R Haar

A Hentges

Ehresmann

K Jänkälä

Ph V Demekhin

S Heinäsmäki

I Haar

R Hentges

A Ehresmann

Theoretical description of dissociative 2 scattering in diatomic molecules

The paper describes the theoretical formulation of dissociative γ → 2γ scattering in fixed-in-space and randomly oriented diatomic molecules in the dipole approximation for excitation and 2γ production. Based on perturbation theory the double differential cross section and two-photon angular correlation function of the process is derived. The developed theory is applied to study the angular correlation function of the two Lyman-α photons produced via decay of the doubly-excited Q 2 1 Π u (1) state of the H 2 molecule.

Introduction

Doubly-excited molecular states, where two electrons are promoted from the ground state into initially unoccupied orbitals, usually lie energetically above the first ionization and dissociation thresholds (see, e.g. the reviews [1,2] and [3]). These states are sometimes called 'superexcited' [4] and may decay either via autoionization or neutral dissociation accompanied by photon emission [1]. Due to the variety of different relaxation pathways, strong couplings with the electronic continua, and fairly small excitation cross sections, these states are generally difficult to study both, experimentally and theoretically [3]. During the last decades, superexcited states have been studied experimentally in diatomic and polyatomic molecules using several different techniques [2], such as ion spectroscopy [5], electron-energy-loss spectroscopy [6], and one fluorescence photon [7] or two fluorescence photon coincidence spectroscopies [8,9]. Particularly for the latter type of experiments theoretical approaches as scarce [10]. Therefore, in the present paper, the main goal is an extended theoretical description of the two-photon coincident emission from dissociation fragments of onephoton superexcited states in diatomic molecules.

One-photon doubly-excited molecular states in diatomic molecules leading after neutral dissociation to an emission of two photons may be divided in two groups. In the first group, dissociation of a molecular superexcited state results in excited atomic fragments, with one in a metastable state. The decay of the Q 1 1 Π state of H 2 into the H(2p) and H(2s) fragments [7] is an example of this type. The radiative decay of the metastable state takes place mainly due to collisional quenching (see, e.g. [11] and references therein). Therefore, the angular correlation between the two emitted photons in this case does not carry precise information on the entangled atomic fragments. In the second group, dissociation of a superexcited state forms two atomic fragments in excited states decaying via dipole transition operator. The Q 2 1 Π u (1) superexcited state of H 2 dissociating into the H(2p) and H(2p) fragments [8] belongs to this second group of processes. Such a two-photon emission process is very interesting, because the two photons carry now detailed information on the structure and de-excitation dynamics of the doubly excited molecular state. The two photons are entangled, since they are emitted from an entangled excited-atom pair, even if the atomic fragments are far from each other. Importance of entanglement of atomic fragments and its impact on correlation in the two photons emission has been pointed out in [10]. Below we focus on the decay of superexcited states within this second group.

The possibility to experimentally detect the two photons in coincidence emitted via the dissociation of superexcited states into two excited fragments were shown in the late 1980's by Arai et al [8] in their pioneering experiments. The authors recorded the absorption profile of the Q 2 1 Π u (1) state of H 2 using the two-photon coincidence technique. The experiment is very demanding mainly due to the requirement of coincidence detection and small excitation cross section. Thus, the technique was unused for more than 25 years. Higher photon flux of the modern synchrotron radiation sources provides more feasible framework for this method, and the task was recently revived by Odagiri et al [9] by repeating the original experiment on H 2 . Since then, similar photon-photon coincidence experiments have been carried out by the same group also for the homonuclear molecules N 2 [12,13] and O 2 [14], and the heteronuclear molecule NO [15]. Recently, this group published also the first experimental work devoted to the measurement of the angular correlation function (ACF) of the two photons emitted after the neutral dissociation of the Q 2 1 Π u (1) state of H 2 [16]. The theoretical description of doubly-excited states in molecules is demanding, and so far mainly carried out for H 2 . As an example, Guberman [17] has calculated energies and Bottcher [18] potential energy curves and transition moments for several doublyexcited states in H 2 . Calculations of the exciting photon energy dependence of the absorption cross section based on semiclassical approximation were reported in [9,19]. The full quantum mechanical treatment of the problem is presented by Borges and Bielschowsy in [20]. To the best of our knowledge, so far only one theoretical work has been devoted to the derivation of the ACF of two photons emitted after dissociation of a superexcited molecular state [10]. The two-photon ACF obtained by Miyagi et al [10] is based on the second-order correlation function in quantum optics and was used to study solely H 2 . Their approach, based on the analysis of the symmetry of the dissociating doubly-excited state is elegant. However, it is challenging to extend it to take into account different polarizations of the exciting radiation, general molecular states, orientations of a molecule and decays to different spin-orbit components.

In this paper we present a theoretical description of one-photon excitation of doubly excited molecular states, their dissociation into two excited atomic fragments, and the following two-photon emission of diatomic molecules. Our model is based on perturbation theory and can be readily applied to all diatomic molecules. The model allows calculations with variable polarization of the incoming radiation, and polarization sensitivities of the photon counters. The theory is applied to study the ACF of the two Lyman-α photons produced via excitation and decay of the Q 2 1 Π u (1) state of the fixedin-space and randomly oriented H 2 molecules. The ACF is studied in different cases and compared to the existing theoretical and experimental data. Atomic units are used throughout the paper unless otherwise stated.

Theory

In the following section we formulate the double differential cross section for one photon excitation and two photon decay of doubly-excited states of diatomic molecules. We are, thus, interested in the following process:

AB + hν i → (AB) * * → A * + B * → A + B + hν f + hν f , (1)
where the incoming photon hν i excites molecule AB from the initial state to the doublyexcited state (AB) * * . The state dissociates into the excited atomic fragments A * and B * , which emit subsequently two photons proceeding into stable states A and B. Thereby, we assume that the radiative decay is slow compared with the dissociation. It is also assumed, that the doubly-excited resonance is isolated allowing lifetime interference effects to be neglected. A schematic view of the coordinate system used in the calculations is shown in figure 1.

Wave functions

The molecular wave functions are described in Hund's coupling case (a) [START_REF] Lefebvre-Brion | The Spectra and Dynamics of Diatomic Molecules[END_REF][START_REF] Brown | Rotational Spectroscopy of Diatomic Molecules[END_REF] by

|JMνΩ = 2J + 1 8π 2 D J * M Ω (ω)|ν |Ω , (2)
where Wigner's rotation matrix D J M Ω (ω) describes the molecular rotational motion, |ν denotes the vibrational wave function, and |Ω is the electronic part of the total wave function. Quantum numbers J and M are the total angular momentum of the molecule and its projection on the quantization axis. In Hund's case (a) Ω = Λ + Σ, where Λ and Σ are projections of the total orbital angular momentum and total spin on the molecular axis. The theory can be directly applied to other Hund's cases by using the basis transformations given elsewhere (see, e.g. [START_REF] Singer | [END_REF]). The molecular wave function (2) does not describe a defined parity of a molecular state, but due to the closure relation, the use of parity adapted wave functions becomes superfluous.

In the limit of large internuclear fragment separation, the adiabatic electronic wave functions are written as linear combinations of atomic wave functions [24] as

|CΩ r→∞ -→ jaΩaj b Ω b M(j a Ω a j b Ω b |CΩ)|c a j a Ω a |c b j b Ω b , ( 3
)
where j is the total angular momentum of an atomic fragment (a or b) and Ω is its projection on the molecular quantization axis. The atomic quantum numbers j a Ω a and j b Ω b (including their signs) are strictly selected in the sum (3) by the entanglement of the two electronic states of atoms in the overall molecular electronic state |CΩ , including signs and values of all quantum numbers of the molecular state (i.e. spin S, its projection Σ, orbital angular momentum L and its projection Λ = Ω -Σ) labeled here by index C. The coefficients M(j a Ω a j b Ω b |CΩ) are given explicitly via Clebsch-Gordan coefficients in [24].

Transition matrix elements

We start by defining the transition amplitudes for initial excitation, fragmentation and subsequent two-photon emission by a two-step cascade by

α f ; hν m hν n |H|α i ; hν i = (8π 3 ω i ω n ω m ) 1/2 × αmαnα n α f | * m • T (1) |α m α m | * n • T (1) |α n α n |H v |α n , r 0 α n , r 0 | i • T (1) |α i , r 0 (E -E m + iΓ m /2)(E -E n + iΓ n /2)(E -E n + iΓ n /2) , (4)
where α| • T (1) |α are the dipole transition matrix elements, |α denotes general molecular states, and H v is the Hamiltonian for the nuclear dynamics describing (pre)dissociation [25,26]. In (4) we have neglected the propagation of atomic fragments between the emission of the first and second photon. The exciting and emitted photons are described using a second-quantized vector potential for the electromagnetic field characterized by the creation and annihilation operators of photons with specific polarization and momentum. In the calculations of the matrix elements, the vector potential is taken in the dipole approximation. The first two terms in the denominator of the transition amplitude (4) are the diagonal matrix elements of the Green's function with the radiative linewidths Γ n and Γ m [25,26,27]. Γ n is the total decay rate of the superexcited state |α n , equal to the sum of its autoionization and (pre)dissociation rates [START_REF] Lefebvre-Brion | The Spectra and Dynamics of Diatomic Molecules[END_REF]27] neglecting direct fluorescence deexcitation. Energies E n and E m are equal to

E n r→∞ -→ E at n + ε k n and E m r→∞ -→ E at m + ε k m + hν n = E at n + ε k m
, where E at and ε k are the total electronic and kinetic energies of the fragments when r → ∞. E is the total energy of the system equal to E i + hν i .

