling is clearly identified. We conclude that the chaotic behaviour takes place for α ≥ 2. This last result allows for delimiting the region of control parameters where the operation of the homogeneously-broadened single mode laser can be chaotic. To do so, we assume that 2C 2th is equal to 2C2. From this identity, we derive a characteristic equation of ℘ according to κ:

64κ -64κ 3 + (96 + 208κ + 92κ 2 -40κ 3 + 12κ 4 )℘ + (120 + 242κ + 274κ 2 + 70κ 3 + 6κ 4 )℘ 2 +(24 + 41κ + 76κ 2 -5κ 3 )℘ 3 + (15κ + 29κ 2 )℘ 4 = 0 (8)
We have solved Eq. ( 8) using the Mathematica™ software to found the analytical expressions of its roots. Accepting

Analytical periodic solutions

The analytical development described in section 3 permits to establish the analytical expression of the electric field. This was done in a previous work (see appendix A in Ref. [7]) to derive the expressions of the first and third-order harmonics of the field:

E1 = -2CP1 = 2 (1 + ∆ 2 )(4 ∆ 2 + ℘ 2 )(1 + κ) √ ℘ (℘ + 2 κ + 2℘ κ -∆ 2 (4 + ℘ + 2 κ)) (9)
E3 = -2CP3 = - T 3d (℘ 2 (1 -3∆ 2 ) -8℘ ∆ 2 ) E 3 1 4 1 -T 1d (℘ 2 (1 -∆ 2 ) -4℘ ∆ 2 ) E 2 1 4 (10)
The expressions of E1 and E3 are related to the decay rates ℘ and κ, and to the excitation parameter 2C through their dependence on the pulsing frequency ∆. Using the previously described analytical features, we now construct a typical sequence of analytical solutions being the counterparts of the numerical one, presented on Fig. (3a).

The long term operating frequency is estimated from Eq. ( 4), while the first and third order field components are evaluated from Eqs. ( 9) and [START_REF] Sparrow | The Lorenz Equation: Bifurcation, Chaos and Strange Attractors[END_REF], respectively. The values of these components, for κ = 3, ℘ = 0.1 and 2C = 10, are E1 = 5.19 and E3 = 1.72 and the corresponding frequency is ∆ = 0.42. Thus, to the third order, the analytical field expansion takes the following form:

E (t) = 5.19 cos(0.42 t) + 1.72 cos(3 × 0.42 t) (11)
The temporal evolution of Eq. ( 11) is illustrated in Fig. (3b). One may notice the resemblance with its numerical counterpart depicted in Fig. (3a), however, differences remain between the analytical and the numerical solutions.

The pulses peak at En = 7.75 in the long-term time trace of Fig. (3a), while from the analytical solution expressed by Eq. ( 11) we find E ≈ 6.95. This difference can be attributed to the limitation of a third-order-only development.

To verify this hypothesis, we have extended the calculations towards fifth order in field amplitude [START_REF] Meziane | Atomic, Molecular and Optical Physics[END_REF][START_REF] Ayadi | Semiconductor Lasers and Laser Dynamics III[END_REF]. Hence, we expand Eqs. (3) to fifth order (for n = 2) for the electric field and polarization and adopt the same procedure as in section 3. After calculations, we obtain the fifth components in terms of the first and the third order field amplitudes:

P5 = Γ d Γ5 f E 5 1 + g E 2 1 E3 + h E 4 1 E3 + q E1E 2 3 + s E 3 1 E 2 3 ( 12
)
E5 = -2CP5 = -2CΓ d Γ5 f E 5 1 + h E3E 4 1 + s E 2 3 E 3 1 + g E3E 2 1 + q E 2 3 E1 (13)
The parameters Γ d , Γ5, and the weight functions f , g, h, q and s, are written as:

f = - 1 16 T 3d Γ4 ℘ α 3 4∆ -2℘ ∆α4 -5∆α3 - 5 ℘ 2 ∆α4 2 (14) g = - 1 4 Γ4 - ℘ 4 ∆ + 6 ∆ + 5 ℘ ∆ 4 + Γ2 - ℘ 4 ∆ + 6 ∆ + 5 ℘ ∆ 4 (15)
h = 1 16       -Γ4T 1d ℘α 1 4∆ -2℘∆α2 -5∆α1 -5℘ 2 ∆α 2 2 -Γ2T 1d α1 -5∆ + ℘ 2∆ + 2Γ2T 1d α2℘∆ 1 + 5℘ 2 -Γ2T 3d α3 -5∆ + ℘ 2∆ + 2Γ2T 3d α4℘∆ 1 + 5℘ 2       (16)
q = Γ2 ℘ 8 ∆ -∆ + 5 ∆ ℘ 8 (17)
s = 1 16 -Γ2 T 1d α1 ℘ 2 ∆ + 2 ℘ ∆ α2 -5 ∆ α1 -α2 ℘ 2 (18)
where:

Γ2 = 2℘ ∆ ℘ 2 + 4∆ 2 , Γ4 = 4℘ ∆ ℘ 2 + 16∆ 2 , Γ5 = 1 1 + 25∆ 2 , Γ d = 1 1 + ∆ 2 + E 2 1 2 (19) T 1d = 1 (1 + ∆ 2 ) (℘ 2 + 4∆ 2 ) , T 3d = 1 (1 + 9∆ 2 ) (℘ 2 + 4∆ 2 ) (20) α1 = ℘ 2 (1 -∆ 2 ) -4℘ ∆ 2 , α2 = 1 + ℘ -∆ 2 , α3 = ℘ 2 (1 -3∆ 2 ) -8℘ ∆ 2 , α4 = 1 + 2℘ -3∆ 2 (21)
For κ = 3, ℘ = 0.1 and 2C = 10, we computed E5 to get E5 = 1.23. Thus, to the fifth order, the analytical field expansion writes as: Eqs. ( 3) to further higher orders should lead to an even better agreement between the numerical solution and the analytical expansion method we have adopted.

E (t) =

Higher order expansion

We therefore now extend the calculations above the fifth order. At first, we calculate the seventh-order components, by expanding the development in Eqs. (3) to n = 3 for the field and polarization, and inserting the obtained relations in Eqs. (1). The calculations are time-consuming but straightforward with the help of the Mathematica™ software, and one obtains the seventh-order component for the field in the form:

E7 = -2C[W7 E 7 1 + V7 E 6 1 + F7 E 5 1 + H7 E 4 1 + S7 E 3 1 + R7 E 2 1 + Q7 E1 + Z7] (23)
The complete expressions of W7, V7, F7, H7, S7, R7, Q7 and Z7 are given in Appendix A, for the interested reader who would like to make use of these developments. With these values, we obtain E7 = -3.70. Therefore, the analytical field expansion is given by the expression:

E (t) =
The field expansion then becomes:

E (t) =
The time evolution of the expression given by Eq. ( 26) is represented in Fig. (3e). Discrepancies between this semi-analytical solution and the purely numerical one given on Fig. (3a) are still noticeable, while one expected indeed a better correspondance, thanks to the extended expansion. An attempt explanation for these remaining discrepancies is that they are in fact inherent to our procedure, which is iterative, based on the successive introduction of the orders of the electric field, order after order. For example, to find E3 it is necessary to calculate E1 (as E3 depends on E1), and similarly to have the value of E5, it is required to know the value of E1 and E3 (see Eq. ( 13)). As a consequence, the residual error on a specific term calculation increases with increasing term order, and (E11 >> E1, E13 >> E1), because the dependence of one term on the previous terms is non-linear. For example, E5 shows a dependence on E1 at the fifth power, and on E3 at the second power.

The Lorenz-Haken equations form a strongly nonlinear system, and therefore, little errors on the first terms are greatly amplified on the next terms, which explains that the analytical method will diverge from the numerical one.

