Emiliya Dimova

Olivier Morizot

Guillaume Stern

Carlos L Garrido Alzar

Andrea Fioretti

Vincent Lorent

Daniel Comparat

Hélène Perrin

Pierre Pillet

Continuous transfer and laser guiding between two cold atom traps

Keywords: . 07.77.Gx, Atomic and molecular beam sources and detectors -32.80.Lg, Mechanical effects of light on atoms, molecules, and ions -32.80.Pj, Optical cooling of atoms; trapping

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Introduction

The realization of degenerate quantum gases requires the production of an initial dense and cold trapped atomic sample. The lifetime of the trapped atoms must be long enough to allow for appropriate evaporative cooling ramps, lasting up to several tens of seconds. A standard vapour Magneto-Optical Trap (MOT) setup cannot always satisfy this last condition because of the relatively high background pressure of the atomic vapour in the cell. The use of a dispenser [1] or of a desorption source [2,3] to load the MOT does not usually provide a trap lifetime longer than a few seconds. To obtain the required lifetime, the MOT has to be placed in an ultra-high vacuum chamber and loaded from a cold atom source, in general a slow and cold atomic beam. One of the demonstrated and widely used methods to create a cold atomic beam is the Zeeman slower technique. However, this solution requires an important technical development of different experimental techniques than the one implied in a MOT setup. In this paper, we will then concentrate on the transfer of atoms from a first cold source to a trap situated in a second high vacuum chamber. a e-mail: Emiliya.Dimova@lac.u-psud.fr b e-mail: morizot@lpl.univ-paris13.fr There are several ways to transfer atoms from a cold atomic source to the high vacuum chamber. Mechanical devices [4] or magnetic guides [5] are used to implement an efficient transfer of atoms initially in a MOT directly to either magnetic, electrostatic or atom chip traps. Other techniques, based on quasi-resonant light forces, allow a faster transfer to a recapture MOT. Beam velocities low enough to allow the capture in a MOT in an ultra-high vacuum chamber can be obtained by the pyramidal MOT [6,7], the conical mirror funnel [8] or the two-dimensional MOT [9,10,11,12]. Even simpler devices exist such as the low velocity intense atomic source (LVIS) [13,14,15]. Very high flux, up to 3 × 10 12 atoms/s, have been reported with a transversely cooled candlestick Zeeman slower type of setup [16]. However, the counterpart of this large flux is a higher atomic velocity, 116 m/s in this last experiment, which is by far too high to load a second MOT. A pulsed multiple loading, starting from a three-dimensional MOT, has been performed in Ref. [17]. The atoms are pushed by a near resonant laser beam resulting in a high number of atoms 1.5 × 10 10 in second MOT, loading rate 2 × 10 8 atoms/s and allow lower velocity 16 m/s. However, the transfer is based on using an hexapole magnetic field, produced by a current above 60 A, which complicates the experiment. Simpler devices, without magnetic guiding, have achieved similar result by using continuous transfer [18,19,20]. In these experiments, a thin extraction column is created in the centre of a MOT and, due to a radiation pressure imbalance, a continuous beam of cold atoms is produced. It is possible to couple these simple devices with a distinct dipolar atomic guide [21]. We propose here to use the same laser beam for pushing and guiding the atoms, resulting in an even simpler setup.

This paper reports on a double MOT setup combining the ability of a pushing laser to extract the atoms from a first trap (MOT1) and to guide them to a second trap (MOT2). The idea is to merge the leaking MOT technique [13,18,19] with the red-detuned far off-resonance optical dipole guide technique [22,23,24]. Two experiments have been simultaneously performed in two different laboratories, with different atoms: 133 Cs at Laboratoire Aimé Cotton and 87 Rb at Laboratoire de physique des lasers. Our setups are as simple as the one used in the leaking MOT techniques, but provide a higher flux and a lower atomic beam velocity. We can achieve a transfer efficiency up to 70 % with a mean atomic velocity 4 to 12 m/s depending on the pushing beam parameters. Our setups are very robust against mis-alignments of the pushing and guiding laser beam, and small variations of its detuning or power. The only requirement is a sufficiently high laser power (tens of mW) to produce a significant dipolar force to guide the atoms during their flight.

This paper is organized as follows: in section 2 we give details on the experimental realization of the beam and discuss the role of MOT1 parameters. In section 3 we describe theoretically the pushing and guiding processes during the atom transfer. Section 4 discusses the experimental parameter dependences of the setup as compared with the theory. Finally we present a comparison with other available techniques.

Experimental realization 2.1 Experimental setup

The vacuum system is similar in both experiments, except for a slight difference in the design of the differential vacuum tubes and the MOT2 cells.

For the cesium (resp. rubidium) experiment the setup consists of two cells vertically separated (see Figure 1). The distance between the two traps is D = 57 cm (resp. D = 72 cm). A reservoir connected to the upper source cell supplies the atomic vapour. The recapture chamber is a glass cell with 1 × 1 × 10 cm 3 (resp. 1.25 × 7.5 × 12 cm 3 ) volume. A differential pumping tube located 3 cm (resp. 10 cm) below MOT1 provides a vacuum within the 10 -11 mbar range in the bottom MOT2 cell while in the MOT1 cell it is in the 10 -8 -10 -9 mbar range. For the cesium experiment, the tube is 18 cm long and has a conical shape (3 mm diameter at its top and 6 mm at its bottom part) whereas it is cylindrical, 12 cm long and 6 mm diameter in the rubidium experiment.

In both cases, MOT1 runs in a standard magneto-optical trap configuration with a magnetic field gradient around 15 G/cm along the horizontal axis of the MOT1 coils. All the laser beams have a 2.5 cm diameter (clipped by the mounts of the quarter-wave plates) and are provided by laser diodes. In the rubidium experiment, the laser is divided into 3 retroreflected beams carrying 10 mW laser power. They are 10 MHz red-detuned from the 87 Rb 5s(F = 2) → 5p 3/2 (F ′ = 3) transition. In the cesium experiment, two 5 mW radial beams are retroreflected and make an angle ±45 • with the vertical axis. Each of the two (non reflected) axial beams carries 10 mW laser power. They are 15 MHz red-detuned from the Cs 6s(F = 4) → 6p 3/2 (F ′ = 5) transition. The 5 mW repumping light, with a frequency on resonance respectively with the Cs transition 6s(F = 3) → 6p 3/2 (F ′ = 4) and the 87 Rb transition 5s(F = 1) → 5p 3/2 (F ′ = 2), is mixed with all the trapping beams. In MOT2, the trapping beams are limited to about 2R = 8 mm in diameter in both experiments due to the cell dimensions and in order to reduce the scattered light.

In addition to these trapping lasers, the linearly polarized pushing-guiding beam, red-detuned from the MOT (F -→ F + 1) transition (F = 4 for Cs, F = 2 for 87 Rb) with maximum power of P 0 = 63 mW for Cs (resp. P 0 = 21 mW for Rb), is aligned vertically into the trap. The parameters used in both experiments are summarized in Table 1. In contrast with the similar setups reported in [18,19], the pushing lasers are not frequency-locked in our experiments. The detuning is chosen to optimize the transfer efficiency and is found to be such that the number of atoms in MOT1 is roughly reduced by a factor ten when the "pushing-guiding beam" is present. The beam is focused at position z 0 = -34 cm (resp. z 0 = -13 cm) before MOT1 by a lens f = 2 m (resp. f = 1 m). It is not perfectly Gaussian, however the waist at position z is still given by w(z) = w 0 1 + (z -z 0 ) 2 /z 2 R , where w 0 = 200 µm (resp. 300 µm) is the measured minimum waist and z R = 110 mm is the estimated Rayleigh length (resp. z R = 260 mm, Table 1. Pushing beam parameters used in cesium and rubidium experiments (see text and Figure 1). All distances are given in mm, the laser power P is in mW. w0, w1 and w2 are the pushing beam radius at 1/e

Flux from MOT1

Experimentally, the main features of the atomic beam are deduced from the loading characteristics of MOT1 and MOT2, where the number of atoms is determined using a calibrated photodiode monitoring the scattered MOT light. The main goal is to have the highest possible recapture rate of atoms in the MOT2 region. This ingoing flux is obviously related to the characteristics of MOT1.

The extraction process can be summarized as follows [18,19]. In MOT1 hot atoms are first decelerated by the MOT radiation pressure, then slowly moved to the centre of the trap where they are extracted by the pushing laser. In addition to its pushing effect, the laser beam shifts the atomic levels by a few natural linewidths so that a transverse cooling of the atomic beam takes place during extraction, limiting the initial atomic temperature to about 25 µK for Cs (40 µK for Rb). Moreover, the trapping forces are reduced and the pushing beam becomes dominant. Hence, atoms are extracted from the trap and accelerated in the direction of MOT2. After the transfer to the second chamber, the atoms are finally recaptured in MOT2 by radiation pressure.

In a first set of experiments, we study the flux of atoms extracted from the upper chamber. This outgoing flux depends on the number of captured atoms in MOT1, which is related to the background pressure of the alkali vapour. As there is no direct access to the background pressure value, we have measured the loading time of MOT1, which, in a large regime of operating parameters, is inversely proportional to the atomic pressure in the source cell. The number of atoms in a MOT in the stationary regime is [25]

N = L γ + γ p + βn , (1)
where L represents the loading rate of the MOT, γ is the loss rate due to background collisions, γ p gives the loss rate induced by the pushing laser, β is the rate of the cold twobody collisions between the trapped atoms, and n is the average atomic density in the MOT. The density in MOT1 is limited to about 10 10 atoms/cm 3 , so that the term βn is negligible in both setups.