The double differential cross section is calculated in the lowest order of perturbation theory utilizing the Fermi's golden rule dσ

2 nm dΩ n dΩ m = 2πρ α f | α f ; hν m hν n |H|α i ; hν i | 2 δ(E -E at f -ε k f -hν n -hν m ), (5)
where ρ is the product of densities of the incoming and outgoing radiation modes [26]. We proceed by inserting the wave functions (2) and matrix elements (4) into (5), and integrating over the kinetic energy of the fragments (see, e.g. [28]). It is assumed, that all symmetry quantum numbers are conserved during the dissociation and that the Franck-Condon approximation is valid on each step of the process. Averaging over the degenerate M i states and dividing by the incident photon flux we obtain on the energy shell dσ

2 nm dΩ n dΩ m = α 4 ω i ω 3 n ω 3 m 4π 2 c 3 Ĵ2 i χ diss M i J f M f Ω i Ω f JnJmMnMmΩnΩm J f M f Ω f | * m • T (1) |J m M m Ω m E t f -E t m + hν m + iΓ m /2 J m M m Ω m | * n • T (1) |J n M n Ω n E t m -E t n + hν n + iΓ n /2 × ν n , r 0 |ν i , r 0 J n M n Ω n , r 0 | i • T (1) |J i M i Ω i , r 0 2 , (6)
where the assumption of a fast homogeneous dissociation yields the branching ratio

χ diss = 2π | J n M n Ω n |H v |J n M n Ω n , r 0 | 2 /Γ n ,
which is assumed to be the same for all |J n M n Ω n states. The Franck-Condon factor for the initial excitation is denoted by ν n , r 0 |ν i , r 0 , α = 1/137 is the fine structure constant, and notation x stands for √ 2x + 1.

In order to calculate the double differential cross section for randomly oriented molecular samples, the first step consists in the analytical integration over the rotational parts. Applying (A.1) from the appendix to the dipole transition matrix elements in (6) we obtain

dσ 2 nm dΩ n dΩ m = α 4 ω i ω 3 n ω 3 m 4π 2 c 3 Ĵ2 i χ diss M i J f M f Ω i Ω f Γ -1 JnJmMnMmΩnΩm p i pnpm q i qnqm (-1) p i +pn+pm -p i * -pn * -pm Ĵf Ĵ2 m Ĵ2 n Ĵi × (-1) M f +Mm+Mn-Ω f -Ωm-Ωn ν n , r 0 |ν i , r 0 × J f 1 J m -M f p m M m J m 1 J n -M m p n M n J n 1 J i -M n p i M i × J f 1 J m -Ω f q m Ω m J m 1 J n -Ω m q n Ω n J n 1 J i -Ω n q i Ω i × Ω f |T (1) qm |Ω m Ω m |T (1) qn |Ω n Ω n |T (1) q i |Ω i 2 . ( 7
)
Index p identifies tensor components in the laboratory coordinates and index q in the molecule fixed frame [START_REF] Brown | Rotational Spectroscopy of Diatomic Molecules[END_REF]. Components -p describe polarizations as well as directions of the incoming and outcoming photons and

Γ = (E t f -E t m + hν m + iΓ m /2)(E t m -E t n + hν n + iΓ n /2). (8)
Summations over all M quantum numbers were carried out analytically by applying (A.2) twice to the sum over M n and M m and using the orthogonality of the 3j symbols in summations over M f and M i . We obtain then the result

dσ 2 nm dΩ n dΩ m = α 4 ω i ω 3 n ω 3 m 4π 2 c 3 χ diss J f Ω i Ω f K Q Ĵ2 f K 2 Γ -1 K F Q KK JnJmΩnΩm ν n , r 0 |ν i , r 0 × (-1) J f +2Jm+Jn-Ωm-Ωn Ĵ2 m Ĵ2 n 1 1 K J i J m J n 1 K K J i J f J m × J f 1 J m -Ω f Ω f -Ω m Ω m J m 1 J n -Ω m Ω m -Ω n Ω n J n 1 J i -Ω n Ω n -Ω i Ω i × Ω f |T (1) qm |Ω m Ω m |T (1) qn |Ω n Ω n |T (1) q i |Ω i 2 , (9)
where

q i = Ω n -Ω i , q n = Ω m -Ω n , q m = Ω f -Ω m
and the following designation has been used

F Q KK = K2 p i pnpm (-1) p i +pn -p i * -pn * -pm 1 1 K p i p n -(p i + p n ) K 1 K p i + p n p m Q . ( 10
)
Tensor F Q KK contains all necessary information about polarizations and directions of the incoming and outcoming photons in the laboratory frame, and is the two-photon generalization of the function F K introduced in [25] for the one-photon decay. One should note that after summing over indices M i and M f , summation over index K must be performed coherently, and over K incoherently, which differs from the one-photon case where only the incoherent summation is required [25]. This leads to interference between F Q KK components with different K.

The radiative decay widths of excited atomic fragments are usually much smaller than experimental fluorescence detection resolution. This requires an integration of the double differential cross section (7) over the fluorescent photon energies hν n and hν m . This integration will eliminate the Γ -1 factor (8) from ( 7) yielding the constant 4π 2 /(Γ n Γ m ) prefactor in the integrated double differential cross section. This, however, will not change the final result for the ACF discussed below.

When only the shape of the two-photon emission pattern (i.e. ACF) is of interest and the J dependence of the vibronic molecular wave functions can be neglected, the double differential cross section can be approximated by considering only the electronic parts of the molecular wave function. Under these assumptions the double differential cross section (9) provides the complete information on the ACF, which reads

dσ 2 nm dΩ n dΩ m ∝ f (θ n , φ n , θ m , φ m ) = dω l Ω i Ω f | ΩnΩm p i pnpm (-1) p i +pn+pm -p i * -pn * -pm × D (1) p i q i (ω l )D (1) pnqn (ω l )D (1) pmqm (ω l ) Ω f |T (1) qm |Ω m Ω m |T (1) qn |Ω n Ω n |T (1) q i |Ω i | 2 , ( 11
)
where ω l = (φ l , θ l , 0) specifies the orientation of the internuclear axis with respect to the laboratory coordinates (see, figure 1). Equation ( 11) is the two-photon generalization of the original one-photon emission result by van Brunt and Zare [29] (see also [25]).

Description of photons

An explicit expression for of the polarization dependent two-photon ACF can be calculated by introducing the general polarization vector = cos ξˆ 1 + sin ξe iδ ˆ 2 (12) in the detector frame [30] (see, figure 1). Terms ξ and δ parametrize the polarization response of the detector in such a way that ξ is the angle between the polarization vector and ˆ 1 , and δ is the phase between ˆ 1 and ˆ 2 shown in figure 1. Choosing δ = 0 denotes detector sensitivity to linear polarization with polarization angle ξ, and δ = ±π/2 with ξ = π/4 gives the right-or left-handed circular polarization. Integration over the polarization sensitivities ξ and δ corresponds to polarization insensitive detector. Basis vectors ˆ 1 and ˆ 2 are the spherical unit vectors

ˆ 1 = cos θ cos φx + cos θ sin φŷ -sin θẑ, ˆ 2 = -sin φx + cos φŷ ( 13
)
in the laboratory frame. Introducing the usual definition of the helicity basis vectors

ê±1 = ∓ 1 √ 2 (x ± iŷ); ê0 = ẑ, (14)
polarization vector can be expressed in the helical basis as

= - 1 √ 2 (cos ξ cos θ -i sin ξ)e -iφ ê+1 + 1 √ 2 (cos ξ cos θ + i sin ξ)e iφ ê-1 -cos ξ sin θê 0 . (15)
The same equation can be also used to describe the direction and polarization of the incoming radiation.

Results and discussion

As a particular example we study the

| 1 Σ g + hν i → | 1 Σ u + hν m + hν n dissociative inelastic scattering via |Q 2 1 Π u (1)
doubly-excited state in the homonuclear diatomic molecule H 2 . We discuss the two-photon ACF for the decays of the H(2p 0 ) and H(2p ±1 ) fragments formed via the dissociating |Q 2 1 Π u (1) molecular state. The energy of one-photon excitation cross section of the | 1 Σ g → |Q 2

1 Π u (1) state is about 35 eV and the dipole transition moment has been calculated to be about 0.064 a.u. at r 0 = 1.4 a 0 [20], being about three times smaller than in [18]. The complete excitation and two-step decay process in H 2 can be written as

|r 0 , 1 Σ g + hν i t 0 → | 1 Π u * * t 0 →t 1 ⇒ | 1 Π g * | 1 Σ g * + hν n t 1 →t 2 → | 1 Σ u + hν m . ( 16
)
The molecule is excited to a dissociative |r 0 , 1 Π u * * state at time t 0 . The first photon

hν n is emitted at t 1 resulting in a | 1 Π g * or | 1 Σ g * intermediate state.
Both of these states decay subsequently to the same final state | 1 Σ u via emission of the second photon hν m at time t 2 .