This establishes the limit of the method: with the approach originally depicted in Refs. [6,7,[START_REF] Meziane | Atomic, Molecular and Optical Physics[END_REF], expansion above the fifth order does not bring noticeable improvement to accuracy of the analytical expressions of the field.

We however believe that one could obtain a better accuracy between the analytical and the numerical solution if developing Eqs. (3) up to the thirteenth order (as the field spectra obtained by FFT exhibit components up to the 13 th term), taking into account all terms without truncation, and inserting the obtained expansion in the system of equations Eqs. (1). Future work will explore this hypothesis.

Conclusion

We have expanded the analytical procedure, introduced in previous work [6,7], that describes the self-pulsing regime of the single-mode homogeneously broadened laser operating in bad cavity configurations. The inclusion of the third-order harmonic term in the field expansion allows for deriving the analytical expression of the pumping rate 2Cα, that leads the laser to display period-doubling sequence. We have shown that for α < 2, the dynamical behaviour is periodic. It is important to note that constructing analytical or semi-analytical boundaries between the various periodic regions is not an easy task, because of the non-linear behaviour of the laser. Despite these difficulties, our method allows for a satisfactory localization of the periodic solution (predictable zone). We have also highlighted a divergence between the numerical solution and the analytical one when we extend the evaluation above the fifth order in the expression of the laser field. Analytical expansion is not meant to replace numerical integration of the Lorenz-Haken model, which is usually the prefered procedure, but development up to the third order gives accurate results, and may be used as an alternate to direct numerical solutions, with the advantage of greater ease of use. In that case, our approach may help to adress the physics of laser chaos, especially for nonspecialists. It is interesting to mention that the Lorenz equations represent a challenge, indexed by Smale [START_REF] Smale | [END_REF]12] in its list of the fourteen problems, which constitute, according to him, a serious challenge to mathematicians.

V7 = Γ6Γ7E5 ℘ 2 2 (-f1 -f3 + 7∆ (f2 + f4)) + ℘∆ (f2 + f4 + 7∆ (f1 + f3 )) + Γ10Γ7    ℘ 2 2 (-E5f1 -E3f3 -h5 + 7∆ (E5f2 + E3f4 + h6)) +℘∆ (3 (E5f2 + E3f4 + h6) + 21∆ (E5f1 + E3f3 + h5 ))    + Γ8Γ7E3    ℘ 2 2 (-f3 -f5 + 7∆ (f4 + f6)) +℘∆ (2 (f4 + f6) + 14∆ (f3 + f5 ))    (A11) Q7 = Γ6Γ7E5    ℘ 2 2 (-E3 q1 -E3 q5 -z3 + 7∆ (-E3q6 -E3 q2 + z4)) +℘∆ (-E3 q2 + E3 q6 + z4 + 7∆ (z3 + E3 q1 + E3 q5))    + Γ8Γ7E3    ℘ 2 2 (-z3 -E3 q1 -E5 q1 + 7∆ ( z4 + E3 q2 -E5q2)) +℘∆ (2 (E3q2 -E5 q2 + z4) + 14∆ (E3 q1 + E5 q1 + z3))    + Γ10Γ7E3 ℘ 2 2 (7 z4∆ -z3) + 3℘∆ ( z4 + 7z3∆) (A12) H7 = Γ6Γ7E5           ℘ 2 2    -E3 h1 -E5h3 -E3 h5 -h1 -s3+ 7∆ (-E3 h2 + E3 h6 -E5h4 + E5s2 + s4)    +℘∆    s4 -E3 h2 -E5h4 + E3 h6 + s2 + 7∆ (E3h1 + s1 + s3 + E5h3 + E3h5)              + Γ10Γ7    ℘ 2 2 (-r5 -E5s1 -E3s3 + 7∆ ( E5s2 + r6 + E3s4 )) +℘∆ (3 ( r6 + E5 s2 + E3s4) + 21∆ ( r5 + E5s1 + E3 s3))    + Γ8Γ7E3       ℘ 2 2 (-s3 -s5 -E3h1 -E5h1 + 7∆ (E3h2 -E5h2 + s4 + s6)) +℘∆    2 (E3h2 -E5h2 + s4 + s6) +14∆ (E3h1 + E5h1 + 2s3 + 2s5 + E3h1 + E5 h1)          (A13) S7 = Γ6Γ7E5    ℘ 2 2 (-r1 -r3 -E3 s1 -E5s3 -E3s5 + 7∆ (r4 -E3s2 -E5s4 + E3s6)) +℘∆ ( -E5s4 + E3 s6 -E3 s2 + r4 + 7∆ ( r1 + r3 + E3 s5 + E5s3))    + Γ10Γ7    ℘ 2 2 (-E3r3 -q5 -E5r1 + 7∆ (q6 + E3r4)) +3℘∆ (q6 + E3r4) + 7∆ (q5 + E5r1 + E3r3)    + Γ8Γ7E3    ℘ 2 2 (-r5 -E3s1 -E5s1 -r3 + 7∆ (r4 + E3s2 -E5s2 + r6)) +℘∆ (2 (E3s2 -E5s2 + r4 + r6) + 14∆ ( r3 + E5 s1))    (A14) R7 = Γ6Γ7E5    ℘ 2 2 (-q1 -E3r1 -E5r3 -E3r5 + 7∆ (q2 -E5r4 + E3r6)) +℘∆ ( q2 -E5 r4 + E3r6 + 7∆ (E3 r1 + E5r3 + E3 r5 + q1))    + Γ10Γ7E5 ℘ 2 2 (-q1 + 7∆q2) + 3℘∆ (q2 + 7∆q1) + Γ8Γ7E3    ℘ 2 2 (-q5 -E3r1 -E5r1 + 7∆q6) +℘∆(2 q4 + 3 q6 + 14∆ ( q5 + E3 r1 + E5r1))    (A15) Z7 = Γ6Γ7E 2 5 2 ℘ 2 (-z3 -7 z4∆) -2℘∆( z4 + 7z3∆) (A16) f3 = Γ3Γ d       Γ2    T 1d -℘α 1 32∆ + 3 α 1 ∆ 16∆ + ℘ α 2 ∆ 8 + 3 ℘ 2 α 2 ∆ 16 +T 3d -℘ α 3 32∆ + ℘α 4 ∆ 8 + 3℘ 2 α 4 ∆ 16 + 3 α 3 ∆ 16    +Γ4T 3d 3 α 3 ∆ 16 -℘ α 3 64∆ + ℘ α 4 8 + 3℘ 2 α 4 ∆ 32       (A34) h3 = Γ3Γ d T 1d E3    Γ2 -℘ α 1 32∆ + 3 α 1 ∆ 16 -℘ α 2 ∆ 8 -3 ℘ 2 α 2 ∆ 16 +Γ4 ℘ α 2 ∆ 8 + 3 ℘ 2 α 2 ∆ 32 -℘ α 1 64∆ + 3 α 1 ∆ 16 -α 1 8    (A35) s3 = Γ3Γ d - E 2
+ ℘ α 4 ∆ 2 8 -Γ 3 α 3 ∆ 8       (A40) s2 = Γ1Γ d       Γ2    1 4 + γ 4 + T 1d    E 2 3 α 1 16 -℘ α 1 32 -℘ 2 α 2 16 -℘ α 2 ∆ 2 8 -∆ 2 4       +Γ4T 1d E 2 3 -α 1 16 -℘ α 1 64 -℘ 2 α 2 32 + ℘ α 2 ∆ 2 8 + ∆ 2       (A41) q2 = Γ1 Γ d E
-℘ α 3 32∆       (A43) h1 = Γ1Γ d E3       Γ2    T 1d -℘ α 1 16∆ -℘ 2 α 2 ∆ 8 + T 3d -℘ α 3 32∆ -α 3 ∆ 16 + ℘ α 4 ∆ 8 -℘ 2 α 4 ∆ 16    +Γ4T 3d ℘ 2 α 4 ∆ 32 -℘ α 3 64∆ + α 3 ∆ 16 + ℘ α 4 ∆ 8 -α 3 T 3d 8       (A44) s1 = Γ1Γ d       Γ2    ℘ 8∆ -∆ 2 -℘ ∆ 8 + T 1d E 2 3 -℘α 1 32∆ -α 1 ∆ 16 -℘ α 2 ∆ 8 + ℘ 2 α 2 ∆ 16    +Γ4T 1d E 2 3 ℘ α 2 ∆ 8 + ℘ 2 α 2 ∆ 32 -℘ α 1 16∆ + α 1 ∆ 16 + 1 2       (A45) r1 = Γ1Γ2Γ d E3 4 ℘ 1 ∆ + ∆ (A46) q1 = Γ1 -1 + Γ d E 2 3 Γ2 ℘ 8∆ + ∆ 2 - ℘∆ 8 + Γ4 ℘ 16∆ - ∆ 2 - ℘ ∆ 16 (A47)