Loading rates in both MOTs are given by the measured initial slope of the number of trapped atoms in the MOT versus time. In MOT2, this measure is performed after suddenly switching on the pushing laser and waiting for the arrival of the first atoms. In MOT1, the loss rate γ is inferred by measuring the 1/e-loading time τ (γ = 1/τ in a wide range of vapour pressure [25]) or by dividing the loading rate L 1 by the number of atoms N 1 measured when the pushing beam is off. When the pushing laser is switched on, the loss processes in MOT1 increase drastically. If N p 1 is the number of atoms in MOT1 in the presence of the pushing beam, then L out = γ p N p 1 is the flux of atoms leaking out of MOT1 through the optical guide. We deduce it from parameters we already measured via the formula:

L out = L 1 -γN p 1 . (2)
To get the data plotted on Figure 2, τ is tuned by varying the background pressure. Whatever the background pressure, the number N 1 of atoms is approximately constant. At high background pressure (i.e. low values of τ ), the outgoing atomic flux L out increases with the loading time τ because the number of atoms without pushing beam N 1 slightly does. Then at relatively low pressure L out decreases, following the behaviour of the MOT1 loading rate L 1 (inversely proportional to τ ). The loading rate of MOT2 L 2 and the number of atoms N 2 in MOT2 are presented as a function of τ on Figure 2. Their dependence with the MOT1 loading time is similar to that of the atomic flux L out . The overall efficiency of the transfer process is defined by the incoming flux in MOT2 divided by the outgoing flux from MOT1, that is L 2 /L out .

We conclude that for higher MOT2 loading rate we need a relatively high background pressure in MOT1 and a large laser power in the trapping beams (to have higher N 1 value). For our experimental conditions the optimum is at a MOT1 loading time of about 1-2 s. The data presented here were not taken with optimized pushing-guiding beam parameters, the efficiency being here limited to typically 10%. Once these parameters are well set we are able to achieve maximum transfer efficiency of about 70% for Cs (resp. 50% for 87 Rb), without affecting the overall dependence of the different quantities on the MOT1 loading time.

Pushing and guiding processes

After leaving the MOT1 region, the atomic beam is no longer affected by the MOT1 lasers and is guided due to the attractive dipolar force created by the red-detuned pushing-guiding beam. In this section, we describe the guiding process using an analytical model similar to that given in references [22,24]. The total force applied on the atoms is the sum of a radiation pressure "pushing force" F push and of a dipolar "guiding force" F = -∇U , where U is the guiding potential. The gravitational force plays a minor role in the loading process.

A two-level model function of the laser parameters (power, detuning and waist) describes qualitatively the experimental dependence of the transfer efficiency between the two MOTs. A more detailed quantitative analysis of the processes is proposed in the rest of this section.

Two-level model

In this first simple model we neglect gravity, the initial velocity and temperature of the atoms and beam divergence. We consider the atoms as a two-level system with a transition energy hω 0 , a natural linewidth Γ (Γ/2π = 5.2 MHz for Cs, 5.9 MHz for Rb) and a saturation intensity I s = 1 6 hck 3 Γ 2π (1.1 mW/cm 2 for Cs, 1.6 mW/cm 2 for Rb). We use here z as the vertical coordinate along the laser beam propagation with origin in the centre of MOT1 and r for the radial cylindrical coordinate (see Figure 1). For this two-level model, the waist w of the pushing-guiding laser is assumed to be constant and equal to its experimental value at MOT1 position z = 0. The laser beam has a power P 0 , a wave vector k = 2π/λ, and an angular frequency ω detuned by δ = ω -ω 0 with respect to the atomic transition.

The on-axis light shift is given by U 0 = hδ 2 ln(1 + s) where s = (I/I s )/(1 + 4δ 2 /Γ 2 ) is the saturation parameter and I = 2P 0 /(πw 2 ) is the peak laser intensity. As the laser is far detuned, saturation is always very low and one can simply replace ln(1 + s) by s in this expression. In this limit, the guiding potential is

U 0 e -2r 2 w 2 . (3)
As the waist is considered constant, the guide does not affect the longitudinal motion. On the contrary, it is crucial for confining the transverse motion.

The atoms absorb and emit spontaneously photons at a rate

Γ ′ = Γ 2 s 1 + s = Γ 2 I/I s 1 + 4 δ 2 Γ 2 + I/I s , (4)
which gives a pushing force

F push = Γ ′ hk = Γ ′ M v rec , (5)
where v rec = hk/M is the recoil velocity and M the atomic mass. The velocity increases due to photon absorption, and the number of scattered photons to reach the position z is approximately v(z)/v rec = 2Γ ′ z/v rec . The pushing process is also responsible for a heating due to random spontaneous emission in all directions. The mean horizontal kinetic plus potential energy per atom 2k B T in the 2D confining potential is increased by two third of the recoil energy E rec = M v 2 rec /2 = k B T rec /2 at each scattering event [26,24]. This gives rise to a horizontal kinetic temperature

T h (z) = v(z) v rec T rec 6 . (6)
To have an efficient pushing-guiding beam we require in this simple two-level approach that the atoms remain trapped in two dimensions inside the guide during the whole transfer. This condition is

2k B T h (z) < |U 0 | for all z. (7)
As the horizontal velocity spread increases with z, this is equivalent to 2k B T h (D) < |U 0 |. A second constraint is that the beam velocity at the MOT2 position (v beam ) should be lower than the capture velocity (v capture ) of the MOT

v beam < v capture . (8)
The value of v capture is on the order of the maximal velocity for an atom to be stopped on the MOT beam diameter distance 2R, that is v capture = √ Γ Rv rec [25]. As a result, we evaluate v capture to be about 21 m/s for cesium and 30 m/s for rubidium.

The efficiency of the pushing-guiding process is determined by how deep the conditions ( 7) and ( 8) are verified. To describe qualitatively the guiding efficiency in relation with these conditions, we propose to describe each condition by a function f , equal to zero when the inequality is strongly violated and to 1 when it is fully verified, with a continuous transition between these two extremes. The guiding efficiency is then described by the product

f ( 2kB T h (D) |U0| ) × f ( v beam
vcapture ) of the two conditional functions. The result is given for Cs in Figure 3 as function of the laser detuning, with v capture = 21 m/s and the function f chosen arbitrarily to be f (x) = 1 1+x 10 . A comparison of the two-level model with experimental results (see Figure 7, left) presents a good qualitative agreement, reproducing the presence of an optimal red detuning at given laser power. The maximum transfer efficiency increases with the power of the pushing beam while the position of the peak is shifted to larger absolute values of the detuning. This simple model is sufficient to derive the main conclusion: the transfer is more efficient with a far red-detuned and intense laser beam. However, the theory predicts a peak further from resonance than observed experimentally. Moreover, the sensitivity to the laser power is much more pronounced than observed in the experiment. This motivates a more detailed analysis of the processes operating during the travel of the atoms from MOT1 to MOT2. In particular, the effect of optical pumping to the lower hyperfine state has to be considered.

Optical pumping

The absorbed photons can lead to optical pumping between the two hyperfine levels of the ground state which have different laser detuning with respect to the pushing laser. Indeed, very quickly after leaving the MOT1 region, the atoms are pumped essentially in the lower ground state F = 3 for cesium (resp. F = 1 for rubidium) as there is no repumping laser light superimposed with the pushing laser beam. This optical pumping is essential for a good transfer efficiency, as it greatly reduces the final velocity of the atomic beam (see sections 3.6 and 4.2). However, a small population in the other ground state is still present, typically 1 to 3 percent for a linearly polarized beam, as we shall see [27]. As the radiation pressure is much larger for atoms in the upper ground state (about 100 times larger for detuning values discussed here), even this small fraction plays a role and both ground state populations have to be taken into account for the estimation of the pushing force. On the contrary, the dipolar force may be estimated by assuming that the atoms are only in the lower ground state, as this force is only about 10 times smaller than in the upper ground state, which is 100 times less populated.

An estimate of the populations in the ground states is obtained by assuming an equal detuning for the transitions from the upper hyperfine ground state to all the hyperfine excited states. We define an "effective" detuning δ ≈ δ + ∆ ′ HFS /2, where ∆ ′ HFS is the total width of the hyperfine structure in the excited state (∆ ′ HFS ≃ 600 MHz for Cs and ∆ ′ HFS ≃ 500 MHz for 87 Rb respectively). Using this mean detuning δ we calculate the pumping rates between the two hyperfine ground states. This is fairly good for large detunings (above 1 GHz from the cycling transition). To illustrate our results we will choose the following typical values: δ/2π = -2 GHz from the (F=4→F'=5) transition of the Cs (i.e. δ/2π = -1.70 GHz) and δ/2π = -1 GHz (i.e. δ/2π = -750 MHz) from the (F=2→F'=3) transition of the 87 Rb. We also define ∆ HFS as the hyperfine structure interval in the ground state (2π × 9.2 GHz for Cs, 2π × 6.8 GHz for 87 Rb) (See [START_REF] Daniel | Alkali D Line Data[END_REF]).