Fixed-in-space H 2 and polarization insensitive detectors

As a first example we study the ACF in the case of fixed-in-space H 2 . The molecule is excited by linearly polarized light and the outgoing photons are counted with polarization insensitive detectors. The ACF in the considered case can be most conveniently calculated using equation (11). The coordinate system was chosen so that the incoming photon propagates along the laboratory x-axis and is polarized into the direction of the z-axis, corresponding to vertical polarization in the laboratory frame. The molecular quantization axis coincides with the laboratory y-axis, which corresponds to the geometry used in [10]. In the molecule fixed frame, summation over positive and negative projections of the angular momenta Ω gives four pathways leading to the same final state

Ω i = 0 q i =±1 -→ Ω n = ±1    qn=0 -→ Ω m = ±1 qm=∓1 -→ qn=∓1 -→ Ω m = 0 qm=0 -→ Ω f = 0. ( 17
)
The ACF was calculated by replacing the molecular wave functions at large internuclear separation by linear combinations of atomic wave functions (3). The atomic dipole transition matrix elements were calculated assuming pure LS-coupling scheme and applying the Wigner-Eckart theorem [START_REF] Edmonds | Angular Momentum in Quantum Mechanics[END_REF]. As a result the ACF reads

f (θ n , φ n , θ m , φ m ) = 1 2 | n 1 m 0 + n 0 m 1 + n -1 m 0 + n 0 m -1 | 2 × | 2 9 ( 2
√ 2 -3) 1s T (1) 2p 2 1 Π Λ=|±1| u |T (1)
q i =|±1| | 1 Σ g | 2 , ( 18
)
where the first term contains all angular and polarization information of the emitted two photons, and the second term is only a constant factor. In the following, for convenience, ACFs are normalized so that their maxima correspond to unity.

Inserting the vector components of ( 15) into (18) and integrating over the polarization sensitivities ξ n , ξ m , δ n and δ m we found that

f (θ n , π/2, θ m , 3π/2) = sin 2 (θ n -θ m ). ( 19
)
Here, choosing φ n = π/2 and φ m = 3π/2 aligns the detectors to opposite sides in plane perpendicular to the direction of propagation of the incoming radiation (i.e. the dipole plane). The ACF ( 19) is depicted in figure 2. We note that the function in figure 2 coincides with the weighting factor I 00 for the T 00 multipole calculated for two-photon production of electron-impact excited atoms [START_REF] Wang | [END_REF]. Function ( 19) is exactly the same as obtained in [10], and it produces the identical result if the angle θ m takes values between [-π, 0] as in [10]. However, with the use of the usual definition of polar angles θ n and θ m running in between [0, π], our result leads to different interpretation of the ACF (19). According to figure 2 the zero line of the ACF is along the θ n = θ m diagonal, and not on the θ m = π -θ n diagonal as suggested in [10].

In order to show how the interference influences the ACF (18), figure 3 depicts the two-photon ACF computed for the same case without accounting for the interference. This means, that the four components of the first term of ( 18) are summed up incoherently. Comparing figures 2 and 3 one can see that the interference has the strongest influence along the diagonals and no influence at the sides, if one detector is aligned parallel to the polarization vector of the incident light (θ n/m = 0 • or 180 • ). In these cases, a single cut from the surface has a clear nonconstant shape, but it is not due to the interference. On the other hand, figures 2 and 3 indicate that the zero diagonal in figure 2 is due to complete destructive interference. If one needs to study interference experimentally in the discussed case by taking a single cut from the surface with one detector fixed, it should be placed preferably perpendicular to the electric field vector of the linearly polarized incident radiation.

Randomly oriented H 2 and polarization insensitive detectors

We may simplify (9) assuming that the radial parts of the dipole transition matrix elements are independent of J, and using, that for the initial and final states of the process Ω i = Ω f = 0. Summations over J m and J n can be carried out by applying (A.3) twice, and the sum over J f using the orthogonality of the 3j symbols. Equation ( 9) is then reduced to the form

dσ 2 nm dΩ n dΩ m = α 4 ω i ω 3 n ω 3 m 4π 2 c 3 χ diss K Q K 2 Γ -1 K F Q KK ΩnΩm (-1) K-Ωm-2Ωn ν n , r 0 |ν i , r 0 × 1 1 K Ω m -Ω n Ω n -Ω m K 1 K Ω m -Ω m 0 × Ω f |T (1) qm |Ω m Ω m |T (1) qn |Ω n Ω n |T (1) q i |Ω i 2 . ( 20
)
Performing summations over K, K , Ω n and Ω m , and simplifying the dipole transition matrix elements as discussed in the previous section, the ACF takes the form similar to (18). Now, however, the information about all photons (including incident radiation) is expressed by the

F Q KK terms f (θ n , φ n , θ m , φ m ) = Q 3| 2 3 F Q 01 - 1 3 F Q 11 + 1 15 F Q 21 | 2 + Q 7| 2 5 2 7 F Q 23 | 2 . ( 21
)
The form of equation ( 21) is independent of the coordinate system, polarization of the incident light and polarization sensitivities of the detectors. Only K = 1, 3 values remain, which can be interpreted as follows. In the molecular frame the process leads to an emission of one linearly and one circularly polarized photon. The fact that the first photon may be linearly or circularly polarized in the molecular frame is reflected in the interference between the F Q K1 terms. Only one F Q K3 component is presented because K = 2 and K = 3 is the only combination allowed by the triangular conditions in the dipole approximation for K = 3. All experimentally observable results presented below can be obtained from [START_REF] Lefebvre-Brion | The Spectra and Dynamics of Diatomic Molecules[END_REF], or equally from ( 11) by integrating over the molecular orientations.

Linearly polarized incident light

The case of linearly polarized incident light was treated in the geometry illustrated in figure 1. Inserting explicit values of the F Q KK given via (10) in [START_REF] Lefebvre-Brion | The Spectra and Dynamics of Diatomic Molecules[END_REF], the ACF takes form

f (θ n , φ n , θ m , φ m ) = 1 328π 4 (19| n 0 m 1 + n 1 m 0 | 2 + 19| n 0 m -1 + n -1 m 0 | 2 + 8| n 1 m 1 | 2 + 8| n -1 m -1 | 2 + 8|2 n 0 m 0 + n -1 m 1 + n 1 m -1 | 2 ). ( 22
)
Comparison of ( 22) and (18) shows that, if the molecular sample is randomly oriented, terms. The analytic form of the ACF ( 22) can be calculated inserting components (15) to [START_REF] Brown | Rotational Spectroscopy of Diatomic Molecules[END_REF], and integrating over the polarization sensitivities of the detectors. The normalized ACF then takes the form

f (θ n , π/2, θ m 3π/2) = 1 164 (111 -25(cos 2θ m + cos 2θ n ) + 3 cos 2(θ m + θ n )), (23)
which is depicted in figure 4. The surface plotted in figure 4 shows that the shape of the ACF for the special orientation plotted in figure 2 is mostly smeared out. However, the ACF exhibits still a clear dependence on the emission angles, which can be verified experimentally. One may see that in contrast to the fixed-in-space case, the highest coincidence probability is obtained when the two detectors are facing each other perpendicular to the electric field vector of the incident light. The function in figure 4 resembles somehow the ACF from [10] weighted with initial excitation probability of sin 2 θ (see, figure 4a' in [10]). Clear differences can be still pointed out. The present ACF has a clear isotropic background and the double peak structure obtained in [10] is missing. The background in the present calculations explains, without invoking finite opening angles of the detectors, why the photon-photon coincidences were observed in [9] despite the experimental geometry was set to a zero point of the ACF computed in [10].

In order to analyze the ACF [START_REF] Brown | Rotational Spectroscopy of Diatomic Molecules[END_REF] in more detail, figure 5 depicts three functions summed in figure 4. The uppermost panel of figure 4 shows the contribution from the first two terms in [START_REF] Brown | Rotational Spectroscopy of Diatomic Molecules[END_REF], the middle panel shows the contribution from the third, and fourth terms, and the lowermost panel the fifth term. The first and second, as well as third and fourth terms were summed because they result in equal dependencies, respectively. The function shown in the uppermost panel of figure 5 for a random orientation is the same as in figure 4a' in [10] (but rotated by 90 • around the vertical axis piercing the horizontal plane at θ n = θ m = 90 • , which is due to different definitions of spherical angle θ discussed above). Additional investigations showed that the ACF derived in [10] and averaged with sin 2 θ does not reproduce the full solution of the problem for randomly oriented molecules. We assume that this is because rotation of the two-photon state obtained from the second-order correlation function does not produce n j m j and n j m -j components. Therefore, the approximate approach in [10] gives only a part of the full solution [START_REF] Brown | Rotational Spectroscopy of Diatomic Molecules[END_REF]. We emphasize that the angular conservation requires that |q i | = |q n + q m | = 1 holds always in the molecular frame, and it is the transformation to the laboratory frame which produces the additional terms in [START_REF] Brown | Rotational Spectroscopy of Diatomic Molecules[END_REF]. From figure 5 we also see that the linear combination of the three parts does not always have to be symmetric for the two diagonals. It is by chance that the sum is symmetric in the case of linearly polarized light shown in figure 4.

Results of the present calculations of the ACF are compared to the experimental 22). Uppermost panel: Function obtained from the first two terms. Middle panel: Function obtained from the third and fourth term. Lowermost panel: Function obtained from the fifth term. The scales show the relative contributions of these functions to the ACF [START_REF] Brown | Rotational Spectroscopy of Diatomic Molecules[END_REF] shown in figure 4. [16] and theoretical [10] ones in figure 6. The figure shows a cut from the surface depicted in figure 4 along the θ n = π -θ m diagonal. As one can see, the coincidence rate function computed in [10] has a minimum and vanishes at θ m = 0 • (dashed curve). [16]. Solid line: Present calculation. Dash-dotted line: Present calculation convolved with the detector opening angles 0.64 sr used in [16]. Dashed line: Calculation [10]. Crosses: Calculation [10] convolved with the detector opening angles 0.64 sr. All results are normalized to the unity at θ m = 90 • .