First, the laser

  displays a symmetric and period-one solution (Fig. (1a)). This solution has been called symmetrical because of the symmetry of E(t) with respect to E = 0: the projection of the trajectory onto the (E, D) plane produces symmetrical loops as shown in Fig. (1b). For increasing excitation level 2C (2C = 18.4), the previously symmetric, period-one solution becomes asymmetric. The field amplitude undergoes a symmetry-breaking transformation with different positive and negative excursions (Figs. (1c,1d)). For 2C = 29, the laser exhibits a period-two, asymmetric solution (Figs. (

Figure 1 :

 1 Figure 1: Examples of Eqs. (4) solutions for (κ = 3,℘ = 0.1). Left column: time dependence of the electric field amplitude. a) Symmetric solution with period one, obtained with 2C = 18.4. b) Asymmetric, period-one solution, obtained with 2C = 20. c) Asymmetric solution with period two, obtained with 2C = 29. d) Asymmetric solution with period four, obtained with 2C = 29.7. e) Chaotic solution. Right column: attractor projection onto the (D, E)-plane. One can distinguish a cascade of period doublings, which leads to chaos.

Figure 2 :

 2 Figure 2: a) Predictable and unpredictable zone. b) Exemple of periodic solution in the region C2 < C 2th . c) Exemple of chaotic pulsation for C2 > C 2th .

  5.19 cos(0.42 t) + 1.72 cos(3 × 0.42 t) + 1.23 cos(5 × 0.42 t) (22) and is represented in Fig. (3c). Compared to the signal of Fig. (3a), the time trace of Fig. (3c) exhibits thinner and higher peaks, with values approaching those of their numerical counterpart in Fig. (3a). A priori, expanding

For κ = 3 ,

 3 ℘ = 0.1, and 2C, one obtains: W7 = 8.27 × 10 -9 , V7 = 1.06 × 10 -5 , F7 = 5.19 × 10 -5 , H7 = 5.21 × 10 -4 , S7 = 1.84 × 10 -3 , R7 = -1.46 × 10 -2 , Q7 = -5.80 × 10 -2 , Z7 = -3.64 × 10 -3 .

  [START_REF] Bougouffa | [END_REF].19 cos (0.42 t) + 1.73 cos (3 × 0.42 t) + 1.23 cos (5 × 0.42 t) -3.70 cos (7 × 0.42 t) (24)The evolution of this expression against time is shown in Fig.(3d). We note that there still is a disagreement between Fig.(3a) and Fig.(3d), owing to the negative sign of the seventh component. To correct this variance, one should take into account higher order components in the analytical expression of the field. Analytical evaluation of the higher terms of the field is however very lengthy and tedious, so we propose to continue the evaluation in a semi-analytical, semi-numerical way. This implies that we inject the values of parameters κ, ℘ and 2C in Eqs.(1) and Eqs. (3), and then evaluate the field components numerically by adopting the same procedure as before. The field spectra obtained by FFT exhibit components up to the 13 th term. We therefore compute the numerical components up to this order. For κ = 3, ℘ = 0.1 and 2C = 10, such a procedure directly yields the higher-order terms: E9 = -64.97, E11 = 1275.46 and E13 = 8.53 × 10 9

Figure 3 :

 3 Figure 3: a) Long-term time dependence of the electric-field amplitude. Analytical solutions representation for b) third-order, c) fifth-order, d) seventh-order, and e) thirteenth-order for laser-field amplitude for κ = 3, ℘ = 0.1 and 2C = 10.

  5.19 cos (0.42 t) + 1.73 cos (3 × 0.42 t) + 1.23 cos (5 × 0.42 t) -3.34 cos (7 × 0.42 t) -64.97 cos (9 × 0.42 t) + 12759.46 cos (11 × 0.42 t) + 8.53 10 9 cos (13 × 0.42 t)

+

  T 3d -α 3 16 -℘ α 3 32 -℘ 2 α 4 16 + ℘ α 4 ∆ 2 +T 3d Γ4 -α 3 16 -℘ α 3 64 -℘ 2 α 4

				3 T 3d α3 8	+ Γ2	℘ 8∆	-	3℘ ∆ 8	-∆	(A36)
	r3 = Γ3Γ d E3	1 2	+ Γ2	℘ 8∆	-	∆ 2	+	3 ℘ ∆ 8	+ Γ4 -	3∆℘ 16	+	℘ 16∆	-∆	(A37)
	z3 = -E3Γ3														(A38)
														
	f2 = Γ1Γ d	  Γ2	 	T 1d -α 1 16 -℘ α 1 32 + ℘ α 2 ∆ 2 8	-℘ 2 α 2 16	  -	T 1d α1∆ 8	 	(A39)
															8
	h2 = Γ1Γ d E3	     	Γ2	  	16 + ℘ 2 α 2 8 T 1d -℘ α 1 T 3d α 3 16 -℘ α 3 32 + ℘ 2 α 4 + 16	8 + ℘ α 4 ∆ 2	  
															32

Appendix A: Derivation of the Seventh-Order Field Component

We give here the weight functions of the analytical expression of the electric field E(t) at seventh order E7(t) given by Eq. ( 23). This expression have been derived from Eq. ( 3), expanded up to n = 3, inserting these equations into the Lorenz-Haken equations Eqs. (1a-1c) and identifying the same order-harmonic terms in each relation using the Mathematica™ software. We find a hierarchical set of algebraic relations:

The seventh order field component is related to the first component through:

with:

Where:


F Mauger

A D Bandrauk

Electronic dynamics and frequency-dependent effects in circularly polarized strong-field physics

Keywords: numbers: 32.80.Rm, 32.80.Wr, 42.50.Hz

We analyze, quantum mechanically, the dynamics of ionization with a strong, circularly polarized, laser field. We show that the main source for non-adiabatic effects is connected to an effective barrier lowering due to the laser frequency. Such non-adiabatic effects manifest themselves through ionization rates and yields that depart up to more than one order of magnitude from a static-field configuration. Beyond circular polarization, these results show the limits of standard instantaneous -static-field like -interpretation of laser-matter interaction and the great need for including time dependent electronic dynamics.