The ratio of populations in the upper hyperfine ground state N F +1 and in the lower one N F may then be estimated as:

η = N F +1 N F + N F +1 ≈ N F +1 N F = α δ δ -∆ HFS 2 (9)
with α = 2F + 3 2F + 1 = 9/7 for Cs (F = 3) 5/3 for 87 Rb (F = 1) .

The factor α is simply the ratio between the number of substates in the F + 1 and F ground states, to which N F +1 /N F should be equal at a detuning large as compared to the hyperfine structure ∆ HFS ; the term involving the detuning is related to the ratio of excitation rates from the two hyperfine ground states. The formula leads to η = 3.2 % of the atoms in the Cs(6s, F = 4) state and η = 1.6 % in the Rb(5s, F = 2). This value is in excellent agreement with a full calculation taking into account all the different detunings with the hyperfine excited states.

Pushing force

Another factor should be considered: the laser mode shape. Indeed, a relatively strong divergence is needed in order to both efficiently push and guide atoms in the MOT1 region and not affect the MOT2 operation. The guiding beam waist varies with position, according to

w(z) = w 0 1 + (z -z 0 ) 2 /z 2 R . (10)
The depth U 0 is then modified along the atomic trajectory due to the change in the laser intensity and, taking into account the results of the previous section, the pushing force in the centre of the beam may be estimated as follows:

F push (z) = Γ 2 hks(z) (1 -η) + η δ -∆ HFS δ 2 ≃ Γ 2 hks(z)(1 + α), (11)
where s(z) is the saturation parameter calculated for the lower ground state, at detuning δ -∆ HFS . We take into account the linear polarization of the pushing beam by multiplying s by a factor 2/3 in all calculations. We neglect however the small change in δ due to the light shift, which is even reduced when the waist w(z) becomes larger.

The mean pushing force is reduced due to the Gaussian transverse profile of the pushing beam and the finite size of the atomic cloud. This may be taken into account approximately by dividing the force by a factor 2 [24]. Note that this underestimates the initial pushing force, when the atoms are still well guided (rms radius less than w(z)/2), and overestimates it when the cloud size becomes larger than half the waist. As the mean pushing force now depends only on z, it may be written as the derivative of a "pushing potential" U push :

U push (z) = Γ 2 hks 0 z R (1 + α) arctan z -z 0 z R . ( 12
)
where s 0 = s(z 0 ). The velocity at each point is then easily calculated by energy conservation (z axis is oriented downwards):

v(z) = v 2 0 + 2gz+ (13) Γ v rec s 0 z R (1 + α) arctan z -z 0 z R + arctan z 0 z R 1/2
, where v 0 is the input velocity in the guide. The effect of gravity is not dominant, but was taken into account by the 2gz term. v 0 can be estimated as the output velocity of the MOT1 region. We have calculated it using formula (13) assuming that the atoms in the MOT1 region have a zero initial velocity and are in the upper hyperfine ground state due to the presence of the repumping light (η = 1). For instance, using a travel distance z roughly equals to the MOT1 region radius (10 mm) and a laser power of 21 mW, we find that atoms enter the guide with a velocity v 0 ≈ 9 m/s for Rb; for the Cs parameters, we obtain in the same way v 0 ≈ 3.1 m/s. From Equation ( 13), we also infer the traveling time as ∆t = D 0 dz v(z) .

Guiding condition

As previously discussed, the light shift of the lower ground state is dominant in our case. The atoms leaving MOT1 are thus guided by the on-axis light shift potential given by

U 0 (z) = h( δ -∆ HFS ) 2 s(z). ( 14
)
Equation ( 7) is still the strongest constraint for the choice of the parameters and becomes more and more difficult to fulfil as z increases, because |U 0 | is reduced due to the beam divergence. The horizontal kinetic temperature T h (z) is evolving due to two opposite effects: photon scattering [29] is responsible for an increase of T h while adiabatic cooling tends to lower it as the waist increases. The adiabaticity condition |dω p /dt| ≪ ω 2 p , where ω p is the transverse oscillation frequency of the guide, is well fulfilled in both experiments except when the atoms move in the non harmonic part of the potential. This break-down of the adiabaticity occurs only when the atoms are close to leave the guide. This only marginally affects the guiding condition and will not be taken into account here. ω p varies with the inverse squared waist, and one has ω p (z) = ω p (0)w 2 (0)/w 2 (z) = ω p (0)

z 2 0 +z 2 R (z-z0) 2 +z 2 R
. To obtain an expression for T h (z), valid while the atoms remain guided, we write the change in T h for a small change δz in z. As the phase space density is conserved during this adiabatic cooling, the cooling contribution is proportional to the inverse squared waist. Spontaneous scattering is responsible for a supplementary heating term, proportional to the number of photons scattered during δt = δz/v:

T h (z + δz) = T h (z) w 2 (z) w 2 (z + δz) + Γ 2 s(z) T rec 6 δz v(z) . ( 15
)
The temperature increase is T rec /6 for each spontaneous scattering event. s(z) is proportional to 1/w(z) 2 , just like the oscillation frequency. Using the dependence in w(z), we obtain the following differential equation for T h :

dT h dz = -T h (z) 2 w(z) dw dz + T rec 6 w 2 (0) w 2 (z) Γ 2 s(0) 1 v(z) (16)
Using the expression of w(z), the solution of this equation reads:

T h (z) = z 2 0 + z 2 R (z -z 0 ) 2 + z 2 R T 0 + T rec 6 Γ 2 s(0) z 0 dz ′ v(z ′ ) ( 17
)
where T 0 is the initial temperature at the guide entrance. The integral in the last term is the time necessary for an atom to travel to position z. In the range of parameters explored in our experiments, the sum of these two terms decreases with z, but slower than the trap depth. As can be seen on Figure 4 (left), the mean horizontal energy 2k B T h becomes larger than the trap depth at some position z out before reaching MOT2. However, as will be discussed below, this partial guiding is sufficient for limiting the size of the atom cloud to below the MOT2 beam diameter.

Recaptured atoms

For a good transfer efficiency, two main criteria have to be fulfilled. First, the atomic beam should stay roughly collimated on a distance long enough to pass through the differential tube, and then the transverse cloud radius at the end should be comparable to the capture radius of MOT2. This means that even if they leave the guide before reaching MOT2, the atoms can still be recaptured. Second, the final longitudinal velocity of the atomic beam must not exceed the capture velocity of MOT2. As the atomic beam velocity is in any case lower than the capture velocity of MOT2, the recapturing mechanism is mostly limited by the matching between the atomic beam size and the size of the capturing region of MOT2. The point zout where the atoms leave the guide is marked by a vertical line. Left: mean horizontal energy 2kBT h (thin lines, Equation 17) and trap depth |U0| (bold lines, Equation 14). Right: rms radius of the guided atomic beam. The radius of the MOT2 beams is marked by a horizontal line.

The capture size of MOT2 is limited by the radius R = 4 mm of the collimated trapping laser beams. According to the former considerations about heating of the guided atoms (see 3.4), the mean horizontal energy of the cloud is lower than the guiding trap depth over a distance z out = 38 cm for a laser power of 63 mW and an initial temperature T 0 = 25 µK in the case of Cs (resp. z out = 28.5 cm with T 0 = 40 µK and a laser power of 21 mW in the case Rb) (see Figure 4). For simplicity we consider hereafter that all the atoms remain pushed and guided up to that point and then undergo a free ballistic expansion as they keep falling. Including this assumption in our model, we can evaluate the size of the atomic cloud ∆r f as it reaches MOT2. While the atoms remain trapped, the cloud size is of the order of ω -1 p (z) k B T h /m. The guiding step ends when k B T h reaches |U 0 (z)|/2, such that the rms size at the guide output is ∆r out = ω -1 p (z) |U 0 (z)|/2m = w(z)/ √ 8, that is ∆r out = 470 µm for Cs (resp. ∆r out = 200 µm for Rb). We assume a fixed temperature for the falling atoms, as the adiabatic cooling is not efficient for a non trapped cloud and the heating rate is also very low after z out . T h is about 10 µK for Cs and 25 µK Rb. After the remaining falling time of 36 ms (resp. 36 ms for Rb) the atomic beam has a typical standard deviation for the transverse Gaussian atomic density distribution of ∆r f ≈ 1 mm for Cs and ∆r f ≈ 1.75 mm for Rb, smaller than MOT2 radius, meaning that almost all the atoms are recaptured in MOT2 for both experiments. Note that this model allows to predict ∆r(z) at any position z, as shown on figure 4, right.

Transfer efficiency

We come back now to an estimation of the transfer efficiency as discussed in section 3.1 and presented in Figure 3. Within the frame of the refined model presented now, we are able to compute a transfer efficiency in the same spirit. As we have seen in the previous section, the guiding is not required until the end for the whole cloud to be recaptured. We thus retained the two following conditions: (i) the arrival velocity has to be smaller than v capture and (ii) the cloud size must be less than the MOT2 beam waist. We then calculate the efficiency f [∆r(D)/R]× f [v(D)/v capture ], with the function f previously used in section 3.1, and plot it on Figure 5. The model predicts a good efficiency in a detuning range between -0.5 GHz and -1.6 GHz, the width of the large efficiency region being reduced with a smaller laser power. These predictions have to be compared with the rubidium experimental data of Figure 7, right. The agreement is qualitatively good, and reproduces the main features. The two limits of the large efficiency region have different origins: On the large detuning side, the efficiency drops due to an increase in the atomic cloud size, as the guiding potential is weaker. On the lower detuning side, close to resonance, the efficiency becomes limited by the final velocity, which is larger than the capture velocity of MOT2. On this side, theory fails to predict the less measured efficiency at lower laser power, as the mean detuning δ approach (section 3.2) is not valid any more. In particular, the efficiency should drop to zero at the resonance with the rubidium 5S 1/2 (F = 2) → 5P 3/2 (F ′ = 1) line, situated at δ/2π = -424 MHz and marked with a vertical line on Figure 5.