The minimum in the experimental coincidence rate [16] (dots) is, however, not so deep, even if one accounts for the finite detector opening angles in the theory (crosses). This disagreement was explained in [16] by the large pressure effect. It was suggested in [16] that experimental data may saturate to the theoretical predictions (crosses) with decreasing of pressure. One can see from figure 6, that the presently computed coincidence rate (solid curve) has the same shape but a smaller dip in the minimum around θ m = 0 • . Due to the constant background in the presently computed ACF [START_REF] Brown | Rotational Spectroscopy of Diatomic Molecules[END_REF], the rate for detecting both photons parallel to the polarization vector of the incident light has a limit of about 0.4 (not zero as suggested in [10]). The finite detector opening angles used in [16] additionally enlarges the coincidence rate around θ m = 0 • (dashdotted curve). Thus, we may conclude that the lower pressure of 0.13 Pa of the H 2 sample used in [16] approaches already the limit where collisions do not considerably destroy the entanglement in the system. Slight asymmetry of the experimental ACF [16] and its disagreement with the presently computed ACF (dash-dotted curve) at θ m = 110 • , 0 • and -20 • might be due to entangled atom-pair reactions suggested in [16].

As a final remark, we note that the linear dichroism pattern [START_REF] Rodger | Circular Dichroism and Linear Dichroism[END_REF], defined as the difference between the ACF recorded with the incident light polarization along the y-axis and the ACF with the incident light polarized along the z-axis, has the same shape as the function shown in figure 4. The only difference is that the linear dichroism function takes values between about (-0.4,0.4). This is a direct consequence of the symmetry and periodicity of function [START_REF] Singer | [END_REF].

Circularly polarized incident light

The most convenient form of the ACF in the case of circularly polarized incident light is obtained when the incoming light is chosen to propagate along the z-axis in the laboratory frame (see figure 1). By inserting the components of the right-handed circularly polarized incident light into [START_REF] Lefebvre-Brion | The Spectra and Dynamics of Diatomic Molecules[END_REF], we obtain

f (θ n , φ n , θ m , φ m ) = 1 76π 4 ( 8 3 | n 0 m 1 + n 1 m 0 | 2 + 5| n 0 m -1 -n -1 m 0 | 2 + 8| n 1 m 1 | 2 + 52 3 | n -1 m -1 | 2 + |2 n 0 m 0 + n -1 m 1 + n 1 m -1 | 2 ). ( 24
)
The left-handed circularly polarized case can be described by exchanging simultaneously the numerical prefactors of the first and second terms, and third and fourth terms. Note, that the function (24) has the same terms as in ( 22), but with different weights, which are not symmetric for the first and the second, and the third and the fourth terms. However, since the functions produced by these pairs are equal, as mentioned above, the total ACFs for right-and left-handed circularly polarized light are equal. This means that, even if the detectors are polarization sensitive, one cannot observe circular dichroism [START_REF] Rodger | Circular Dichroism and Linear Dichroism[END_REF] in the two-photon ACF from H 2 . The function ( 24) is explicitly given by Figure 7 shows the two-photon ACF (25) for randomly oriented molecules excited by circularly polarized incident light. One can see that the ACF in this case is almost constant and does not change by more than 5 %. The figure also shows that the maximum of the function corresponds to the φ m = π + φ n diagonal. This result is expected and physically meaningful because, if the incident light is circularly polarized, there is no preferred direction in the dipole plane when both detectors are facing each other. We can also conclude that, in contrast to the case of linearly polarized light, the function in figure 7 has additional maxima in the two corners, where the photons are detected in the same direction.

f (π/2, φ n , π/2, φ m ) = 1 38 (37 + cos 2(φ m -φ n )). (25)

Randomly oriented H 2 and polarization sensitive detectors

As an example of an ACF measured by polarization sensitive detectors, the case of molecules excited by linearly polarized light is studied. The two photons are counted by detectors sensitive to linear and circular polarization, such that detector sensitive to the linear polarization is at the direction θ n . Selecting the detector at θ n to be sensitive to linear polarization parallel ( ) or perpendicular (⊥) to the direction of propagation of the incident light, we obtain from ( 22) two functions

f (θ n , π/2, θ m , 3π/2) = 1 102 (64 -10 cos 2θ m -25 cos 2θ n + 3 cos 2(θ m + θ n )) (26) f ⊥ (θ n , π/2, θ m , 3π/2) = 1 62 (47 -15 cos 2θ m ). (27)
The choice of the sensitivity between left-and right-handed circular polarization of the detector at θ m do not alter the above functions. The functions ( 26) and ( 27) are plotted in the upper and lower panels of figure 8, respectively. The function in the lower panel shows that the ACF is independent of the emission angle of the linearly polarized light which is perpendicular to the direction of propagation of the incident radiation. On can also see that the coincidence detection is most probable if the circularly polarized photon is detected perpendicular to the electric field vector of the incoming radiation (θ m = 90 • ). On the other hand, the upper panel of figure 8 shows clear variations as a function of both angles. We note that the sides of the surface are slightly asymmetric. As an example, the coincidence detection is a bit more probable, if the detector sensitive to the linearly polarized photon is in the perpendicular direction of the electric field vector of the incoming light θ n = 90 • and θ m = 0 • , than if the detector sensitive to circular polarization is in the corresponding direction θ m = 90 • and θ n = 0 • . This behaviour was indicated already by analysis of the lowermost panel of figure 8, since functions ( 26) and ( 27) do not depend on the detector sensitivity to the direction of polarization of the circularly polarized photons. Therefore, if functions (26) and (27) are summed before normalization one obtains a function, which is proportional to the case of polarization insensitive detectors [START_REF] Singer | [END_REF].

As a final example, the two-photon coincidence probability as a function of the angle of linear polarization sensitivity ξ n is shown in figure 9 for two special geometries (see caption of figure 9). For both geometries we obtained that the coincidence detection probability show clear measurable variations as a function of ξ n . One can see that the two cases show opposite behaviour. If the linearly polarized photon is detected parallel to the electric field vector of the incident light, the maximal coincidence rate corresponds to the detection of the photons polarized parallel to the direction of propagation of the incident light. On the other hand, if the linearly polarized photons are detected in the direction perpendicular to the electric field vector of the incident radiation, the maximal coincidence rate corresponds to the detector sensitive to photons polarized perpendicular to the direction of propagation of the incident light.

Conclusion

We have provided a general theoretical framework to simulate the double differential cross section and ACF of two photons emitted by two entangled atomic fragments after neutral (pre)dissociation of a superexcited diatomic molecule. The theory can be applied to the studies of fixed-in-space and randomly oriented molecules with different polarizations of the incoming light and polarization sensitivities of the photon counters. The presented model was used to calculate the analytical forms of the ACF of two photons following the dissociation of the Q 2 1 Π u (1) superexcited state of H 2 in different cases. In the case of a fixed-in-space H 2 molecule, we demonstrated how the interference influences the two-photon ACF. In the case of randomly oriented molecules and polarization insensitive detectors, the two-photon ACFs induced by linearly and circularly polarized incident light were obtained. The present theory provides more complete description of the process than the previously published calculations [10] and it agrees with the recent experimental data [16] considerably better. It was shown that in the studied case, the linear dichroism pattern has the same shape as ACF obtained by using horizontally polarized incident light, and that there is no circular dichroism in the ACF. In the case of polarization sensitive detectors we obtained that the shape of ACF depends strongly on the linear polarization angle of the detected photon, but shows no difference between detector sensitivity to the left-and right-handed circular polarization.

Figure 1 .

 1 Figure 1. The coordinate system used. Positions of the detectors as well as the molecular orientation are described by polar θ ∈ [0, π] and azimuth φ ∈ [0, 2π] angles. The incoming photon propagates along the x-or z-axis depending on the chosen polarization. The polarization vectors of the outgoing photons are described in the basis spanned by spherical unit vectors ˆ 1 and ˆ 2 .

Figure 2 .

 2 Figure 2. Surface and contour plot of the dipole plane angular correlation function(19) for fixed-in-space H 2 molecules as a function of θ n and θ m . The molecules are aligned perpendicular to the electric field vector and the direction of propagation of the incoming radiation.

Figure 3 .

 3 Figure 3. Surface and contour plot of the dipole plane angular correlation function of fixed-in-space H 2 molecules computed without interference. The molecules are aligned perpendicular to the electric field vector and direction of propagation of the incoming radiation.

  are present in the ACF, which are absent for the special orientation discussed in the previous section. In addition, we note that the summation over M i and M f smears out the interference between n

Figure 4 .

 4 Figure 4. Surface and contour plot of the dipole plane angular correlation function (23) for randomly oriented H 2 molecules excited by linearly polarized radiation.

Figure 5 .

 5 Figure 5. Surface plots of the three different functions summed in equation (22). Uppermost panel: Function obtained from the first two terms. Middle panel: Function obtained from the third and fourth term. Lowermost panel: Function obtained from the fifth term. The scales show the relative contributions of these functions to the ACF (22) shown in figure 4.

Figure 6 .