Strong-field physics, which analyzes the interaction between very strong ultra-short laser pulses and atoms or molecules, attracts an increasing interest as new tools to probe matter at the spatio-temporal resolution of the electronic dynamics in these systems in the nonlinear non-perturbative regime [1][2][3]. The intrinsic coherence of the recollision mechanisms [4,5] now allows for tomographic orbital imaging [6][7][8] and promises for retrieving spatial geometrical properties [9][10][11] of molecules: Upon electronic recollision, an instantaneous snapshot of quantum structures is imprinted in the sub-products of the laser-matter interaction. Yet, since the laser has the ability to distort orbitals, the question of what is imaged exactly remains. With linear polarization, this difficulty is often eluded by arguing that recollision, i.e., the time at which the imaging occurs, mostly corresponds to a zero of the field [6] and unperturbed orbitals are retrieved. In this Letter, using the specific example of circular polarization (CP) excitation ionization, we show that one cannot disregard dynamical effects due to the overall interaction with the laser and that these effects manifest themselves through frequency-dependent non-adiabatic effects.

A large body of literature has been dedicated to the computation of strong-field ionization rates, following the seminal works of Keldysh [12], Perelomov-Popov-Terentev [13], Ammosov-Delone-Krainov [14] and later including empirical corrections guided by numerical integration of the time-dependent Schrödinger equation (TDSE) [15,16]. For typical experimental configurations, the difficulty for defining ionization rates comes from the constantly changing amplitude and/or direction of the laser field. Following the aforementioned instantaneous picture of the laser effect, a usual shortcut consists of an adiabatic approximation and to consider, at each time, as if the system were subjected to a static field. Experiments with near circular polarization and Ar have revealed ionization statistics [17] incompatible with such an adiabatic picture [18] while other experiments with He tend to validate it [19]. In this Letter, we reconcile these two results by showing that non-adiabatic effects are indeed at play but might be too weak to be observed with He.

Among all laser polarizations, CP is unique in that the amplitude of the laser is constant and only its direction changes with time. As a consequence, an adiabatic dynamics would result in ionization rates and yields that do not depend on the laser frequency. Far from that, in this Letter, we make use of this symmetry to identify real laser-induced non-adiabatic effects, which are defined as any deviation from a purely static (DC) field with similar amplitude. More specifically, we show that Coriolis effects associated with the laser rotational dynamics induces an effective barrier lowering, as illustrated in Fig. 1, and we study its impact on ionization.

In the dipole approximation, and using atomic units (unless otherwise specified), the Hamiltonian operator corresponding to the interaction between an atomic or molecular system and an external laser field reads

Ĥ = - ∆ 2 + V (x) + f (t) E 0 √ 1 + ε 2 (cos ωt x + ε sin ωt ŷ) , (1)
where f is the laser envelope, I 0 ∝ E 2 0 its intensity with ellipticity ε (0 ≤ ε ≤ 1 spans linear to circular polarization) and frequency ω. In this Letter, we consider rotationally symmetric potentials V (x) = V (|x|), where |•| is the Euclidean norm. We use soft-Coulomb potentials [20] to model atomic systems in reduced two dimensions V (|x|) = -Z eff / |x| 2 + a 2 where the effective charge Z eff and softening parameter a are chosen such as to reproduce the appropriate eigenstate in the free-field configuration [START_REF]Atomic data[END_REF]22]. A quick glance at Eq. ( 1) reveals two possible sources for dynamical effects in the associated electronic evolution, each one with a different timescale. The laser envelope f defines the absolute pulse duration and therefore corresponds to the "slow" timescale; comparatively, the laser frequency defines a "fast" time-scale. In what follows, we investigate the respective role of each time-scale on the ionization dynamics and show that the main source for non-adiabatic effects is frequency related.

For the sake of simplicity, we treat separately the analysis of the two time-scales and begin with the "fast" one, associated with the laser frequency. For this purpose, we consider the idealistic situation of a constant envelope (f = 1) such that all effects related to the "slow" timescale are canceled. For a CP laser field (ε = 1), moving the system into a frame rotating with the laser field maps the Hamiltonian (1) to

Ĥ = - ∆ 2 + V (x) -ω Lz + E 0 √ 2 x, (2)
where tildes stand for rotating frame coordinates, and Lz = -i x∂ ỹ -ŷ∂ x is the angular momentum. The price to pay for moving the analysis into the rotating frame is the introduction of a Coriolis effect (-ω Lz ) but it has the invaluable advantage of removing all explicit time dependence and make the Hamiltonian operator autonomous. In such a configuration, from the theoretical point of view, ionization rates can be defined rigorously and are related to specific generalized eigenstates with complex energy E r = E -iΓ/2 called resonances [START_REF] Reinhardt | [END_REF].

The scalar E is usually referred to as the energy of the resonance while Γ (≥ 0) is its width or ionization rate. Indeed, for a resonance |ψ r , combining the generalized eigenstate relation Ĥ |ψ r = E r |ψ r with the Schrödinger equation leads to an exponential decrease of the electronic density with rate Γ. In this Letter, all resonances are computed numerically through a partial diagonalization of a discretized representation of the Hamiltonian operator (2) using complex coordinates [START_REF] Reinhardt | [END_REF] and a thick restart Arnoldi strategy [24]. Resonances are found by continuously varying the laser parameters (E 0 and ω) from the relevant free-field eigenstates. Ionization rates are further confirmed by numerical integration of the TDSE, in the static frame, using the Hamiltonian operator (1) (see insets of 4) and continuous curves to the best fit between low and high intensity regimes fits (see text). For comparison, markers show the ionization rates directly computed from numerical integration of the TDSE with Hamiltonian (1).

Hamiltonian (2) corresponds to the free-field problem to which are added the laser frequency and amplitude effects, respectively parametrized by ω and E 0 . It explains how resonances can be followed from free-field eigenstates in the limit ω → 0 and E 0 → 0. Besides, for a fixed laser frequency, i.e., a given laser source, two intensity regimes are expected in the ionization dynamics: In the low intensity regime (E 0 ≪ ω) Coriolis effects dominate while in the high intensity regime (E 0 ≫ ω) static field-like effects take the lead, as we shall see in what follows. The transition between the two regimes depends on the target and initial state, through the principal (state symmetry, n) and magnetic (angular momentum, m) quantum numbers. We attribute those effects to strong non-adiabatic manifestations in the electronic dynamics, which cannot be neglected in standard experimental setups.

We compare, in Fig. 2, the ionization rates for He (s state, zero initial angular momentum m = 0) and Ar ion (p state, nonzero initial angular momentum) models for static and 800 nm wavelength with CP. Non-adiabatic effects manifest themselves through ionization rates that depart up to more than one order of magnitude from static-field results, in the low intensity regime. We attribute these overwhelming effects to an effective bar-rier lowering induced by the rotational motion associated with CP (see Fig. 1). This effect is best understood considering the classical counterpart to Eq. ( 2). In the rotating frame, the classical Hamiltonian reads [26,27] H (x, p) = |p|

2 2 + V (x) -ω Lz + E 0 √ 2 x,
where x and p are canonically conjugated position and momentum, and the angular momentum Lz = xp yỹ px . From there, the limits of the classically accessible domain to the electron dynamics are charted by the socalled zero-velocity surface (ZVS) [26][27][28] of equation

Z ω (x) = - ω 2 2 |x| 2 + V (x) + E 0 √ 2 x, (3)
which is deduced from the classical Hamiltonian through a non-canonical change of variables to position-velocity coordinates [Z ω (x) = H x, ẋ = 0 ]. We see that the laser frequency induces a barrier lowering (-ω 2 |x| 2 /2 term), increasing with the frequency, irrespective of the polarization direction, left or right: Here, the rotational symmetry of the potential is incompatible with circular dichroism -defined as an asymmetry between left and right polarizations -and that, here, can only arise from dynamical effects, e.g., through the initial state configuration. More precisely, the ionization barrier is defined by a saddle point, in phase space [26,27], whose position corresponds to a saddle on the ZVS [28]. We define Z * ω as the energy of the ionization barrier and ∆Z ω = Z * 0 -Z * ω the barrier lowering induced by the laser frequency, compared to the static field configuration ω = 0 (see Fig. 1). Numerical analysis of reduced two-dimensional models reveals that non-adiabatic effects can be factorized into a correction to static field ionization rates

Γ ω (E 0 ) ≈ Γ 0 (E 0 ) exp (β∆Z α ω ) , (4)
for some constants α and β that depend on the intensity regime (low intensity, Coriolis effect dominating, or high intentity, static-field like) and on the initial state (quantum numbers), i.e., the atomic properties, for each species. The simplicity of formula (4) along with its robustness as the wavelength and target species are varied [START_REF]For the sake of shortness of this Letter, we only[END_REF] show the central role played by the barrier lowering effect associated with the CP induced rotational electronic dynamics. The exponential dependence, further amplified by the fact that ionization yields themselves depend exponentially on the rates, is at the heart of frequency induced non-adiabatic effects in strong field physics.