Experimental results

In this section, we present the experimental study of the guiding process and compare it with the above theoretical model. The dependence of the recaptured atom number on the pushing beam parameters are first investigated. We then measure the mean atomic velocity and the traveling time. During the experimental investigation the atom vapour pressure in MOT1 is kept constant.

Pushing Beam Parameters

The parameters of the pushing beam that we have experimentally optimized are its divergence, waist, power and detuning.

Divergence and waist In order to optimize the atomic beam characteristics we have first investigated the role of the laser beam waist, related to the divergence of the pushing beam and to the pushing force. It is clear that the pushing-guiding beam should diverge, to have a significant effect on MOT1 without disturbing MOT2. Moreover, this divergence provides an horizontal adiabatic cooling of the guided atoms. We have used three different lenses (f = 0.75 m ; 1 m ; 2 m) to focus the pushing beam. For each lens the transfer efficiency is studied as a function of the focus distance from MOT1. The position of the lens is more critical than its focal length. The optimum is obtained with a lens f = 2 m for the Cs experiment (resp. f = 1 m for 87 Rb) and distance from MOT1 34 cm (resp. ≈ 13 cm), where the beam diameter on the MOT1 region is ≈ 1.3 mm (resp. ≈ 0.6 mm). The measured waist at the focal point is 200 µm (resp. 300 µm). It leads to a divergence w 0 /z R of about 2 mrad (resp. 1 mrad).

In conclusion we found that the best transfer efficiency occurs when the pushing beam, focused before MOT1, has a diameter smaller than 1 mm in MOT1 and a divergence such that the beam diameter at MOT2 position is less than 3 mm. In this sense our results are similar to the one found in references [18,19].

Power and Detuning

The recaptured number of atoms into MOT2 at different laser powers of the pushing beam and at different detunings for the two elements Cs and 87 Rb is shown resp. on left and right of Figure 6.

It is first obvious that the best experimental conditions are achieved with a laser frequency red-detuned with respect to all atomic transitions (curve (a) in Figure 6, left). The transfer efficiency is larger for a red-detuned laser frequency than for the other laser frequencies due to the fact that after leaving the MOT1 area the atoms feel the pushing light also as a guide. For such detunings, the atomic flux as well as the number of recaptured atoms N 2 in MOT2 increase when the power of the pushing light increases, and saturates at large power when all the atoms are efficiently guided to MOT2 (see also Figure 6,right). At a given detuning, an increase of the laser power leads to a decrease of the transfer efficiency, due to an excessive final velocity, a strong perturbation of both MOTs, and a large heating of the atoms.

In order to optimize the conditions for the atomic beam, the influence of the detuning of the pushing light was investigated in more details (see Figure 7 left and right resp. for Cs and 87 Rb). For a frequency close to resonance (corresponding to the best conditions found in reference [18,19]) the number of recaptured atoms into MOT2 is much smaller than the one we could achieve with a much more red-detuned light and a higher power. In conclusion we find that the best loading of MOT2 is at highest possible power of the pushing laser beam and, given this power, at the value of red detuning optimizing the flux.

Atomic beam velocity

For a high recapture efficiency, a relatively slow and collimated atomic beam is required (see section 3.5). After the pushing and guiding process, the atoms reach MOT2 within a time delay ∆t. This time has been measured in two different ways. First, one can record the MOT2 fluorescence after having suddenly removed the atoms in MOT1 (the MOT1 laser beams are stopped by a mechanical shutter). In this case, one observes the delay after which the number of atoms in MOT2 starts to drop. The second method consists in pulsing the pushing beam through a permanently loaded MOT1. Both methods lead to the same result ∆t ≈ 130 ms for the Cs experiment at 63 mW power. In the Rb experiment presented in Figure 8, the measured time delay as a function of the pushing beam power is obtained by using the second method. A similar dependence on the pushing beam power is observed in the cesium experiment. The two-level model is not sufficient to describe accurately the atomic beam velocity, the predicted transfer time ∆t being by far too short (see Fig- ). The theoretical calculations are done for both the two-level model approximation (blue lower curve) and for the more detailed model described above (red upper curve). In the calculations the radius of MOT1 trapping region is 10 mm. ure 8, lower curve). On the contrary, the theoretical model presented in section 3.3, Equation (13) describes well the experimental results as demonstrated on Figure 8. From the model, we also deduce the final longitudinal velocity of the atomic beam v ≈ 5.5 m/s (resp. 12.6 m/s for Rb). Note that this final velocity is not very different from the mean velocity D/∆t, as the acceleration stage takes place essentially in the MOT1 zone, where the atoms remain in the F + 1 state thanks to the repumping MOT beams.

Conclusion

In our work we have studied a very efficient setup to transfer cold atoms from a first MOT to a second one. Our setups have a similar geometry to the ones described in references [18,19], but due to the higher laser power (tens of mW) we could achieve a partial dipolar guide for the atoms at a larger detuning (1 GHz typically). As a result, the mean longitudinal velocity of the atomic beam is lower (4.3-12 m/s) than in these previous experiments (15 m/s). Moreover, thanks to the lower sensibility of the method to the frequency of the pushing laser, its frequency does not need to be locked (see for instance the detuning dependence in Figure 7, right) and the setup is much more robust to small mis-alignments of the pushing beam. The atomic flux is limited only by the number of atoms loaded into MOT1. We estimated the transfer efficiency to MOT2 which is about 70 % for the 133 Cs experiment and about 50 % for the 87 Rb experiment.

We used a two-level system model to describe the processes during the atomic transfer. A good qualitative agreement between theory and experiment was found. The transfer efficiency is maximum for a large red detuning, and this maximum efficiency increases with the laser power. A more detailed discussion of the pushing, guiding and recapture processes is presented for a better understanding of the atomic transfer between the two traps. Our theoretical description, which takes into account the optical pumping, the pushing force and the guiding potential nicely reproduces the experimentally observed traveling time.

In conclusion, we experimentally described and theoretically modelled a method to transfer cold atoms between two traps. Two different setups lead qualitatively to the same optimized parameters -a large laser power (tens of mW), ≈ 1 GHz detuning, 300 µm waist. The implementation of this technique in our setups brought in both cases a much better stability and improved loading efficiency, with respect to the use of a near resonant laser beam.

This work was partially supported by European RTN networks COMOL, QUACS, FASTNet and Atom Chips. LAC and LPL are members of IFRAF. LAC is UPR 3321 of CNRS associée à l'Université Paris-Sud and member of LUMAT FR 2764, and LPL is UMR 7538 of CNRS and Université Paris 13. We are very grateful to Brigitte Mercier for giving us the seminal idea of this pushing-guiding scheme. The authors also thank Laurent Vernac for helpful discussions on optical pumping and for providing us a computing code on the calculation of the populations in the magnetic sublevels.

Fig. 1 .

 1 Fig. 1. Scheme of the experimental setups. The parameters used in the discussion (f , D, z0, w0) are labeled on the picture. The vertical z axis is oriented downwards.

Fig. 3 .

 3 Fig. 3. Efficiency (see text) of the pushing-guiding processes versus laser detuning δ for a 650 µm waist laser beam with different laser power 10 mW (dashed line ×1000) and 46 mW (solid line). The other parameters are: initial temperature T0 = 25 µK, Isat = 1.1 mW/cm 2 , Γ = 2π ×5.2 MHz, vrec = 3.5 mm/s, Trec = 0.2 µK and the two MOT cells are separated by 57 cm (the values used are those of the Cs experiment).

Fig. 4 .

 4 Fig.4. Evolution for the87 Rb experiment of different parameters with the traveling distance z between the two MOTs, at different powers: 15 mW (dashed lines) and 21 mW (solid lines); the initial temperature at the guide entrance was set to T0 = 40 µK. The point zout where the atoms leave the guide is marked by a vertical line. Left: mean horizontal energy 2kBT h (thin lines, Equation17) and trap depth |U0| (bold lines, Equation14). Right: rms radius of the guided atomic beam. The radius of the MOT2 beams is marked by a horizontal line.

Fig. 5 .

 5 Fig. 5. (Color online) Efficiency (see text) of the pushingguiding processes versus laser detuning δ, calculated for 87 Rb for the parameter of the pushing beam given in text, with different laser power 10 mW (thin solid line, black), 15 mW (dashed line, red) and 21 mW (thick solid line, blue). The maximal capture velocity in MOT2 has been fixed to vcapture = 30 m/s, the initial temperature to T0 = 40 µK and the MOT2 beam radius R to 4 mm. The corresponding experimental values are shown on Figure 7, right.

Fig. 6 .

 6 Fig. 6. Dependence of the number of recaptured atoms N2 in MOT2 on the pushing beam power. Left: Cs data. The experiment is done at four different frequencies (see the diagram in the centre). Right: 87 Rb data, recorded at -1 GHz detuning from the 5S 1/2 (F = 2) → 5P 3/2 (F ′ = 3) transition.