 6 Figure 6. Single cut of the dipole plane two-photon ACF for randomly oriented H 2 molecule excited by linearly polarized incident light. The detectors are facing each other with relation θ n = π -θ m . Dots: Experimental data from[16]. Solid line: Present calculation. Dash-dotted line: Present calculation convolved with the detector opening angles 0.64 sr used in[16]. Dashed line: Calculation[10]. Crosses: Calculation[10] convolved with the detector opening angles 0.64 sr. All results are normalized to the unity at θ m = 90 • .

Figure 7 .

 7 Figure 7. Surface and contour plot of the dipole plane angular correlation function (25) for randomly oriented H 2 molecules excited by circularly polarized light.

Figure 8 .

 8 Figure 8. Surface and contour plots of the dipole plane two-photon ACF detected by polarization sensitive detectors. The H 2 molecules are randomly oriented and excited by linearly polarized light. Upper panel: The detector on the θ n direction is sensitive to the linear polarization parallel to the direction of propagation of the incident light.Lower panel: The detector on the θ n direction is sensitive to linear polarization perpendicular to the direction of propagation of the incident light. Detector on the θ m direction in both cases is sensitive to the left-or right-handed circular polarization.

Figure 9 .

 9 Figure 9. Coincidence intensity obtained in the dipole plane by detectors sensitive to circular and linear polarization as a function of the linear polarization angle ξ n in two orientations. Dashed line: The detector sensitive to the linear polarization is parallel (θ n = 0 • ) and the detector sensitive to the circular polarization is perpendicular to the electric field vector of the incident light (θ m = 90 • ). Solid line: The detector sensitive to the linear polarization is perpendicular (θ n = 90 • ) and detector sensitive to the circular polarization is parallel (θ m = 0 • ) to the electric vector of the incident light.

Acknowledgments

K J would like to acknowledge the Finnish Academy for financial support under research grant Nr. 125480. Ph V D gratefully acknowledge the Marie Curie fellowship (PIIF-GA-2008-219224) by the European Commission.

Appendix

In the following we list some identities utilized in the present derivation. The analytical integration over the rotational parts of the wave functions is well known and can be found e.g. in [START_REF] Brown | Rotational Spectroscopy of Diatomic Molecules[END_REF]. The matrix elements are

where index p refers to the laboratory coordinates and q to the molecular frame. Summation over M m and M n in ( 9) can be carried out analytically with the aid of (2.19) of Rotenberg et al [START_REF] Rotenberg | The 3j-and 6-j Symbols[END_REF]. In our case it reads:

Summation over J n and J m in (20) was carried out using (4.16) from Zare [START_REF] Zare | Angular Momentum[END_REF], which in our case reads:


I Liontos

S Cohen
email: scohen@uoi.gr

A Lyras

Multiphoton Ca 2+ production occurring before the onset of Ca + saturation: Is it a fingerprint of direct double ionization?

Keywords: . 32.80.Rm, 32.80.Zb, 32.50.+d

Singly and doubly charged Ca ions have been produced by multiphoton excitation of Ca vapor with 5 ns, low intensity ( 3×10 11 W/cm 2 ) dye-laser pulses in the 675 -685 nm wavelength range, including the four-photon 4s 2 1 S 0 4p 2 1 S 0 excitation. The intensity and wavelength dependence of the ion yields was recorded as well as the fluorescence emission from excited states of the Ca ion in an effort to identify the excitation pathways leading to single and double ionization. Unambiguous evidence for the absorption of at least two photons above the first ionization threshold was recorded, in agreement with earlier results for Mg and Sr obtained under similar conditions. However, certain characteristics of the process differ significantly from those earlier results, despite the apparent similarity in both the atomic structure and the excitation scheme. The most striking and unexpected finding is that for a certain wavelength the Ca 2+ yield is observable well before the saturation intensity of Ca + and, moreover, it grows with intensity and saturates in parallel with the Ca + yield. Possible mechanisms behind this outcome are discussed in detail as well as their implications for the multiphoton multiple ionization of complex atoms. Our tentative conclusion is that the occurrence of doubly charged ion production before the singly charged ion saturation should not be considered as "synonymous" to direct (or non-sequential) multiphoton double ionization.

Introduction

The multiphoton single and double ionization of alkaline earth atoms (Mg, Ca, Sr and Ba) is a process of fundamental interest for the field of laser-atom interactions because it can provide crucial details on the role of electron correlation in the process of multiphoton multiple electron ejection from complex atoms. The dense spectrum of autoionizing (AI) states above the first ionization threshold and all the way up to the double ionization threshold is a common feature of all alkaline earth atoms that is critically determined by correlation effects of the two optically-active outer electrons.

Early work on single and double ionization of those atoms [1][2][3][4][5][6] with ns, moderate peak intensity ( 10 12 W/cm 2 ) laser pulses, revealed that the number of photons absorbed above the first ionization limit depends upon the nonlinearity of the process. For example, if the minimum number of photons required for single ionization was three or less, the number of photons absorbed above-threshold was rarely found to be more than one. On the other hand, high excitation/ionization nonlinearities lead to resonant or near-resonant ladder excitation schemes formed within the dense AI spectrum. The latter's role, however, was not very thoroughly investigated (in most cases due to the lack of tunability). More recently, studies on Mg [7] and Sr [8] re-examined the effect by intentionally connecting the ground state ms 2 1 S 0 (m = 3, 4 and 5 for Mg, Ca and Sr respectively) to the highly correlated doubly excited states mp 2 1 S 0 , located just above the first ionization threshold, via four photons. In those experiments, it was unambiguously established that at least two (Mg and Sr) and possibly up to four (Sr) photons are absorbed in the structured continuum above the first ionization threshold. Subsequently, fast AI decay populates the ground state of the ion as well as a number of excited ionic states from which efficient double ionization occurs by further multiphoton absorption within the same laser pulse. These results on double ionization were in agreement with the body of existing fragmented experimental evidence [1][2][3][4][5][6][9][10][11] and offered a clearer and more systematic account of the details of excitation/ionization pathways. Despite the apparent similarities of the spectra of Mg and Sr and although common general features of the process were identified in both systems, certain non-trivial differences also emerged, obviously related to the specific structure of each atom. In view of this, we presently extend our study to the Ca atom, for the purpose of acquiring further insight into the particulars of the underlying mechanism. In the past, multiphoton excitation of Ca atoms with ns laser pulses at infra-red (1064 nm) and visible (532nm) wavelengths has provided evidence for excess photon absorption above the first ionization threshold [11], in accordance to other studies based on detection of fluorescence from excited Ca + states [12]. Recent results on double ionization of Ca by shaped femtosecond intense laser pulses [13] have hinted to the possible enhancing contribution of AI states to the process, but the evidence was partly inconclusive. In the present study, laser pulses of ~5 ns duration, intensity up to 3×10 11 W/cm 2 and wavelength in the range 675 -685 nm were employed, in accordance with excitation conditions in our earlier experiments with Mg and Sr. Recording Ca + and Ca 2+ ion yields as a function of wavelength and intensity, as well as fluorescence emission from excited ionic states, we were able to confirm unambiguously the absorption of at least two photons above the first ionization threshold as well as AI-induced population of certain excited ionic states. The double ionization of Ca was confirmed to occur by further multiphoton excitation within the ion for an extended range of wavelengths within the window scanned, as expected from our earlier findings. A signature of this sequential process, easily discernible from the intensity dependence of the ion yields at specific wavelengths, is that the onset of double ion production occurs at intensities higher than the saturation intensity of the singly charged ion. However, we have also identified a wavelength (~678.5 nm) for which the Ca 2+ yield is significant and clearly observable well before Ca + starts to saturate. In fact, the two ionic signals start to saturate at comparable intensities. In the context of our experimental conditions this is an extremely rare [2] and unexpected feature. With sub-ns pulses, the simultaneous occurrence and saturation of the singly and doubly charged ionic signals was always considered as a signature of direct or non-sequential double ionization [14][15][16][17], the two terms being used, almost indiscriminately, to describe a process in which the doubly charged ion is formed without any intermediate single ion production. However, this process has been mostly recorded for systems quite different from the alkaline earths (mostly heavy rare gases and He, whose structure is quite different from that of the alkaline earths, despite being the prototypical two-electron system) excited by short, intense infrared laser pulses. Under such conditions, double ionization is commonly attributed to a tunneling and rescattering mechanism [18]. Another criterion, often employed under such conditions, is the value of the Keldysh parameter, γ = (IP/2U p ) 1/2 , where IP is the ionization threshold and U p the ponderomotive potential [16]. For γ 1 the tunneling and rescattering mechanism is considered dominant, while for γ>>1 the multiphoton mechanism prevails. Under our experimental conditions, the latter is the case, as explained below. Therefore, it is clear that from every point of view the conditions of the present experiment are vastly different from the ones described above and a tunneling and rescattering mechanism cannot be evoked to interpret the observed behavior. In the following paragraphs, after analyzing the available data as thoroughly as possible, we offer a tentative interpretation of this surprising finding, while awaiting further insight from future experiments.