In order to substantiate the factorized expression of formula (4), we begin the analysis with the simplest configuration of a symmetric (s) initial state and take a He model atom. This state is non-degenerate and has zero angular momentum (m = 0). Therefore, looking at Hamiltonian (2) no circular dichroism in the ionization rate is expected. Numerical simulations confirm that the ionization rate only depends on the magnitude of the laser frequency ω and field amplitude E 0 . In this configuration, analysis of the ionization rates shows that the barrier lowering effect enhances ionization for all intensities and the difference gets more pronounced for low intensities (see markers in the left panel of Fig. 2). At 800 nm, the fitting parameters in Eq. ( 4), obtained through a linear regression of ionization rates deduced from the resonances, are α = 3.17 and ln (β) = 11.7 and yield almost a perfect match with a direct computation of the ionization rate (compare markers and the curve in the Fig. ). Further analysis reveals that, for a fixed laser intensity, nonadiabatic effects get stronger when the laser frequency (resp. wavelength) is increased (resp. decreased), as can be expected from Eq. ( 2).

We now consider the situation of a degenerate p state with non-zero angular momentum and consider Ar ion model. In this situation, Hamiltonian (2) shows that the polarization direction, i.e., the sign of ω, matters in the ionization dynamics and circular dichroism is expected in ionization rates. Equation ( 4) suggests comparison of the ionization process with the static field situation. Static fields lift the state degeneracy and an aligned (p x ) orbital is preferred compared to the orthogonal (p y ) state configuration for ionization (see light markers in the right panel of Fig. 2). Turning on the laser frequency shows that negative, m = -1, (resp. positive, m = 1,) angular momentum is connected to the p y (resp. p x ) eigenstate: Examining the corresponding generalized eigenstates shows that they change their shape to converge towards resonances with the identified orientation (x-or y-alignment). As introduced previously, the low intensity regime is dominated by Coriolis effects such that the barrier lowering induces ionization rate enhancement, irrespective of the angular momentum. However, the effect is far more pronounced when the Coriolis effect and natural electron rotation coincide, i.e., for negative angular momentum (m = -1, with ω > 0), than it is with opposite effects, leading to higher ionization rates (see dark markers) [17,18]. At 800 nm, the fitting parameters in Eq. ( 4) are α = 1.66 and ln (β) = 6.56 for negative angular momentum (m = -1) and α = 5.23 and ln (β) = 14.6 for positive (m = 1) one. On the other hand, the high intensity regime is dominated by staticfield like effects, such that ionization rate curves intersect and reverse their order. For p y states (connected to negative angular momentum, m = -1) the laser induced and natural electron rotation still coincide, leading to a favorable ionization situation and thus a positive β parameter. At 800 nm, we find α = 3.69 and ln (β) = 12.98 as a best fit to Eq. ( 4). On the contrary, for a p x state (connected to positive angular momentum, m = 1) the two rotational effects counteract, leading to a slightly smaller ionization rate and thus negative β parameter.

At 800 nm, we find α = 3.41 and ln (-β) = 9.39 as best fit to Eq. ( 4). Comparing markers and their corresponding curves on the right panel of Fig. 2, we notice the good agreement with the result of Eq. ( 4) both in low and high intensity regimes. Similarly to the initial s state configuration, further analysis shows that all the aforementioned effects get more pronounced as the laser frequency is increased, in agreement with Hamiltonian (2). Finally, as can be expected from the expression of Hamiltonian (2), the transition between low intensity regime, dominated by Coriolis effects, and high intensity regime, where a static-field like configuration takes the lead, shifts towards higher intensities when the laser frequency is increased. To conclude, the present "fast" time-scale analysis clearly shows that strong, frequencydependent, non-adiabatic effects are commonly at play in strong-field physics.

Finally, we consider the "slow" time-scale and reintroduce the envelope f (t). The change of coordinates into the rotating frame yields a non-autonomous system [the amplitude E 0 / √ 2 is replaced with f (t) E 0 / √ 2 in Hamiltonian (2)] and the rigorous ionization rate definition, based on resonances, breaks down. Yet, the slow variation of the envelope advocates for an adiabatic treatment where the instantaneous effective ionization rate is defined as the one with corresponding laser frequency and amplitude Γ (t) = Γ ω (f (t) E 0 ). Then, neglecting the probability for recapture, the ionization probability is solution of Ṗ (t) = (1 -P (t)) Γ (t) leading to

P (t) = 1 -exp - t -∞ Γ (s) ds , (5)
assuming zero initial ionization. It is important to note that, here, the adiabatic approximation is taken on the envelope solely while frequency-dependent non-adiabatic effects are fully included. In Fig. 3, we compare ionization yields with a cosine square envelope and various pulse durations to direct integration of the TDSE with Hamiltonian (1). Overall we notice the very good agreement with Eq. ( 5); only for the shortest pulses do the results slightly depart. For He, frequency-dependent nonadiabatic manifestations, in the near infrared regime, are limited (see insets) such that they might not be accessible to current experimental setups [19]. On the other hand, for Ar ion, because of the non-zero angular momentum, non-adiabatic effects associated with the laser frequency play a dramatic role, eventually leading to orders of magnitude differences in the ionization yields (compare dark and light data) and eventually leading to experimentally observable manifestations [17].

In summary, we have shown that non-adiabatic effects, mostly due to the laser frequency, are responsible for dramatic changes in ionization rates and yields with CP. It clearly shows that for typical experimental configurations -near infrared laser, rare gas target -an instantaneous representation of the quantum mechanical system, FIG. 3. (color online) Ionization yield for cosine square envelope for Ar ion (main panels, blue and red data online) and He (insets, green) models. Marker code (color and shade) is the same as in Fig. 2 and corresponds to numerical integration of the TDSE with Hamiltonian (1). For comparison, curves correspond to the ionization yields of Eq. ( 5).

as if the laser field is static, is limited. Dynamical effects resulting from the overall interaction with the laser cannot be disregarded. Beyond CP, our results call for further and more careful analysis of laser-matter interaction, both from the theoretical and experimental points of view, and that for all polarizations, including linear as is commonly considered in strong-field physics.

FIG. 1 .

 1 FIG. 1. (color online) Barrier lowering induced by a CP laser field. Insets display the ZVS Eq. (3) for static (left) and rotating (right) frames. Main panels show the corresponding section ỹ = 0 (thick red curve on the insets) passing by the ionization barrier (saddle, see text).

Fig. 2 )

 2 [25].

FIG. 2 .

 2 FIG. 2. (color online)Main panels: Ionization rate for He (left) and Ar ion (right) models. In each panel, dark (resp. light) markers correspond to 800 nm wavelength (resp. static field) and curves draw the ionization rate given by Eq. (4). Circles (green online) correspond to s initial state, triangles (blue) correspond to px (connected to positive angular momentum, m = 1, see text) initial state while inverted triangles (red) correspond to py (connected to negative, m = -1, angular momentum) initial state. Insets: Ionization rate fit curves. Dashed curves correspond to Eq. (4) and continuous curves to the best fit between low and high intensity regimes fits (see text). For comparison, markers show the ionization rates directly computed from numerical integration of the TDSE with Hamiltonian (1).