Fig. 7 .

 7 Fig. 7. (Color online) Number of atoms recaptured in MOT2 N2 vs. pushing beam detuning for different optical powers. The vertical lines indicate the position of hyperfine resonance frequencies. Left: Cs data. The detuning is given with respect to the 6S 1/2 (F = 4) → 6P 3/2 (F ′ = 5) transition. The pushing beam power is 46 mW (squares), 10 mW (stars) or 2 mW (circles). Right: 87 Rb data. The frequency is measured relatively to the 5S 1/2 (F = 2) → 5P 3/2 (F ′ = 3) cycling transition. Pushing beam power: 10 mW (squares), 15 mW (circles), and 21 mW (triangles).

Fig. 8 .

 8 Fig.8. (Color online) Experimental (points) and theoretical results (solid lines) for the traveling time ∆t between MOT1 and MOT2 for different pushing beam powers. The beam is red-detuned by 1 GHz from the cycling transition of 87 Rb, 5S 1/2 (F = 2) → 5P 3/2 (F ′ = 3). The theoretical calculations are done for both the two-level model approximation (blue lower curve) and for the more detailed model described above (red upper curve). In the calculations the radius of MOT1 trapping region is 10 mm.


Diana M Lewis

A comparable-corpus based approach to the expression of obligation across English and French

Keywords: deontic modality, obligation, contrastive genre analysis, comparable corpora, political discourse, French, English

A French-English comparable corpus of political discourse is used to investigate the expression of obligation across the two languages. The aims of the study are to look at the expression of obligation in the comparable genres in the two languages and to compare the contexts of use of deontic modal and semi-modal verbs, i.e. comparing their frequencies in contextual frames characterized by agentivity, polarity and event type.

The focus is on the modal and semi-modal verbs must, need to, have to, devoir and falloir. While have to and falloir are more frequent than must and devoir respectively in spontaneous spoken language, the reverse is the case in the political speeches. The five verbs are found to occur in similar contexts within and across the two languages in the genre in question. The study highlights the potential impact of genre on frequency and distribution and the interactions between genre-based patterns and ongoing change in the wider languages.

Introduction

The increased ease of creating genre-specific corpora, together with the recent 'revival' of contrastive analysis over the turn of the 21 st century, have given a new impetus to contrastive genre analysis, shedding light on the functioning of genres across languages.

In his pioneering work on corpus-based genre analysis, Biber (1988:70) drew a crucial distinction between genre and text type1 . In essence, genres are defined by their sociocultural characteristics and text types by their linguistic characteristics. A genre is therefore to be conceived of, in an extensional definition, as the language actually used across events belonging to a recognized sociocultural event type. An event type in this context is some socially established, conventional task. It arises out of what Hyland (2009:211) calls "perceived repeated situations" and is characterised mainly in terms of form (the participants and the sequence of actions that compose the event), distribution (times, frequency, place) and purpose (objectives of the event). A text type, again extensionally, is a set of texts that share certain distributional patterns of linguistic features to a degree not found in other texts. This theoretical distinction between genre and text type is important in at least three respects. Most obviously, it is behind good practice in setting up the criteria for text selection when building genre-specific corpora; such criteria exclude linguistic features. A text belongs to a particular genre to the extent that its context of production and reception can be identified as an exemplar of a some socially-recognized event type. Secondly, if it is in principle possible to independently establish genres and text types, we can discover to what degree they line up, how and to what extent a given genre is linguistically distinguished from others, or is homogenous or heterogeneous. And thirdly, of course, it is sociocultural parameters that provide the situational tertium comparationis that enables contrastive genre analysis to be done at all.

The present study focuses on the expression of obligation in the political speech genre across English and French. The aims are threefold: (i) to look at the distribution of the commonest expressions of strong obligation in the comparable genres in the two languages, (ii) to examine the contexts of use of these modal and semi-modal verbs of obligation, and (iii) in the light of the polysemy of the quasi translation equivalents must and devoir, to compare the use of these verbs in the genre.

Patterns of usage may be genre-dependent. Where there is a 'marked' or atypical distribution of modal markers in a particular genre, there may also be an atypical distribution in the 'equivalent' genre in another language. Testing such a hypothesis would require comparisons among a wide range of matched genres that are beyond the scope of this paper. We shall look out for discourse-pragmatic similarities across the two languages English and French in one matched genre, that of political discourse, to see how they interact with linguistic choice.

In the light of the above discussion of 'genre', it is clear that each genre belongs to a particular discourse community or 'community of practice', and that we cannot therefore strictly speak of a 'political-speech genre'. For the purposes of the present study, the term 'language genre' will be used for convenience to refer to 'equivalent' genres in different language communities.

Section 2 outlines the notional area of obligation and looks in particular at its expression in English and French. The corpus on which the present study is based is described in section 3, and section 4 presents the findings of the study of three English verbs of obligation (must, need to and have to) and two French ones (devoir and falloir). Section 5 provides a conclusion.

Obligation markers in contrast 2.1 The notional area of obligation

This section considers the nature of obligation as an area of meaning, or semantic category, and the issues that arise in attempting cross-linguistic comparisons of the linguistic expression of obligation.

Obligation is traditionally studied under the rubric of modality. However, there is no general agreement among linguists on how to cut up the semantic space of modality into types of modality, or even on the boundaries of modality. A distinction is traditionally made between root modality, which pertains to the degree of necessity of the proposition in an utterance, and epistemic modality, which pertains to the degree of probability of the proposition (table 1).2 Other perspectives on modality have flourished, however, such as a division into internal and external modality, or into subjective and objective modality (v. van der [START_REF] Van Der Auwera | Modality's semantic map[END_REF][START_REF] Van Der Auwera | Modality: domains, layers and parts of speech[END_REF][START_REF] Palmer | Mood and Modality[END_REF], Heyvaert 2003:90ff;[START_REF] Depraetere | Mood and modality in English[END_REF] provides a brief and useful summary). Likewise, within the traditional framework, there is no generally-agreed approach to the sub-categorization of root modality. Nevertheless, there is a fair consensus that it is useful to identify an area of modality that involves human-generated precepts about the desirability, or otherwise, of people bringing about particular states of affairs, and this area is usually referred to as deontic obligation and permission3 .

Root modality

Deontic necessity Obligation Permission

Non-deontic necessity

Dynamic modality

Epistemic modality

Probability

Evidentiality

Table 1. The place of obligation in a traditional schema of types of modality

From the perspective adopted here, obligation and permission can be seen as poles at either end of a continuum of desirability, stretching from moral necessity (obligatory) to moral acceptability (permissible). 'Moral desirability' can be thought of as a broad space ranging from legal requirements and widely accepted social norms to personal ethics, opinions and wishes. Obligation is then a scalar category, ranging from strong obligation to absence of obligation, and the other side of the coin is permission.

Deontic modality as a semantic category can be defined, according to Nuyts, "as an indication of the degree of moral desirability of the state of affairs expressed in the utterance, typically, but not necessarily, on behalf of the speaker" (2006:4). The 'deontic source', i.e. the creator of the obligation, has traditionally been considered an important parameter in modality studies (v. Lyons 1977:825ff). The deontic source may be 'internal' (the obligation is created by the speaker) or 'external' (the obligation is imposed by a third party and reported by the speaker). A difference in source is potentially reflected in a difference in linguistic expression, i.e. grammaticalized or lexicalized. The speaker can choose her expression to indicate the degree to which she assumes responsibility for the obligation. Thus, the imperative is normally understood to express a speaker-created obligation. The English modal verb must tends to be associated with speaker-created obligation, and the semi-modal have to with external, speaker-reported obligation. However, the internal-external distinction is often blurred, and the correlation with must and have to is far from systematic.

For van der Auwera and Diewald, the investigation of "which markers express which meanings and why", and the discovery of how much variation there is across languages is the most important reason for studying modality (2012:123).

The expression of obligation in English and French

Obligation in both English and French is semi-grammaticalized, being typically expressed by a range of grammatical and lexical means, most notably modal auxiliaries, semi-modals (such as English have to, have got to or need to), modal verbs, modal adjectives and adverbs, as well as by imperatives and speech act verbs of the type demand or require.

Obligation, as seen above, stems ultimately from someone's desire or will that some state of affairs (event or situation) be (or not be) realized, and it therefore concerns unrealized situations, especially -arguably exclusively --future ones. As Lyons points out, "there is an intrinsic connexion between deontic modality and futurity " (1977:824). Once the desired action or event is realized, the modality is lost. A distinction is to be made, then, between an affirmation of obligation and a deontically modalized proposition of obligation. This distinction is partially grammaticalized in English by the choice of verbal expression, as can be seen in the following example constructions: [Subject had to V] (affirmation, past), dynamic, such as an accomplishment.

[Subject ought to have V-en] (counterfactual, past) and [Subject must V] (deontically modalized, future). But there is no clear-cut correspondence between linguistic expression and the realized/unrealized divide. In French, the distinction may be made by the choice of verbal form, as can be seen in the following example constructions: [Subject a dû V]' (affirmation, past), [Subject aurait dû V] (counterfactual, past) and [Subject doit/devra V] (deontically modalized, future).