Experimental procedure

The experimental setup and procedure has been previously described in detail [7,8] so here their presentation will be brief. Multiphoton ionization experiments are performed in an atomic beam apparatus whilst the fluorescence studies in a vapor cell. In both experiments the light source is a Nd:YAG pumped dye laser delivering linearly polarized pulses of ~5 ns duration and ~0.1 cm -1 linewidth at a repetition rate of 10 Hz. The laser system operates within the 675 -685 nm wavelength range. The optogalvanic signal of an Ar-filled commercial hollow cathode lamp, into which a small part of the dye laser beam is directed, is used as an absolute wavelength calibration standard. The laser beam pulse energy (with amplified spontaneous emission (ASE) content less than 1.5%) is controlled by a variable attenuator without any spatial misalignment and measured by a digital joule-meter. Its maximum value is ~20 mJ. The latter corresponds to a maximum power density of I~3×10 11 W/cm 2 (under the assumption of a diffraction limited Gaussian beam profile) when the beam is focused by an f 10 cm focal length lens.

Either the atomic beam chamber or the vapor cell is pumped by a 250 lt/min turbo molecular pump and a rotary pump system resulting to a background pressure of ~10 -6 mbar. In the atomic beam experiment Ca vapor is produced in an electrically heated stainless steel oven at the top of the chamber. The oven temperature is ~700 o C resulting in an atomic density of ~10 8 atoms/cm 3 at the interaction point. The laser-atom interaction takes place within two circular stainless steel plates. One of the plates is grounded while a positive voltage is applied to the other for creating a ~70 Volts/cm static electric field. The produced singly-and doubly-charged Calcium ions are pushed by the field into a direction perpendicular to both the atomic and laser beams. The ions pass through the grounded plate via a ~2 mm hole and towards a 30 cm long time-of-flight (TOF) tube where they are discriminated according to their mass-to-charge ratio and detected by a dual micro-channel plate detector (MCP). The TOF system resolves the most abundant 40,42,44 Ca + and 40,42,44 Ca 2+ isotopes (see figure 1). The MCP signal is monitored by a digital oscilloscope, processed by a boxcar integrator and fed to a chart recorder and a personal computer.

For the fluorescence experiments the laser beam is focused into a quartz T-shaped tube where Ca undergoes continuous evaporation. Coating of the quartz windows with Ca is prevented by the use of a metallic mesh rolled around its main body and by the presence of 3 -7 Torr He buffer gas (slightly higher than the Ca vapor pressure). An f 11 cm focusing lens is mounted inside the tube.

The metal vapor fluorescent light is collected at 90º with respect to the laser beam direction and analyzed by a monochromator -photomultiplier tube (PMT) system, sensitive within the 200 nm -550 nm range. The PMT signal is processed by the same recording/storage system employed for the ions.

Excitation and ionization pathways

The partial -in scale -energy level diagram of figure 2 depicts the most relevant atomic and ionic levels involved in the present study. The aforementioned wavelength range is chosen so as to include the four-photon excitation of ground (4s 2 1 S 0 ) state Ca atoms to the 4p 2 1 S 0 doubly excited state [19,20] above the first ionization potential. Owing to the dipole selection rules and the given initial state, four-photon absorption by linearly polarized light allows excitation of even parity states with total angular momentum J=0, 2 and 4. According to available spectroscopic data [19,21], two [3d6d] J=2,4 and two [3d5g] J=2,4 neighboring states may also be excited by four-photon absorption, the relevant wavelengths lying between 679 nm and 682 nm. Within the third photon energy range only the bound doubly excited 3d4p 1 P 1 state may be reached, expected at ~682.7 nm [22]. On the contrary, at the fifth photon level (where excitation to odd parity states with J=1, 3 and 5 is dipoleallowed) several members of the 4pns 1,3 P 1 (n=11-15), 4pnd 1 P 1 (n=10-14) and [4pnd] J=3 autoionizing series [23] may be excited. All these states possess fairly large autoionization widths (tenths or even hundreds of cm -1 ), particularly the 4pnd ones. There are no spectroscopic data for J=5 o levels.

No resonances are expected around the sixth photon range, while the doubly excited state spectrum corresponding to the seventh-or higher-photon ranges is largely unknown. However, dense doubly excited 4fnl and 6fnl Rydberg series are expected around the eighth and ninth photon respectively, since the corresponding Ca + 4f and 6f ionization thresholds lie within the corresponding wavelength range. Additionally, above the ninth photon energy range there exist the 8s and 7d thresholds with respect to which the relevant effective quantum numbers are v ~6 -~11 and v ~7 -~20, respectively. Therefore, the energy level structure around the ninth photon should be even denser and more complicated. Hence, the ladder system formed by 3d4p 1 P 1 (third photon), 4p 2 1 S 0 (fourth photon) and [4pnl] J=1 o with l=s,d and n~11 (fifth photon), as well as other unknown states lying even higher in energy could facilitate absorption of a number of photons larger than that expected on the basis of long-pulse/low-intensity multiphoton ionization studies (as implied also by our recent experiment in Sr [8]). Consequently, under the assumption of sequential double ionization, a number of excited ionic states would acquire population. Apart from Ca + 4s ground state, our experimental findings prove (directly or indirectly) the formation of 3d and 4p Ca + ions. For the presently available laser intensities, we expect a fairly weak Ca + 4s ground state contribution to the double ionization yield, since the latter would stem from either a seven-photon non-resonant or a 6+1-photon resonant ionization process though the 8d Ca + level. Both of these two processes should have very low cross sections. Therefore the largest, by far, contributions to sequential Ca 2+ production must originate from 3d and 4p states. Six-photon ionization of the former level may proceed either non-resonantly or through the (possibly AC-Stark shifted) 5d and 7f ionic excited states (four-and five-photon resonance respectively). As for the Ca 2+ yield stemming from the 4p states, it is of non-resonant character (five-photon ionization), even by taking into account possible AC-Stark or ponderomotive shifts of excited ionic states in the proximity of each absorbed photon.

For the purpose of comparison between the required number of photons necessary for Ca 2+ production by the sequential pathways (a)-(c) and the direct one (Ca Ca 2+ ), the latter (d) is also included in figure 2. It is evident that, assuming that both Ca + and Ca 2+ yields are unsaturated, Ca 2+ production requires absorption of ten photons for the direct path and eleven photons for all sequential paths. The distinction between these two high orders of nonlinearity is not always easy as it is usually masked by experimental uncertainties. On the other hand, it is customarily expected that Ca + yield will saturate at intensities much lower than those corresponding to Ca 2+ production due to the lower order of nonlinearity. Consequently, Ca 2+ production through the direct process or other highly nonlinear mechanisms mimicking it, could possibly be assessed more reliably by simply observing if Ca + yield is saturated (or not) upon the appearance of Ca 2+ signal.

Results and discussion

Laser wavelength dependence of Ca + , Ca 2+ and fluorescence yields

Typical Ca + , Ca 2+ and 4p 3/2 4s 1/2 ionic fluorescence spectra are given in figure 3 for linearly polarized light and the maximum available laser intensity. Below we discuss separately the wavelength dependence of each yield. at ~682.4 nm and ~683 nm respectively, are blended. The latter is assigned to the 4s 2 1 S 0 -4 4p 2 1 S 0 transition since its location and width practically coincide, within experimental uncertainties, to the spectroscopically determined values for the 4p 2 1 S 0 doubly excited state [19,20]. The line is therefore found unshifted both for high as well as low intensities. This fact implies the occurrence of a stabilizing effect, whereby AC-Stark shift contributions from one-photon resonant states below (3d4p 1 P 1 ) and above ([4pnl] J=1 o ) the 4p 2 1 S 0 one, lead to a nearly-zero dynamic dipole polarizability. A similar situation was encountered also in Sr [8]. On the contrary, the maximum of the line at ~682.4 nm (attributed to three-photon resonant, four-photon ionization of ground state Ca through the 3d4p 1 P 1 intermediate state [22]), is found to be blue-shifted by ~0.35 nm with respect to its unperturbed location (~23 cm -1 in total excitation energy). This is to be expected since the 3d4p 1 P 1 state has important AC-Stark shift contributions stemming solely from higher lying levels (4p 2 1 S 0 ).

However, the shift is found to be intensity-independent down to the lowest intensity values allowed by the detection efficiency of our TOF system. In order to explain the above observation, it is first noted that the line is evidently quite asymmetric. It is well-known that under the employed laser intensities and tight focusing conditions such asymmetric profiles stem from the combination of two phenomena: (i) The AC-Stark shift, on one hand and (ii) The collection of ions produced over the entire laser-atom interaction volume on the other. Ions produced at different distances from the focal point sense different laser intensities and therefore undergo different shifts. The latter effect leads to asymmetric profiles and washes-out any shift intensity-dependence after spatiotemporal integration over the interaction volume. By recording the Ca + spectrum with the same pulse energy but under smoother focusing conditions (f~25 cm -not shown here) the line is found much more symmetric and slightly shifted towards its unperturbed location, since in that case the range across the focus over which the intensity suffices to produce Ca + is greatly reduced. However, under such conditions no Ca 2+ was detected.

The weaker line at ~677.1 nm can not be assigned to any three-or four-photon excited level. Moreover, the fact that the line appears also in the fluorescence spectrum as well as the intensity dependence of the line's maximum (both discussed in more detail below) point towards a five photon resonance. Among the known relevant [4p j nl] J=1,3 (n~12-14, l=s,d) levels [23], the 4p14s 1 P 1 one lies closest to the line maximum (within -40 cm -1 ). All of the above states however are quite broad, particularly the 4pnd ones. This fact, in conjunction with the strong blue tail of the three-photon resonance (3d4p 1 P 1 ), may explain why it is the only five-photon excited state observed, while there are several other candidates within the same spectra range. . The weaker and broader resonance (~683 nm) is evidently correlated to three-and four-photon excited atomic states (3d4p 1 P 1 , 4p 2 1 S 0 ) while for the strongest one (~678.5 nm) there is no straightforward connection to any atomic (singly or doubly excited) level.