  F.M. acknowledges enlightening discussions with E. Soccorsi and O. Atabek. The authors thank RQCHP and Compute Canada for access to massively parallel computer clusters and the CIPI for financial support in its ultrafast science program. The authors acknowledge financial support from the Centre de Recherches Mathématiques. F.M. acknowledges financial support from the Merit Scholarship Program for Foreign Student from the MESRS of Quebec. A.D.B. acknowledges financial support from the Canada Research Chair.


R Lopes

A Imanaliev

M Bonneau

J Ruaudel

M Cheneau

D Boiron

C I Westbrook

The 2nd order coherence of superradiance from a Bose-Einstein condensate

Keywords: numbers: 03.75.Kk, 67.10.Jn, 42.50.Lc

We have measured the 2-particle correlation function of atoms from a Bose-Einstein condensate participating in a superradiance process, which directly reflects the 2nd order coherence of the emitted light. We compare this correlation function with that of atoms undergoing stimulated emission. Whereas the stimulated process produces correlations resembling those of a coherent state, we find that superradiance, even in the presence of strong gain, shows a correlation function close to that of a thermal state, just as for ordinary spontaneous emission.

Ever since the publication of Dicke's 1954 paper [1], the problem of the collective emission of radiation has occupied many researchers in the fields of light scattering, lasers and quantum optics. Collective emission is characterized by a rate of emission which is strongly modified compared to that of the individual atoms [2]. It occurs in many different contexts: hot gases, cold gases, solids and even planetary and astrophysical environments [START_REF] Letokhov | Astrophysical Lasers[END_REF]. The case of an enhanced rate of emission, originally dubbed superradiance, is closely connected to stimulated emission and gain, and as such resembles laser emission [START_REF] Siegman | Lasers[END_REF]. Lasers are typically characterized by high phase coherence but also by a stable intensity, corresponding to a Poissonian noise, or a flat 2nd order correlation function [START_REF] Loudon | The quantum theory of light[END_REF]. Here we present measurements showing that the coherence properties of superradiance, when it occurs in an ultracold gas and despite strong amplified emission, are much closer to those of a thermal state, with super-Poissonian intensity noise.

Research has shown that the details of collective emission depend on many parameters such as pumping configuration, dephasing and relaxation processes, sample geometry, the presence of a cavity, etc. and, as a result, a complex nomenclature has evolved including the terms superradiance, superfluorescence, amplified spontaneous emission, mirrorless lasing, and random lasing [2,[START_REF] Siegman | Lasers[END_REF][START_REF] Rehler | [END_REF][7][8][9], the distinctions among which we will not attempt to summarize here. The problem has recently seen renewed interest in the field of cold atoms [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. This is partly because cold atoms provide a reproducible, easily characterized ensemble in which Doppler broadening effects are small and relaxation is generally limited to spontaneous emission. Most cold atom experiments differ in an important way from the archetypal situation first envisioned by Dicke: instead of creating an ensemble of excited atoms at a well defined time and then allowing this ensemble to evolve freely, the sample is typically pumped during a period long compared to the relaxation time and emission lasts essentially only as long as the pumping. The authors of reference [10] however, have argued that there is a close analogy to the Dicke problem, and we will follow them in designating this process as superradiance.

In the literature on superradiance there has been relatively little discussion about the coherence and correlation properties of the light. The theoretical treatments we are aware of show that the coherence of collective emission can be quite complicated, but does not resemble that of a laser [2, 13,20,[26][27][28]. These results, however, were obtained for simple models that do not include all parameters relevant to laboratory experiments. Experimentally, a study performed on Rydberg atoms coupled to a millimeter-wave cavity [29] showed a thermal mode occupation, and an experiment in a cold atomic vapor in free space [24] observed a non-flat 2nd order correlation function. In the present work, we show that even if the initial atomic state is a Bose-Einstein condensate (BEC), the 2nd order correlation function looks thermal rather than coherent.

Such behavior, which may seem counter-intuitive, can be understood by describing superradiance as a four wave mixing process between two matter waves and two electromagnetic waves. The initial state consists of a condensate, a coherent optical pump beam, and empty modes for the scattered atoms and the scattered photons. If we make the approximation that the condensate and the pump beam are not depleted and can be treated as classical fields, the matter-radiation interaction Hamiltonian is given by:

Ĥ = i χ i â † at,i â † ph,i + χ * i âat,i âph,i , (1)
where â † at,i (â at,i ) and â † ph,i (â ph,i ) denote atom and photon creation (annihilation) operators for a specific pair of momenta i fixed by energy and momentum conservation and χ i is a coupling constant. Textbooks [START_REF] Gerry | Introductory quantum optics[END_REF] show that, starting from an input vacuum state, this Hamiltonian leads to a product of twomode squeezed states. When one traces over one of the two modes, α = {at, i} or {ph, i}, the remaining mode β has a thermal occupation with a normalized 2-particle or 2nd order correlator

â † β â † β âβ âβ â † β âβ 2 = 2 , ( 2
)
whereas it is unity for a laser. The problem has also been treated for four wave mixing of matter waves [START_REF] Mølmer | [END_REF]. We emphasize that, when starting from initially empty modes, the occupation remains thermal regardless of the gain. In the experiment, we start from initially nearly motionless atoms of a BEC and observe their recoil upon photon emission. To the extent that each recoil corresponds to the emission of a single photon, we can obtain essentially the same information about the radiation from such measurements as by observing it directly. In doing this, we are following the approach pioneered in experiments such as [10,29] and followed by many others, which uses highly developed atom detection and imaging techniques to glean most of the experimental information about the process. We are able to make time-integrated measurements of the emission, resolved in transverse and longitudinal momentum as well as in polarization, and reconstruct the 2-particle correlation function of the recoiling atoms, or equivalently the 2nd order correlation function of the scattered light. We will show that in the configuration of our experiment, the 2nd order correlation is close to that of a thermal sample, and very different from the correlation properties of the initial, condensed atomic state.

We use helium in the 2 3 S 1 , m = 1 state confined in a crossed dipole trap (see Fig. 1a) with frequencies of 1300 Hz in the x and y directions and 130 Hz in the (vertical) z direction. The dipole trap wavelength is 1.5 µm. The atom number is approximately 50 000 and the temperature of the remaining thermal cloud 140 nK. A 9 G magnetic field along the y axis defines a quantization axis. After producing the condensate, we irradiate it with a laser pulse of 2.4 W/cm 2 tuned 600 MHz to the red of the 2 3 S 1 → 2 3 P 0 transition at λ = 1083 nm and with natural linewidth 1.6 MHz. The excitation beam propagates with an angle of 10 • relative to the x axis and its polarization is linear, with the same angle relative to the y axis (see Fig. 1a). The pulse length is 5 µs and it is applied with a delay τ after switching off the trap. The expansion of the cloud during this delay is a convenient way to vary both the optical density and the anisotropy of the sample at constant atom number. The absorption dipole matrix element is of the σ -form and thus one half of the laser intensity is coupled to the atomic transition corresponding to a Rabi frequency of 56 MHz. The excited atoms decay with equal branching ratios to the 3 ground states. During the pulse, less than 10 % of the atoms are pumped into each of these states. Because of the polarization selection rules, the atoms which are pumped into the m = 0 state cannot reabsorb light from the excitation laser. By focusing on these atoms, we study the regime of "Raman superradiance" [15,32], by which we mean that an absorption and emission cycle is accompanied by a change of the internal state of the atom. When the trap is switched off, the atoms fall toward a micro-channel plate detector which detects individual atoms with 3 dimensional imaging capability and a 10 to 20 % quantum efficiency [33]. A magnetic field gradient is applied to sweep away all atoms except those scattered into the m = 0 magnetic sublevel. The average time of flight to the detector is 310 ms and is long enough that the atoms' positions at the detector reflect the atomic momenta after interaction with the excitation laser. Conservation of momentum then requires that these atoms lie on a sphere with a radius equal to the recoil momentum k rec = 2π/λ. Any additional scattering of light, whether from imperfect polarization of the excitation laser, or from multiple scattering by the atoms, will result in the atoms lying outside the sphere. We see no significant signal from such events, but in order to completely eliminate the possibility for multiple scattering we restrict our analysis of the data to the spherical shell with inner radius 0.8 k rec and outer radius 1.2 k rec .