A major difficulty in contrasting expression of modality across languages is that of maintaining a coherent notional tertium comparationis (TC). The natural TC in this case is the notional space of deontic obligation, but, as is clear from the discussion above, the domain lacks clear-cut boundaries. First, both the more grammatical (e.g. modal verbs) and the more lexical expressions (e.g. modal adjectives) most closely associated with obligation tend to range over other domains too, and so are potentially ambiguous or vague. Such vagueness is often exploited by speakers to save face or to implicate something without committing themselves. Speakers can thus use so-called modal-evaluative adjectives such as It's essential to.., It's important to .., or modal-adverbials such as You had better .. and so on to convey an obligation. Second, speakers can employ expressions having as yet scant 'obligation' sense at all, such as want to and need to, but which, in the right context (as in You want to get your hair cut), the speaker can use to impose an obligation on the hearer. Such strategies may be used from considerations of politeness: expressing an obligation can be face-threatening, so that a speaker may choose to use a weak-obligation expression, a neutral one or even a permissive, for what she perceives as an obligation. Examples include such constructions as, in English, be for <someone> to + V, involves V-ing, the time has come to +V, and similarly in French à <quelqu'un> de +V (roughly 'up to <someone> to V'), il s'agit de +V ('it is a matter of V-ing'), le moment est venu pour +V ('the moment has come to V'). An example of this kind of inferred obligation is given in (1).

(1)

L'avenir consistera pour l'Europe à assumer des responsabilités difficiles. [Alliot-Marie 2002] 'The future for Europe will consist of taking on difficult responsibilities.'

Depending on the contexts and the expressions, the 'obligation' implicatures, in principle defeasible, can be strong. The communicative intent in (1) is clearly to convey that an obligation exists. Deontic modality, then, is not only expressed through lexemes whose main or sole function is modal, but also through a variety of constructions not usually associated with modality. These usages, of course, can presage incipient grammaticalization of lexical constructions into markers of modality. Within deontic obligation, distinguishing internal from external obligation is especially tricky. In English and French, where the same constructions serve for both, the source of the obligation can only be pragmatically inferred, and is frequently interpreted as vague. The speaker's will is often in harmony with some external will, so that the source is simultaneously external and internal. For example, Je dois arriver avant midi ('I must arrive before noon') may mean that the speaker has a self-imposed (internal) obligation to arive before noon or that she has an externally-imposed obigation to do so, or again that she predicts that she will arrive before noon. But it can also mean all of these simultaneously. This paper focuses on the 'central', semanticized modal markers of obligation, not on whether obligation figures to a similar extent in the two language genres. The verbs have been chosen because (a) to the extent that they carry roughly the same 'weight' of obligation they are often interchangeable and (b) they are dictionary equivalents across French and English. Expressions of mitigated obligation such as should or ought to in English or il faudrait or devrait in French are therefore excluded, as are modal adjectives and adverbs. The aim is to look at high-frequency verbal expressions of obligation in each language and to examine their distribution in the texts, with a view to comparing their contexts of use in the two language genres, and the ways in which the aims of the speakers may be reflected in the linguistic patterns of the genres.

The high-frequency verbs most closely associated with obligation are, for French, devoir and falloir and for English must, have to and need to. All of them are also very frequent in the political speech language genres sampled here. The conceptual space occupied by these five verbs is wide: they can all express a range of contiguous modal meaning and lend themselves to wider or narrower pragmatic interpretations depending on context. 4Occurrences that correspond solely or mainly to deontic obligation, therefore, are not always straightforwardly distinguishable in discourse. Figure 1 maps the five expressions on to a partial semantic map of modality. A semantic map is "a geometric representation of meanings or, if one likes, uses, and of the relations between them" (van der Auwera and Plungian 1998:86). Semantic maps are useful for graphically representing polysemy, partial synonymy and cross-linguistic mapping. In fact they have been most extensively used in typological studies to map grammatical functions and word senses across languages. Such mapping allows typologists to build up a representation of conceptual space based on (a) the ways in which it is grammaticalized and lexicalized by different languages and (b) regularities in semantic shifts over time.5 (1998:90-97). The background area represents a postulated conceptual space, on to which we can map the (prototypical) senses of linguistic expressions. Lower case labels refer to semantic notions; upper case labels refer to (prototypical uses of) lexical items of English and French. The central area corresponds to the 'root modality' row of table 1, that is, to the modal notional area of root necessity. The large circle (centre top) represents deontic necessity6 , comprising that emanating from both internal and external sources. The small ring (centre bottom) represents physical, non-deontic necessity (exemplified in ****). The arrows indicate the directions of frequently-attested diachronic developments (semantic shift or semantic expansion and grammaticalization). Thus, the three rings to the left show non-modal notions which cross-linguistically often share a lexical item with a root modality notion. The two rings to the right show the non-root modal notions of epistemic judgment and future time. 7 Again, the arrows show that these notions too often attach to a lexical item expressing a root modality notion. The semantic map therefore provides a rough way of visualising how the senses of lexical items map on to conceptual space and how they relate to each other. For the lexical items that concern us here, we can see in figure 1 that all of them range over two or more notional areas: they are polysemous as a result of semantic expansion. The map also graphically represents cross-linguistic overlap: devoir is used across a wider range of notions than must, for instance.

French 'devoir' and 'falloir'

French makes a basic grammatical distinction between the semi-modal devoir and the impersonal verb falloir, insofar as deontic devoir typically takes a human subject while falloir can only be used with dummy subject il, as in il faut que 'it is necessary that' or il faut + V 'it is necessary to V' (originating from the sense 'want' as in 'be lacking'). But this personal/impersonal distinction does not carry over neatly to the semantics or the pragmatics of the two verbs. Falloir can be personalised by the use of one of two constructions: il + oblique personal pronoun + faut/faudra as in il vous faut V ('you have an obligation to V') and il faut que + person +V subjunctive_mood as in il faut que Jean V ('John has an obligation to V').

In usage, devoir is said to be more solemn or more insistent than falloir, while falloir is more often used in 'subjective contexts' where devoir might be interpreted as epistemic or as expressing futurity (Larreya 2004:748-9). In fact, devoir has grammaticalized into a wide polysemous network of uses, so that ambiguous or vague uses abound. Two main areas of usage beyond Obligation have developed: (i) epistemic modality (deduction, along similar lines to the development of English must) and (ii) future marking (roughly equivalent in sense to the English [be [expected] to V] or [be due to V] constructions).

Overall, deontic devoir is more associated with formal registers, while falloir is more colloquial. 'must', 'have [got] to' and 'need to' By contrast, English must and have to are said to be distinguished along internal/external obligation lines, must being associated with speaker-created obligation, and have to with externally-imposed obligations (v. Palmer 2001:75). While must and have to are both deontic, need to primarily expresses 'objective' modality (non-deontic necessity) in contexts such as You need to be tall to see over this fence (v. Quirk et al 1985:226). This situation, however, is rapidly changing, so that the most salient fact about these obligation markers is the collapse of must over recent decades and its partial replacement by have to and, more recently, need to.

English

The decline of must in spoken English seems to have affected the root and epistemic uses equally, root uses staying at around one third of all uses through the period 1960s -1990s [START_REF] Close | Current change in the modal system of English: a case study of must, have to and have got to[END_REF] This affects both written and spoken English, and both epistemic and root must8 . For written English, [START_REF] Mair | Current changes in English syntax[END_REF] document a sharp decline in the use of must between the 1960s and the 1990s: -29% for written British English and -34% for American -while have to shows a very modest increase and need to increases in their data by a massive 249% for British English and 123% for American. [START_REF] Johansson | Modals and semi-modals of obligation in American English: some aspects of developments from 1990 until the present day[END_REF] shows that for American English need to has now overtaken must. [START_REF] Close | Current change in the modal system of English: a case study of must, have to and have got to[END_REF] find a halving of the frequency of must in spoken British English 1960s-1990s.

It has been suggested that social factors, notably politeness, may be behind the decrease in the use of the 'subjective' forms such as must [START_REF] Smith | Changes in the modals and semi-modals of strong obligation and epistemic necessity in recent British English[END_REF]). Yet the notion that must is subjective may need revising. Collins claims that "deontic must is more often used objectively than subjectively" (2009:37). His data suggest that 'objective' must conveys weak obligation, as in agentless passives "with an unspecified deontic source having no necessary connection with the speaker, where must merely expresses what is thought to be desirable" (2006:38). An example taken from the corpus of political speeches is given in (2).

(2) majority voting must apply across the board [Rifkind 1997]

And Larreya (2004:743n) points out that must is particularly frequent in very formal registers such as political speeches. He suggests that announcements of the decline of must are therefore premature. But, as will be seen, it may be rather that the high frequency in political discourse and other formal registers is due to occurrences of must in particular construction types, and reflects a usage characteristic of the genre, resulting in local pragmatic effects.

In both languages, there is scope for a great deal of ambiguity and vagueness in the use of these modal and semi-modal verbs, due largely to wide-ranging polysemy in the case of French and to major ongoing change in the modal verb system in English.

The corpus

The study is based on a comparable corpus of political speeches from the United Kingdom and from France. The speeches were given by serving government ministers, including prime ministers and, in the case of France, presidents, of the countries. In each case, ministers belonging to governments of different political persuasions are included. The corpus is outlined in table 2.