Within the studied wavelength range there are two groups of transitions connecting the 3d j ionic levels with the 5d j (four photons) and 7f j (five photons) ones before the ions ionize further to produce Ca +2 . The unperturbed locations of these transitions are marked in figure 3(b). It can be seen that they do not match well the maxima of the broad resonances. In order to clarify the role of AC-Stark shift to the disagreement, the relevant dynamic dipole polarizabilities of the above states were calculated by the Dalgarno-Lewis method and the numerical technique developed in [24]. The finestructure splitting is ignored and the computation is restricted to an azimuthal quantum number m=0. The parametric Ca + potential curves given in [25] (see equation 3.26 therein) were used as input for the calculation. The computed dynamic polarizabilities are found to vary smoothly with photon energy so here we provide their frequency-averaged values which are: d a 3 -73 atomic units (au), d a 5 +310 au and f a 7 -247 au. Using the value 3.5095×10 16 W/cm 2 as an atomic unit of light intensity, these values result to -2.64×10 -11 nm/(W/cm 2 ) and +1.03×10 -11 nm/(W/cm 2 ) for the 3d j 5d j' and 3d j 7f j' transitions, respectively. Hence, by taking into account the AC-Stark shifts the agreement between expected and experimental Ca 2+ maxima in figure 3(b) worsens (for both transitions). Therefore, there is no clear assignment for the strong feature around 678.5 nm. On the other hand, its shape, as recorded for different laser intensities (inset of figure 3(b)), implies that it may be composed from several resonances. Indeed, the weak bump at ~674.4 nm and the strongest feature at ~678.5 nm survive at low intensity, while another apparent maximum at ~679.3 nm is emerging for higher intensity values. Note that by computing the dipole polarizabilities of a number of other ionic states we have ruled-out the possibility that any of these levels come into resonance during the laser pulse.

Ca + radiative decay.

The only ionic radiative decay lines detected, by widely scanning the monochromator's wavelength for several fixed laser photon energies, correspond to two resolved Ca + 4p 3/2,1/2 4s 1/2 transitions at 393.4 nm and 396.8 nm respectively. All the data were acquired after ensuring that for our Ca-cell operating conditions the PMT signal varied linearly with Ca density. These experimental findings verify the absorption of six photons from Ca ground state (two of them above the first ionization threshold) and confirm the trend observed earlier in Mg [7] and Sr [8], under similar excitation conditions. It is anticipated that Ca + 3d j states are also populated by five-photon ionization and/or by the unobserved ionic 4p 3/2,1/2 3d j radiative decay. On the other hand, the data do not provide evidence for absorption of more than six photons by the Ca atom, in accordance to Mg experiment but in contrast to Sr data. Therefore, if other ionic states (lying higher than the 4p j ones) are indeed populated, subsequently to, for example, autoionization of doubly excited Ca, these states should be further ionized to Ca 2+ quite efficiently (saturated Ca + Ca 2+ step).

As for the fluorescence's laser wavelength dependence, let us first note that it is identical for the 4p 1/2 4s 1/2 and 4p 3/2 4s 1/2 components, so only the latter is given in figure 3(c). It is remarkable that there is a small non-resonant background even at laser intensities somewhat lower than the maximum ones employed in the photoionization experiment. Moreover, there is a clear spectral fingerprint of the resonances corresponding to the 3d4p 1 P 1 and 4p 2 1 S 0 states. A comparison with Ca + spectra reveals more symmetric fluorescence line profiles as well as the absence of any long blue tail for the 3d4p 1 P 1 resonance, presumably because six-photon absorption occurs only near the la-ser beam focus whereas the (lower nonlinearity) ionization signal is produced over a wider interaction volume. Thus, fluorescence detection experiences spatiotemporal laser-atom effects of much lesser importance. Finally, the spectrum shows two additional broad lines that should be attributed to five-photon excitation of doubly excited 4pnl states having appreciable autoionization widths.

The first at ~677.1 nm is also observed in Ca + spectrum while the other at ~679.5 nm is probably masked under the aforementioned blue tail. Both resonances are located below the 4p 1/2 limit (by ~650 cm -1 and ~900 cm -1 respectively) and they show-up in the fluorescence spectra due to the absorption of an additional (sixth-) photon leading to the population of both 4p 3/2,1/2 states. The ~679.5 nm line is found to be somewhat shifted with respect to the strongest resonance of the Ca 2+ spectrum.

Orders of nonlinearity for singly and doubly charged ion production

Signal vs laser intensity (S vs I) curves for Ca + and Ca 2+ were recorded for the laser wavelengths labeled as A-E in figure 3(b). The apparently unsaturated parts of those curves were fitted to the power-law form S∝I K and the orders of nonlinearity (K + and K 2+ for singly and doubly charged ions, respectively) extracted from the fits are assembled in table I. Two examples of signal vs laser intensity curves are given in figure 4 for 683.7 nm (E) and 678.5 nm (B). The former lies on the red side of the 4p 2 1 S 0 resonance and was chosen for reducing any influence stemming from a possible red tail of the 3d4p 1 P 1 three-photon excitation. Nevertheless, for point (E) as for the 4p 2 1 S 0 -maximum (D) the Ca + order of non-linearity is + ≈4 -a result to be expected for a four-photon excitation/ionization process. In fact, most of the fitted K + values are found to be ≈4. There are two exceptions corresponding to 682.3 nm (3d4p 1 P 1 -maximum, C) and 677.1 nm (maximum of weaker broad line, A). For the latter wavelength, due to the absence of three-or four-photon resonances, the order of nonlinearity K + ~5 clearly implies a five-photon resonance, in accordance to the appearance of this line in the fluorescence spectrum. The K + >4 value for wavelength C can be explained by the presence of the 3d4p 1 P 1 intermediate level. It is well known that in the vicinity of a resonance the observed order of nonlinearity may be found to be higher as well as lower than the true number of photons absorbed by the atom, since in this case this quantity only expresses atomic level shifts [26].

Let us now discuss the fitted K 2+ values. From table I it may be noted that they all fall in the range 7-10 except the one corresponding to 677.1 nm, which exhibits the very low value of K 2+ ~4. Taking into account that for that wavelength Ca 2+ appears when Ca + is already saturated, the low K 2+ value simply reflects the occurrence of ionic 3d 5d transitions shifted into resonance by the laser field.

Among the other wavelengths, particularly interesting are the B and D ones which show high K 2+ ~10 nonlinearities. However, even more important is the fact that of all the data of table I, it is only for B that the Ca 2+ signal appears well below the onset of Ca + saturation. The relevant graph is shown in figure 4(b) and corresponds to the maximum of the main Ca 2+ lobe -see figure 3(b). In fact, this graph shows practically simultaneous saturation of both Ca + and Ca 2+ signals which are, additionally, of comparable magnitude. It is interesting to note that, as mentioned earlier, there are no ionic resonances at that wavelength (since the nearest 3d 5d one is shifted away from the observed maximum). As for the atomic spectrum, it may be postulated that there exist five-photon resonances (particularly the [4p13s] J=1 one -see figures 3(a),(b) -which however does not appear in the fluorescence spectrum) as well as others lying above the sixth photon from the ground state. These resonances may not be exact and therefore it is expected that excitation takes place under (either atomic or ionic) near-resonant conditions.

Synthesis

Having presented our data, we now proceed to their discussion and comparison to the findings from our previous studies on Mg [7] and Sr [8]. In all three experiments the atomic ground state is connected to mp 2 1 S 0 doubly excited states via four photons. These states are strongly one-photon coupled with bound (below) and autoionizing (above) atomic levels. In Mg this strong coupling results to a large nonperturbative shift of the resonance, apparently because there is just a single state at the fifth-photon level. On the contrary, for Ca and Sr the existence of a large number of fivephoton excited autoionizing Rydberg states leads to an overall nullification of the dynamic dipole polarizability and mp 2 resonances experience small (Sr) or zero (Ca) shifts. Nevertheless, the multitude of near-resonant levels establishes in all cases favorable conditions for absorption of two photons within the continuum. In fact, the (fluorescence) data of Sr support the absorption of at least two more photons before the production of singly charged ions.

It is also interesting to compare the signal vs laser intensity plots concerning the doubly charged ions. The Mg 2+ yield appeared above the onset of Mg + saturation (in fact theory predicted it was itself saturated). The same holds for Sr apart of the location of the 5p 2 1 S 0 resonance where the (weak) Sr 2+ signal appeared slightly below the onset of Sr + saturation and it was accompanied by a weakly evident but reproducible inflection point ("knee" structure, see [8] and references therein).