We excite atoms in an elongated BEC in such a way that an allowed emission dipole can radiate along the long axis. In an anisotropic source, collective emission builds up more efficiently in the directions of highest optical thickness. Superradiance is therefore expected to occur along the long axis of the BEC, in so called "endfire" modes [10,[START_REF] Dicke | Quantum electronics; proceedings of the third international congress[END_REF]. An important parameter then is the Fresnel number of the sample [2], F = 2R 2 ⊥ /λR z , where R ⊥ and R z are the horizontal and vertical Thomas-Fermi radii of the condensate. The Fresnel number distinguishes between the diffraction limited (F < 1) and multimode superradiance regimes (F > 1). In our case, R ⊥ ≈ 5 µm and R z ≈ 50 µm, yielding a Fresnel number of about unity.

Typical cuts through the atomic momentum distribution in the yz plane are shown in Fig. 2, for τ = 500 µs (left panel) and τ ≈ 0 (right panel). In both cases, the spherical shell with radius 1 k rec appears clearly. For the short delay, when the atomic sample remains dense and anisotropic, we observe strong scattering in the endfire modes at the top and bottom poles of the sphere. In addition to this change in the profile of the distribution, we measure an increase of the total number of atoms on the sphere by a factor ∼ 5 from τ = 500 µs to τ ≈ 0. Because each atom has scattered a single photon, this increase directly reflects an increase of the rate of emission in the sample and therefore demonstrates the collective nature of the scattering process. At long delays, the condensate has expanded sufficiently that the optical thickness and anisotropy have fallen dramatically, suppressing the collective scattering. By looking at the number of scattered atoms in the x direction (perpendicular to the plane of Fig. 2), we have verified that, away from the endfire modes, the rate of emission varies by less than 10% for different delays [35].

To see the distribution in a more quantitative way, we show in Fig. 3 an angular plot of the atom distribution in the yz plane. Data is shown for three different delays τ before application of the excitation pulse. For 500 µs delay, the angular distribution follows the well-known "sin 2 θ" linear dipole emission pattern with the angles θ = 0 and π corresponding to the orientation of the dipole along the y-axis [35]. For 200 µs delay, the superradiant peaks are already visible on top of the dipole emission profile. For the shortest delay, the half width of the superradiant peaks is 0.14 k rec , or 0.14 rad, consistent with the diffraction angle and the aspect ratio of the source. In the vertical direction, the superradiant peaks are 10 times narrower than in the horizontal direction [35].

In the strongly superradiant case, we observe large and uncorrelated fluctuations of the heights of the two superradiant peaks on a shot-to-shot basis. These fluctuations directly reflect the fluctuations of the population of the superradiant modes. We investigate these fluctuations further by measuring the normalized 2-particle correlation function of the scattered atoms, defined as Here, n is the atomic density and : : denotes normal ordering. In practice, this function is obtained from a histogram of pair separations ∆k normalized to the autoconvolution of the average particle momentum distribution [START_REF] Schellekens | [END_REF]37]. Figure 4 shows the experimentally measured correlation functions integrated over the momentum along two out of three axes, both for the superradiant peaks and on the scattering sphere away from the peaks [35]. We see that in both cases the correlation function at zero separation reaches a value close to 2. This shows clearly that, despite strong amplified emission in the endfire modes, the atoms undergoing a superradiant process have statistics comparable to that of a thermal sample. As underlined in the introduction, these large fluctuations can be simply understood by modeling the superradiant emission as a four wave mixing process; they arise from the fact that the emission is triggered by spontaneous emission. For the superradiant peaks, the correlation actually is slightly larger than 2. Similar behavior has appeared in some models [20,38], but these models may not be directly applicable to our situation.

g (2) (∆k) = : n(k)n(k + ∆k) : n(k) n(k + ∆k) . (3)
Figure 4 also shows that the correlation widths of the superradiant modes are somewhat broader than those of the atoms scattered in other modes. The effect is a factor of about 1.5 in the vertical direction and about 1.25 in the horizontal direction [35]. The broadening indicates that the effective source size for superradiance is slightly smaller than for spontaneous scattering. A decreased vertical source size for superradiance is consistent with the observations of Ref. [39,40] which showed that the superradiant emission is concentrated near the ends of the sample. In the horizontal direction, one also ex-FIG. 4: (color online) Correlation functions along the z (a) and y axis (b) for τ ≈ 0. Blue circles correspond to the superradiant peaks (defined by |kz| > 0.95krec). Orange circles correspond to atoms from the scattering sphere away from the superradiant peaks (defined by |kz| < 0.92krec). Solid lines are Gaussian fits constrained to approach unity at large separation. Gray solid circles correspond to a fraction of the initial condensate transferred to the m = 0 state via a stimulated Raman transfer. The dashed gray line shows unity. Error bars denote the 68% confidence interval. pects a slightly reduced source size relative to the atom cloud since the gain is higher in the center, where the density is higher. The fact that the correlation widths are close to the widths of the momentum distribution [35] indicates that the superradiant peaks are almost single mode as expected for samples with a Fresnel number close to unity [2].

The spontaneous superradiant scattering process should be contrasted with stimulated Raman scattering. In terms of the model described by the Hamiltonian (1), stimulated Raman scattering corresponds to seeding one of the photon modes by a coherent state. In this case, vacuum fluctuations do not initiate the scattering process, and the resulting mode occupation is not thermal but coherent. To study stimulated scattering, we applied the excitation beam together with another beam polarized parallel to the magnetic field and detuned by the Zeeman shift (25 MHz) with respect to the σ-polarized beam, inducing a stimulated Raman transition. The laser intensities were adjusted to transfer a similar number of atoms to the m = 0 state as in the superradiance experiment. The normalized correlation functions in this situation, shown in Fig. 4, are very nearly flat and equal to unity as we expect for a BEC [START_REF] Schellekens | [END_REF]41,42]. The complementary experiment, seeding the atomic mode with a coherent state has also been observed to produce a coherent amplified matter wave [43,44]. As a side remark, we have also observed that the superradiant atom peaks are 2.8 times narrower in the vertical direction than the vertical width of the transferred condensate [35]. We attribute this to a longitudinal gain narrowing effect [27].

We also investigated the influence of several other experimental parameters on the 2nd order coherence of the superradiant emission: We have excited the atomic sample with a longer and stronger pulse (10 µs, 3.2 W/cm 2 ), so that the initial condensate was entirely depleted. We have explored the Rayleigh scattering regime, in which the atoms scatter back to their initial internal state. We also changed the longitudinal confinement frequency of the BEC to 7 Hz, leading to a much greater aspect ratio. These different configurations led to 2-particle correlation functions which were very similar to the one discussed above. We believe that similar fluctuations will occur in superradiance from a thermal cloud provided that the gain in the medium is large enough. We were unable to confirm this experimentally in our system, precisely because of the vastly reduced optical density. However, noncoherent intensity fluctuations have been observed already using magneto-optically trapped atoms [24]. This seems to confirm our interpretation that the large fluctuations of the superradiant mode occupation is an intrinsic property of superradiant emission, reflecting the seeding by spontaneous emission. The only way to suppress these fluctuations would be to restrict the number of scattering modes to one by means of a cavity and to saturate the gain by completely depleting the atomic cloud. The occupation of the superradiant mode would then simply reflect that of the initial atomic sample.