Table 2. The composition of the comparable corpus of French and English political speeches

Although the corpus is balanced in terms of word count, there are fewer English language speeches, due to their being on average noticeably longer (2,922 words) than the French language ones (2,594 words). This is already an indication that there is no total match between the genres, and that the typical function of the political speech may vary across the different contexts. Epistemic occurrences of must, have to and devoir were excluded. Must being modal only, comparison is restricted to deontic modal contexts (i.e. excluding needed to, would have to, a dû, etc.) 9 . Discourse-marking and highly idiomatic uses, such as I must say or dois-je le rappeler were also excluded. For devoir, clear future-time uses were excluded; however, many uses were ambiguous between expectation and obligation and these were included on the grounds that an obligation reading was pragmatically possible in the context.

The figures for the occurrences of the relevant modal and semi-modal verbs of obligation that were extracted from the corpus using WordSmith Tools [START_REF] Scott | WordSmith Tools v.6. Liverpool: Lexical Analysis Software[END_REF]) are given in Table 2. These are the occurrences that were analysed.

The occurrences were coded for the type of subject (of the proposition in the scope of the modal marker); voice; polarity; negative effect (i.e. whether the realization of the proposition affects anyone negatively) 10 .

The expression of obligation in political speeches

Overall frequencies of the five verbs

Table 3 shows the raw and normalised figures for obligation occurrences in the corpus of all five verbs of obligation. These are the occurrences that are taken into account in the analysis. The frequency of deontic (root) must is strikingly high, bucking the well-documented trend described in section 2. It is more than ten times higher than that found by Close and Aarts (2010:176) for spontaneous spoken English of the 1990s. Another salient finding is that need to is twice as frequent as have to. Must, then, looks like a preferred choice for obligation. The situation is similar for French. In a study of the use of modal verbs in the speeches of successive French presidents of the fifth Republic, [START_REF] Labbé | La modalité verbale en français contemporain. Les hommes politiques et les autres[END_REF] found very different patterns from both spoken and literary French. While falloir was more than twice as frequent as devoir in their 2.3million-word spontaneous spoken language corpus (2013 table 15) 11 , the reverse was the case on average for the presidential speeches, where, for the four commonest [modal + V] combinations, devoir was more than twice as frequent. The findings of the present study of ministerial speeches are consistent with those of Labbé and Labbé for presidential speeches. These frequencies may simply reflect a conservative style, but if the distribution (the contexts) also differs significantly from that of other genres, it may be seen as genre-specific rather than conservative. 9 There is only one occurrence of the form have got to in the corpus, perhaps because of its association with informal speech.

English

10 Politicians recycle sections of previous speeches into new speeches and such passages occur in the corpus.

Where a whole sentence or clause complex is reused in a second speech, only one occurrence is counted; where a sub-sentential unit is reused in a different context, two occurrences are counted. 11 Around a fifth of the corpus in question was Canadian and Belgian French.

The five verbs and speaker attitude

For French devoir and English must, obligation is by far the most frequent use: 90% of occurrences of devoir and 96% of occurrences of must. This is markedly different from what has been found for other registers of English. For written language, [START_REF] Collins | Modals and Quasi-modals in English[END_REF] found on average that one third of occurrences of must were epistemic, and for spoken language [START_REF] Close | Current change in the modal system of English: a case study of must, have to and have got to[END_REF] found that more than half were epistemic, and around 40% were deontic (root modality). The situation for devoir is complicated by the overlap between the obligation and future-time uses of the verb, so that many occurrences are ambiguous or vague between what is due to happen and what the speaker desires to happen. Only about 7% of occurrences were epistemic in the sense of speaker estimation of a truth value.

In their cross-linguistic study of obligation, Myhill and Smith rejected speaker attitude towards the proposition as an appropriate parameter for categorising obligation utterances in favour of 'negative effect', that is, whether the carrying out of the obligation will have a negative effect on anyone. The contexts of use of must in the data are overwhelmingly those where the speaker not only urges the rightness of bringing about the state of affairs, but where the state of affairs is couched in terms of universal desirability. Have to by contrast collocates with states of affairs that will affect some party negatively. This may be a necessary evil externally imposed (3), or a speaker-created obligation that will be unpleasant for the agent to carry out (4). In about a third of cases (36%) the speaker can be interpreted as favourable to the obligation. This finding for have to echoes what Myhill and Smith found for their latetwentieth century English drama data and which they term "negative effect" (1995:247ff). They did not include must in their study, but found that negative effect was a significant parameter separating have to from other expressions of obligation. It is in the nature of political discourse to focus on the positive and the desirable and exclude the unpleasant; this in itself may go some way to explaining the relative infrequency of have to in speeches.

(3)

Anyone alleging a violation has to take his case to Strasbourg to obtain a remedy [Irvine 1997] (4) These issues have to be addressed -to ignore them would stoke up fear [Taylor 1996] By contrast, need to, like must, almost always correlates with the speaker being favourable to the obligation, which is by implication external even where we is the subject (5).

(5) a. There are some old ghosts which need to be laid to rest [Mowlam 1998] b. We also need to sort out proper arrangements to appoint Special Envoys [Rifkind 1997] By comparison, the French verbs cover a wider range of obligation types, and their contexts are very heterogeneous. Thus, devoir occurs in contexts of both very strong and very weak obligation, and in both speaker-created and external obligation, with much vagueness. The speaker is generally favourable to the obligation.

Likewise falloir gives rise to vagueness, but again the speaker is usually favourable to the obligation expressed. Falloir differs, of course, from the other four verbs in that it takes an impersonal construction with dummy subject il.

Negation, for the three verbs (must, devoir and falloir) that can be negated to produce a prohibition or obligation not to do something, is rare (less than 5%); speakers focus on the positive 12 .

Types of subject and agentivity

Deontic obligation is typically conceived of as agent-oriented, so that "examples of root must in English would normally be classified as prototypical examples of agent-oriented modality" (Coates 1995:57). Some source of obligation (typically the speaker) desires some agent to effect some state of affairs (typically a dynamic event). The prototypical example given by Coates is (6). ( 6)

You must finish this before dinner. [from Coates 1995:57] The three elements that form the deontic obligation 'frame' are thus a source of obligation (typically the speaker and if so implicit), an agent (typically the addressee) and a desired state of affairs (typically expressed by a dynamic verb phrase). In (5 these three elements are the speaker, 'you' and 'finish this before dinner'.

The aim of examining the types of subject is to look at how the agentivity of obligation is conveyed in political speeches. For all five obligation verbs, second person subjects and first person singular subjects are very rare. Four main types of subject account for the vast majority of occurrences. In order of animacy these are first person plural, collectivities (countries, institutions, companies, professions, and so on), agentless passives, and abstract or inanimate subjects (such as deverbal nouns, mental activities, measures, rules, processes, time periods, and occasional metaphors). Figure 2 shows the percentages of each verb that are accounted for by these patterns. Figure 3 shows the same patterns as occurrences per 100,000 words.

A first person plural subject ( 7) is strikingly frequent across all five verbs, accounting for between 17% (devoir) and 49% (need to) of occurrences (fig. 2). For devoir, must, and need to 'we' represents between 54 and 74 occurrences per 100000 words (fig. 3) 13 . The referent(s) of 'we' are rarely specified and by implication range from the speaker and their immediate entourage through governments, professions and socio-economic sectors, countries, international fora, Europe and the world. This pattern with 'we' conforms to a typical agent-oriented obligation structure (self-obligation); it firmly includes the speaker in the source of obligation.

(7)

a. that is not a discourse marker of the type il faut bien le dire ('it has to be said').

13 For the impersonal verb falloir, occurences of il nous faut + V and il faut que nous + V are counted as 'we' subjects and account for over 20% of all falloir occurrences (Fig. 2).

Fig. 2. Types of subject, %.

Fig. 3. Types of subject and frequencies per 100,000 words

The second most frequent subjects are collectivities. These include the political collective to which the speaker belongs (e.g. the government), the speaker together with his/her addressees, and/or wider social units such as sectors of the population, the country as a whole, Europe, or society generally. But there is a clear division between the unspoken source of the obligation in the speaker and the collective agent enjoined to carry it out. Excepting The third and fourth types of subject reflect two strategies for depersonalizing the obligation by removing the agent.

One is the use of the agentless passive construction (9). The other is the use of an inanimate, abstract noun as subject of the modal verb. In a large proportion of the political speech occurrences of must and devoir not only is the source of obligation barely recoverable, but the agent is missing and unidentifiable from the context, and instead of a dynamic event, we find some situation. The data reveal four sub-types of abstraction.

One way of delinking the obligation from an agent is by nominalizing the state of affairs and using a copula or stative verb. Both devoir and to a lesser extent must lend themselves to this construction (10).

(10) a. une meilleure coopération doit exister entre les acteurs [Voynet 1999] 'better cooperation must exist between the participants' b. Cooperation in the fight against crime must be as instinctive as it is in foreign and defence policy [Major 1994] This strategy can result in an unusual form of redundancy (11).

(11) L'exercise des missions de police doit également s'exercer au plus proche des citoyens (kind of tautology) [Bartolone 1999] 'The carrying out of police duties must be carried out close to the public.'

A second sub-type of depersonalization consists of an abstract noun in subject position, such as an idea, process, rule, time or metaphor (12). The only element of the deontic obligation frame that is expressed is the verb itself which prompts the hearer to recreate an obligation scenario, filling in the missing elements. In (12a), hearers will infer a source as usual (such as the speaker and others) but also an agent (political actors, perhaps including the speaker and the hearers) as well as an event that can bring about the situation where flexibility is a component of the future European construction. The speaker's linguistic choice is to evoke obligation with must, but in an atypical mustconstruction, so that the usual frame can only be retrieved by pragmatic inferencing. The effect, and presumably the aim, is to depersonalize the utterance, so that the source can be everyone and the agent no one in particular. This is a face-saving strategy, perhaps, that does not involve abandoning the use of must. Similar strategies are also found with the other modal verbs.