A more interesting behavior emerges in the Ca experiment. By recording signal vs intensity curves for a number of wavelengths, quite different behaviors are unveiled. In all but one cases, the, by now, trivial observation emerges that Ca 2+ appears well above the saturation intensity of Ca + . There is however a specific photon energy, which cannot be correlated to known doubly excited atomic states (or even ionic states) and at which Ca 2+ appears well before the occurrence of Ca + saturation. Moreover, at that photon energy Ca 2+ and Ca + yields are comparable in magnitude, although the nonlinearity of the process is quite high (K 2+ ~10). Surprisingly, this high order of nonlinearity is compatible with the direct double ionization process. On the other hand, at that photon energy the Ca 2+ spectrum suggests the presence of at least one resonance which, additionally, disappears with circular laser polarization (not shown here). Evidently, doubly excited intermediate states, located necessarily above the fourth-photon level are involved in the process. Nevertheless, by taking into account the long laser pulse duration and its low intensity it is difficult to adopt a direct double ionization scenario. The latter is furthermore not supported by the fluorescence experiment since it did not reveal the presence of population into ionic states lying higher that the 4p j ones. Of course, this fact may be attributed to the low maximum available laser intensity (~3×10 11 W/cm 2 ); the lowest employed in our experiments with alkaline-earths so far. It is probable that fluorescence signals were just too low as compared to our detection efficiency. There exist however other possible, sequential in nature, double ionization scenarios that could mimic the direct process and involve double Rydberg states. Assume, for example, that an atomic resonance is reached at, say, the eighth or ninth photon from the ground state and that the transition is (presumably) unsaturated. One possibility is, of course, that by an additional photon the atom will be doubly-ionized [27]. Alternatively (and far more probably), autoionization may produce highly excited Ca + ions. The latter may subsequently be ionized by one or two photons. The measured order of nonlinearity will simply reflect the excitation process since the one-or two-photon ionization step within the ion is expected to be saturated. Additionally, no highly excited Ca + ions will survive over such a process while its identification by fluorescence detection is quite improbable because ionization will dominate over their radiative decay for the laser intensities considered here. Mechanisms resembling the above scenario have already been observed in Alkaline Earth atoms, albeit within a different, multi-color stepwise excitation scheme [27,28]. There is however another alternative explanation to the above observation. Note, for example, that at wavelength D an order of nonlinearity of ~10 is also recorded while Ca 2+ appears above the saturation intensity of Ca + . Hence, it is possible that the detection efficiency of our TOF system is in the case of wavelength B simply enough for Ca 2+ to be observable below Ca + saturation, while this is not so for the other wavelengths at which the signal is weaker. Even so, this quite different behavior implies different underlying mechanisms. Clarifying this point and the exact role of near-resonant doubly excited states (by, for example, an electron energy analysis experiment) is thus of great interest. Nonetheless, the results of the present study raise another issue of, perhaps, even greater importance. Specifically, it renders the so far considered as a well established criterion on the emergence of direct or non-sequential double ionization of questionable significance and value. Since in the present experiment the multiphoton laser-atom interaction picture applies (as implied by the Keldysh parameter [16] value of ~15>>1) the discussion at present should be restricted to that picture only and no appeal to tunneling or over the barrier ionization mechanisms can be reasonably made.

Conclusion

We have presented the results of an experimental study on single and double multiphoton ionization of Calcium with laser pulses of ns duration, low intensity ( 3×10 11 W/cm 2 ) and a photon energy range that included the four-photon 4s 2 1 S 0 4p 2 1 S 0 transition. The study was based on detection of Ca + and Ca 2+ in an atomic beam apparatus as well as detection of fluorescence stemming from the 4p 1/2 , 3/2 4s 1/2 radiative decay of excited Ca + in a vapour cell experiment. Observation of the particular fluorescence transition constitutes a direct proof of absorption of two photons above the first ionization threshold. This fact signifies that laser-atom interaction is critically mediated by the presence of (autoionizing and bound) doubly excited states forming resonant or near-resonant excitation ladders. For the present experiment, for which the multiphoton picture applies, the process relies on the existence of dense AI structure whose properties are determined by intra-atomic electron corre-lation effects. It, therefore, differs qualitatively from the mechanisms involved when short and intense infrared laser pulses are employed to irradiate rare gases. In that latter case, for which the tunnelling and recollision mechanism is considered valid, the dominance of electron correlation is not critical for observing multiple photon absorption within the continuum and eventually double electron ejection.

Our present findings are in agreement with the trend observed also in Mg [7] and Sr [8] in the vicinity of the corresponding four-photon transitions. Nevertheless, an additional, interesting and rare feature was revealed by the orders-of-nonlinearity for Ca + and Ca 2+ production. For all-but-one wavelengths the onset of Ca 2+ detection occurred at power densities higher than Ca + 's saturation intensity. For a specific wavelength however, Ca 2+ yield was found comparable in magnitude to that of Ca + , it showed a resonant maximum, exhibited an order-of-nonlinearity compatible with a direct double ionization scenario and, finally, the onset of its detection occurred well before Ca + 's saturation intensity. Moreover, the two yields saturated at the same laser intensity. The latter observations were considered in the past as a strong indication of direct double ionization because they imply that singly and doubly charged ions pump out of the same population of neutral atoms [14]. Our dynamic polarizability calculations are also supportive towards such a scenario as they show that the observed resonant structure could not be attributed to any AC Stark-shifted ionic multiphoton transition. On the other hand, the absorption of more than two photons in the continuum is not supported by the fluorescence experiment, contrary to observations in Sr [8] and in accordance to a very low direct double ionization probability expected under our conditions. Therefore, the double ionization mechanism around that specific Ca 2+ resonance remains unclear at present, the only certainty being its coexistence with well understood and unambiguously interpreted, sequential double ionization pathways within the examined laser wavelength range.

Evidently, more helpful for clarifying the above or similar situations would be an electron energy analysis experiment or the employment of presently available experimental techniques of multicoincidence nature, such as COLTRIMS and related ones [29]. The latter may under specific circumstances unambiguously unveil the exact double or multiple ionization mechanisms. The technique however has been mostly applied to noble gasses [29], lacking dense manifolds of doubly excited states just above the first ionization limit. Additionally, these studies are not been customarily carried out within the multiphoton regime. It is for the above reasons that in numerous recent multiphoton multiple ionization studies, performed with short and intense EUV pulses, the findings are still discussed in terms of the observed orders of nonlinearity (see, for example, [30]). Consequently, one of the major conclusions of the present work is that doubly charged ion detection before singly charged ion saturation should no longer be considered as an unambiguous indicator of direct double ionization, particularly under the presence of intermediate resonances. It is anticipated that this remark could hold for a variety of laser wavelength, peak intensity and pulse duration values. Also, it will be worth examining its pertinence under conditions such that the tunnelling and rescattering mechanism prevails. Note that at this wavelength the singly and doubly charged Ca ions are of comparable magnitude.

The Ca + peak is somewhat broader, as compared to the Ca 2+ one, due to weak space charge effects. Most of the (not fitted) saturated parts of the plots are not parallel to the intensity axis but continue to increase with a lower slope. This well-known phenomenon is due to the increase of the focal volume with increasing intensity (see [7] and references therein). The data points denote the ionic signals integrated over the TOF peak profiles. Due to the fact that the width of Ca + signal is larger than that of Ca 2+ , its integrated doubly charged ion signal for 678.5 nm appears to be lower although the peak heights are comparable (see figure 1). In (b) the vertical dashed line separates saturated and unsaturated parts of both ionic signals.

Table 1. The orders of nonlinearity for Ca + (K + ) and Ca 2+ (K 2+ ) extracted by fitting the linear parts of signal vs intensity log-log plots to a power-law form I K and for the indicated wavelengths. The labels are also shown in figure 3

4. 1 . 1

 11 Ca + yield. The highest laser intensity spectrum of Ca + (figure 3(a)) is dominated by three broad resonances structured over a considerable non-resonant background. Two of the resonances,

4. 1 . 2

 12 Ca 2+ yield. The doubly charged ion spectrum consists of two broad resonant-like features and a non-zero background (figure 3(b))

Figure 1 .

 1 Figure 1. Typical TOF mass spectrum recorded for the indicated laser wavelength and intensity.

Figure 2 .

 2 Figure 2. Partial -in scale -energy level diagram depicting the most relevant Ca and Ca + levels involved in the present study. Pathways (a)-(c) refer to sequential double ionization while pathway (d) to the direct one. For the latter, a few thresholds in the vicinity of the eighth and ninth photon have been drawn. Also noted is the observed ionic 4p 1/2,3/2 4s 1/2 radiative decay.

Figure 3 .

 3 Figure 3. (a) Ca + (b) Ca 2+ ionization yields (recorded with laser intensity I~3×10 11 W/cm 2 ) and (c) Ca + , 4p 3/2 4s 1/2 fluorescence (~2×10 11 W/cm 2 ) as a function of laser wavelength. In (a) and (b) the locations of most relevant, unperturbed, atomic and ionic states and resonance transitions are noted.In (b) the inset shows the spectrum around the strong ~678.5 nm resonance recorded with I~(0.9, 1.4, 1.7 and 2)×10 11 W/cm 2 (higher I values correspond to higher Ca +2 signals) while letters A-E mark the wavelengths at which the signal vs laser intensity curves were recorded. In (c) the profile of the dye laser output is shown by the dashed line.

Figure 4 .

 4 Figure 4. Log-Log plots of signal vs intensity for Ca + and Ca 2+ for two indicated laser wavelengths. Most of the (not fitted) saturated parts of the plots are not parallel to the intensity axis but continue

  (b).

	Label	(nm)	K +	K 2+
	A	677.1	4.7±0.5	4.3±0.2
	B	678.5	4.1±0.3	9.8±0.8
	C	682.3	6.7±0.7	8.3±0.8
	D	683.0	4.2±0.2	9.5±0.7
	E	683.7	4.0±0.2	7.1±0.1

Acknowledgments 11

The experiment was performed at the Central Laser Facility of the University of Ioannina.