An interesting extension of the techniques used here is to examine superradiant Rayleigh scattering of a light pulse short enough and strong enough to populate oppositely directed modes [45]. It has been predicted [13,14,46] that the modes propagating in opposite direction are entangled, similar to those produced in atomic four wave mixing [47][48][49]. A similar measurement technique should be able to reveal them.

We acknowledge fruitful discussions with A. Browaeys, J.-J. Greffet and P. Pillet. This work was supported by the IFRAF institute, the Triangle de la Physique, the LABEX PALM, the ANR-ProQuP project, the ERC (Grant No. 267 775) Quantatop, J.R. by the DGA, R.L. by the FCT scholarship SFRH/BD/74352/2010.

I. SUPPLEMENTARY MATERIAL

Distribution of atoms in the xz plane The distribution of scattered atoms in the yz plane showed a vanishing population along the direction of the emission dipole (angles 0 and π in the Fig. 3 of the main text). In the xz plane on the other hand, the angular distribution is, as expected, uniform between the superradiant peaks, see Fig. S1. The signal is zero on one side of each superradiant peak because the atomic cloud in the xz plane is off center with respect to the detector due to the recoil from the excitation laser and the part of the distribution corresponding to k x > 0.4 k rec misses the detector as shown in Fig. S2.

Calculation of the correlation functions The quantity actually displayed in Fig. 4 of the main text is not the correlation function as defined in Eq. 1, but the one defined by Eq. S1: The volume Ω 1 is defined by the boundary conditions

g(2) (∆k z ) = ˆΩ1 d∆k x d∆k y ˆΩV d 3 k : n(k)n(k + ∆k) : n(k) n(k + ∆k) g(2) (∆k y ) = ˆΩ2 d∆k x d∆k z ˆΩV d 3 k : n(k)n(k + ∆k) : n(k) n(k + ∆k) (
|∆k x | < 3 × 10 -2 k rec , |∆k y | < 3 × 10 -2 k rec and Ω 2 by |∆k x | < 3 × 10 -2 k rec , |∆k z | < 3 × 10 -3 k rec .
Integration in momentum space is performed over a specific volume Ω V for each of the three cases showed in Fig. 4:

• superradiant peaks case: |k x | < 0.5 k rec , |k y | < 0.5 k rec and |k z | > 0.95 k rec ;
• scattered sphere away from the superradiant peaks: |k z | < 0.92 k rec and no constraint in the xy plane;

• stimulated Raman transfer: Ω V is the volume centered on the cloud with a width along z of 0.1 k rec and no constraint in the xy plane. Widths of the superradiant peaks In order to obtain the widths of the superradiant peak, we first derive the contribution of ordinary spontaneous emission. Fig. S3 displays a close up of the superradiant peak around -π/2 in the yz plane. The data corresponding to a long delay before application of the excitation pulse (green circles, τ = 500 µs) are well described by a pure spontaneous emission profile sin 2 (θ), where θ is the polar angle in the yz plane (green curve). Since the contribution of the spontaneous emission should be the same for all delays, we subtract this background from the atomic signal before fitting the distribution with a Lorentzian function. The sum of the background and the fit is also displayed in Fig. S3 (blue and red curves). The choice of a Lorentzian fitting function is empirical and we expect the exact shape of the superradiant contribution to be more complex [38]. From this fit we obtain half-widths at half-maximum of 0.14 and 0.25 rad for τ = 0 and 200 µs, respectively.

Table I summarizes the various widths measured in this experiment. The first three lines refer to the widths of the observed atomic distribution in momentum space. The "BEC" entry corresponds to the configuration in which the m = 0 sublevel of the 2 3 S 1 state was populated by stimulated Raman transfer (see main text

FIG. 1 :

 1 FIG.1:(color online) (a) Sketch of the experiment. A 9-G magnetic field B applied along the y axis defines the quantization axis. The excitation beam propagates with an angle of 10 • (not shown) relative to the x axis and its polarization is linear, with the same angle relative to the y axis. After emission, the atoms fall 46 cm to a positionsensitive micro-channel plate (MCP). The atom cloud forms a sphere with enhanced occupation of the endfire modes. (b) Atomic level scheme. The atoms, initially in the 2 3 S1, m = +1 state, are excited to the 2 3 P0 state. From there, they can decay with equal branching ratios to the 3 sub-levels of the ground state. We detect only the atoms which scatter into the m = 0 state.

  FIG. 2: (color online) Momentum distribution of scattered atoms in the yz plane (containing the emission dipole). Both figures show the distribution in the yz plane, integrated between kx = ±0.1 krec and summed over 100 shots. See the supplemental information for a cut in the xz plane [35]. Left: Excitation laser applied 500 µs after the trap switchoff. Only the radiation pattern for a y-polarized dipole is visible. Right: Excitation laser applied immediately after the trap switchoff. Strong superradiance is visible in the vertical, endfire modes.

FIG. 3 :

 3 FIG. 3: (color online) Angular distribution of scattered atoms in the yz plane (containing the emission dipole) for different values of the delay τ before the excitation pulse. The data for τ = 0 and 500 µs are the same as those shown in Fig. 2. The images were integrated along the x-axis between ±0.1 krec and only atoms lying inside a shell with inner radius 0.8 krec and outer radius 1.2 krec were taken into account. The delays τ = 0, 200 and 500 µs correspond to peak densities of ≈ 8, 2, 0.4 × 10 18 m -3 and to aspect ratios of 10, 5 and 2.5, respectively. The endfire modes are located at ±π/2. The half-width at half-maximum of the highest peak is 0.14 rad. Error bars are shown and denote the 68 % confidence interval.

  FIG. S1: (colour online) Angular distribution of scattered atoms in the plane perpendicular to the emission dipole for different values of the delay τ before the excitation pulse. The data shown are the same as those discussed in the main text.

  FIG. S2: (colour online) Momentum distribution of scattered atoms in the plane perpendicular to the emission dipole. Both figures show the atom distribution in the xz-plane, integrated between ky = ±0.1 krec and summed over 100 shots. Left: Excitation laser applied 500 µs after the trap has been switched off. Only the radiation pattern for a y-polarized dipole is visible. Right: Excitation laser applied immediately after the trap has been switched off. Strong superradiance is visible in the vertical, endfire modes.

  FIG.S3: (colour online) Close up of the momentum distribution of scattered atoms around one superradiant peak in the plane of the emission dipole (yz plane). The data shown are the same as those discussed in the main text. Plain lines are fits to the data (see text for details).

  ). * Current address: Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, 1020 Vienna, Austria. † Current address: Laboratoire Kastler Brossel, Univ. Pierre et Marie Curie -École normale supérieure -CNRS, 4 place Jussieu, 75005 Paris, France. R. H. Dicke, Phys. Rev. 93, 99 (1954). [2] M. Gross and S. Haroche, Phys. Rep. 93, 301 (1982).

	‡ Electronic	address:	christoph.westbrook@
	institutoptique.fr
	[1]

TABLE I :

 I Half-widths at half-maximum for density and correlation function in units of krec. The number in parenthesis denotes the uncertainty on the last digit.

	Configuration	vertical horizontal
	BEC density	0.039(1)	0.190(2)
	Superradiance density, τ = 0	0.014(2)	0.14(2)
	Superradiance density, τ = 200 µs	0.032(2)	0.27(2)
	Superradiance correlation, τ = 0	0.021(2)	0.15(1)
	Scattered sphere correlation, τ = 0	0.014(2)	0.12(2)