A third strategy is to employ what looks like a reversal of semantic roles: rather than the usual Subject-Agent + Verb + Object-Patient we find Subject-Patient + Verb + oblique or evoked agent, as in ( 13).

(13) a. the lead must come from the real experts [Taylor 1996] b. cette question aussi devra trouver une réponse [Sautter 1999] 'this question will also have to find an answer / an answer will also have to be found to this question' c. Et je demeure convaincue que la culture doit détenir une place particulière dans notre réflexion. [Trautmann 1999] 'And I remain convinced that culture must hold a special place in our deliberations.'

Finally, there is a pattern in which the 'role reversal' involves an animate beneficiary. Sequences such as devoir pouvoir and must be able to or have to be able to are associated with this pattern, as in ( 14). In each case, the subject of the obligation verb is the beneficiary of the desired state of affairs.

(14) a. People have to be able to protect their children [Taylor 1996] b. We must instil confidence. All of these patterns occur in other genres too, of course. But the parallels between the English and the French in their frequency and distribution are quite salient: 'prototypical', agentive expressions of obligation occur regularly only with the first person plural, where the speaker is part of both the source of obligation and the agent. Otherwise, non-agentive constructions of the four types outlined are preferred in both languages and fairly consistent across different verbs of obligation (fig. 2).

Conclusion

Devoir in French and must in English are both associated with conservative or formal language, as seen in section 2. Both show unusually high frequency in the political speeches. Could it be the case that the very high relative frequency of must in the speeches at a time when must is declining is due to this formal register being conservative and slow to change? It looks unlikely. Perusal of some nineteenth-century political speeches reveals a different pattern of use of must altogether: in addition to discourse-marking uses (I must say), must is used widely across epistemic (predictive) and logical necessity contexts as well as for obligation, with no apparent connotations of positive speaker attitude, rather the opposite. Although this data is inadequate for any conclusions to be drawn, a plausible hypothesis is that must has undergone semantic shift (amelioration) and further subjectification (speakeroriented desirability). Present-day usage of must in political discourse appears to be a recent, possibly genre-related development rather than a hangover from a period before must started to wane.

Both French and English have, on the face of it, a more subjective modal expression for deontic obligation in devoir and must, and a more objective one in falloir and have to, which are both also associated with non-deontic necessity. In both language genres, speakers favour the use of the supposedly more formal and subjective expressions, but use them atypically in impersonalized constructions. Moreover, the desired states of affairs tend to be presented in both langauges through stative or resultative constructions as situations to be brought about rather than as events to be accomplished. Another striking similarity across all five verbs is the proportions of first person plural subjects and rarity of other persons.

Political discourse is characterized by a number of special features: it is often very carefully crafted, every nuance being analysed, and is designed for a wider audience than the immediate hearers; it aims to impress and persuade and may have a hortatory function; it has a ceremonial function that favours rhetorical routines; and above all it deals largely with unrealized states of affairs. Speakers are projecting a vision of realizable states of affairs and therefore have recourse to modal expressions. It is perhaps not surprising, then, that the uses of the main deontic modal expressions in political speeches are similar across English and French. [START_REF] Johansson | Modals and semi-modals of obligation in American English: some aspects of developments from 1990 until the present day[END_REF] found strong genre effects in American English, with need to almost twice as frequent as must in spoken informal language, but the order inversed for academic prose. The contextual overlap in the political speeches between must and need to may therefore reflect some colloquialization or modernization in the English speeches (rather than a flight from a potentially face-threatening expression). [START_REF] Labbé | La modalité verbale en français contemporain. Les hommes politiques et les autres[END_REF] also found a strong genre effect for French presidents' speeches, which showed both less personalisation and a greater use of modal verbs than literary French. Our observation that il nous faut accounts for a fifth of falloir occurrences and appears in the same contexts as nous devons may reflect a move towards a more colloquial style. At the same time it may be a way of emphasizing a strong obligation: the semantic expansion of devoir towards weak obligation and future time results in vagueness regarding degree of obligation.

Overall, we conclude that there are salient similarities in the ways in which each language genre handles the expression of obligation. Such similarities are masked when register-diversified corpora are used to identify ongoing changes in the frequencies and distributions of linguistic patterns, or to compare some notional area or semantic field across two or more languages. It is therefore interesting to undertake contrastive-linguistic studies at a relatively fine-grained level of social context matching: a genre may have its own twist on an evolutionary dynamic, stemming from the strategies and situational constraints guiding its speakers.

Figure 1 .

 1 Figure 1. Notional areas of the five verbs, showing polysemies, partial synonymies and crosslinguistic mappings.

Figure 1

 1 Figure1is inspired by van der Auwera and Plungian's discussion of directionality in modal meaning change(1998:90-97). The background area represents a postulated conceptual space, on to which we can map the (prototypical) senses of linguistic expressions. Lower case labels refer to semantic notions; upper case labels refer to (prototypical uses of) lexical items of English and French. The central area corresponds to the 'root modality' row of table 1, that is, to the modal notional area of root necessity. The large circle (centre top) represents deontic necessity 6 , comprising that emanating from both internal and external sources. The small ring (centre bottom) represents physical, non-deontic necessity (exemplified in ****). The arrows indicate the directions of frequently-attested diachronic developments (semantic shift or semantic expansion and grammaticalization). Thus, the three rings to the left show non-modal notions which cross-linguistically often share a lexical item with a root modality notion. The two rings to the right show the non-root modal notions of epistemic judgment and future time.7 Again, the arrows show that these notions too often attach to a lexical item expressing a root modality notion. The semantic map therefore provides a rough way of visualising how the senses of lexical items map on to conceptual space and how they relate to each other. For the lexical items that concern us here, we can see in figure 1 that all of them range over two or more notional areas: they are polysemous as a result of semantic expansion. The map also graphically represents cross-linguistic overlap: devoir is used across a wider range of notions than must, for instance.

French

  a quarter and a third of occurrences of the five verbs specify a collective agent (8). (8) a. .. new democratic institutions have to be founded on agreement and consent. they have to command the support of both communities here [Mowlam 1997] b. L'ESA doit rester au coeur du dispositif spatial européen [Schwartzenberg 2000] 'The ESA must remain at the heart of European space activities.'

  Les problèmes doivent être évoqués ici, là où ils prennent leur source [Sarkozy 2002]'Problems must be raised here, where they originate' b. The fighting there must be brought to an end. [Major 1995] c. First, the fighting needs to be brought to an end[Hurd 1995]

(

  12) a. That is why an essential component of the future European construction must be flexibility. [Major 1994] b. Le concept de police de proximité répond évidemment aux évolutions sociales, mais il doit avoir un contenu plus fort [Sarkozy 2002] 'The concept of neighbourhood policing is of course a response to social change, but it must have a stronger component.'

  [Moscovici 2001] 'It's a real problem for European citizens, to which we must find solutions.' e. Il nous faut simplifier nos structures nationales pour les faire mieux coopérer au niveau européen. [Fabius 2001] 'We need to simplify our national administrations to make them cooperate better at European level.'

First we have to get agreement in the talks [Mowlam 1997] b. We must achieve a sustainable consensus on pensions policy [Harman 1997] c. We need to pursue agreement with the Russians [Hurd 1995] d. C'est un vrai problème pour les citoyens européens, auquel nous devons trouver des solutions.

Biber subsequently used the term 'register'(Biber 1994, 1995 and later works) rather than 'genre'. This may be considered unfortunate, as 'register' traditionally refers to the variety or style of language conventionally associated with a reucrrent social situation or event rather than the body of texts used in the performance of that social situation or event. Maintaining the distinction allows 'rogue' texts to be included in the genre but not the register.

The term 'root modality' is often used in the Anglo-American literature as a cover term for both deontic modality and dynamic modality (see[START_REF] Coates | The Semantics of Modal Auxiliaries[END_REF],[START_REF] Depraetere | Towards a more explicit taxonomy of root possibility[END_REF],Quirk et al (1985:495). In this tradition, as Rothsetin et al note, "Root modality expresses permission, obligation, possibility, necessity, ability or volition" (2010:115).

For convenience, we will use the term 'state of affairs' for the content of the proposition in the scope of the modality marker, 'situation' for a state of affairs presented as stative and 'event' for a state of affairs presented as

This means that; for instance, in the case of the example Je dois arriver avant midi, the context of utterance may favour a wide interpretation ('the speaker feels some pressure to arrive or some expectation of arriving by noon') or a narrower interpretation such as 'the speaker has been told by someone to arrive before noon'.

[START_REF] Haspelmath | The geometry of grammatical meaning: semantic maps and cross-linguistic comparison[END_REF] provi.des a useful introduction to semantic maps.

This includes obligation and permission, as shown in table 1, but we are not concerned with permission here.

For van der Auwera and Plungian, among others, the notional area of future is not modal; as we are not concerned with future time here, we leave the question open.

The decline of must in spoken English seems to have affected the root and epistemic uses equally, root uses staying at around one third of all uses through the period 1960s -1990s (Close and Aarts 2010).