R Chang

A L Hoendervanger

Q Bouton

Y Fang

T Klafka

K Audo

Alain Aspect

C I Westbrook

D Clément

Three-dimensional laser cooling at the Doppler limit

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I. INTRODUCTION

The seminal proposals for Doppler cooling in 1975 [START_REF] Hänsch | Cooling of Gases by Laser Radiation[END_REF][START_REF] Wineland | Proposed 10 14 ∆ν/ν Laser Fluorescence Spectroscopy on TI + Mono-Ion Oscillator[END_REF] prompted the realization of the first optical molasses in the 1980's [START_REF] Chu | Three-Dimensional Viscous Confinement and Cooling of Atoms by Resonance Radiation Pressure[END_REF], a major landmark in the field of laser cooling and trapping of atoms [START_REF] Cohen-Tannoudji | Nobel Lecture: Manipulating Atoms With Photons[END_REF][START_REF] Chu | Nobel lecture: The Manipulation of Neutral Particles[END_REF][START_REF] Phillips | Nobel Lecture: Laser Cooling and Trapping of Neutral Atomsl[END_REF]. The physical concepts behind the Doppler cooling mechanism are both simple and elegant, and predict the achievement of very low temperatures. They remain to this day the starting point of most courses on laser cooling and degenerate gases [START_REF] Cohen-Tannoudji | Fundamental Systems in Quantum Optics[END_REF]. It is thus ironic that not only, to our knowledge, quantitative predictions of this simple model have not been experimentally validated, but moreover, experimental studies have found results quantitatively and even qualitatively different from the predictions of the model [START_REF] Lett | Observation of Atoms Laser Cooled below the Doppler Limit[END_REF][START_REF] Lett | Optical Molasses[END_REF][START_REF] Weiss | Optical Molasses and Multilevel Atoms: Experiment[END_REF][START_REF] Salomon | Laser Cooling of Cesium Atoms Below 3 µK[END_REF].

Today, it is well known that for most atoms with a degenerate ground state manifold, the complex multilevel structure gives rise to new mechanisms which come to dominate the cooling process, yielding ultimate temperatures far below those predicted by Doppler theory [START_REF] Cohen-Tannoudji | Fundamental Systems in Quantum Optics[END_REF][START_REF] Dalibard | Laser Cooling below the Doppler Limit by Polarization Gradients: Simple Theoretical Models[END_REF][START_REF] Ungar | Optical Molasses and Multilevel Atoms: Theory[END_REF]. Accounting for these sub-Doppler mechanisms allows one to understand the qualitative and quantitative experimental observations. In contrast, threedimensional laser cooling of an atomic sample in agreement with the celebrated Doppler model is still lacking [START_REF]Agreement with the Doppler model has only been reported for experiments in one-dimension[END_REF]. Recent work with alkaline-earth and rare-earth atoms, which exhibit a non-degenerate ground state, have provided natural candidates to study cooling in a purely Doppler regime. Yet, to our knowledge, all experiments with these atoms have found temperatures above the expected Doppler limit in magneto-optical traps (MOTs) [START_REF] Zinner | Photoassociation of Cold Ca Atoms[END_REF][START_REF] Curtis | Quenched Narrow-Line Laser Cooling of 40 Ca to Near the Photon Recoil Limit[END_REF][START_REF] Grünert | Optimizing the Production of Metastable Calcium Atoms in a Magneto-Optical Trap[END_REF][START_REF] Dammalapati | A Compact Magneto-Optical Trap Apparatus for Calcium[END_REF][START_REF] Binnewies | Doppler Cooling and Trapping on Forbidden Transitions[END_REF][START_REF] Th | High-Resolution Spectroscopy With Laser-Cooled and Trapped Calcium Atoms[END_REF][START_REF] Sengstock | Optical Ramsey Spectroscopy on Laser-Trapped and Thermal Mg Atoms[END_REF][START_REF] Riedmann | Beating the Density Limit by Continuously Loading a Dipole Trap from Millikelvin-Hot Magnesium Atoms[END_REF][START_REF] Katori | Magneto-Optical Trapping and Cooling of Strontium Atoms down to the Photon Recoil Temperature[END_REF][START_REF] Xu | Dynamics in a Two-Level Atom Magneto-Optical Trap[END_REF][START_REF] Kuwamoto | Magneto-Optical Trapping of Yb Atoms Using an Intercombination Transition[END_REF][START_REF] Loftus | Probing Magneto-Optic Trap Dynamics Through Weak Excitation of a Coupled Narrow-Linewidth Transition[END_REF][START_REF] Dorscher | Creation of Quantum-Degenerate Gases of Ytterbium in a Compact 2D-/3D-Magneto-Optical Trap Setup[END_REF]. This discrepancy has been attributed to the presence of additional heating mechanism [START_REF] Choi | Three-Dimensional Analysis of the Magneto-Optical Trap for (1+3)-Level Atoms[END_REF][START_REF] Chaneliere | Extra-Heating Mechanism in Doppler Cooling Experiments[END_REF].

Here we report the three-dimensional laser cooling of metastable Helium-4 gases in the Doppler regime. We monitor the temperature of the gas as a function of the laser detuning δ, as shown in Fig. 1, finding excellent agreement with Doppler theory. The temperature reaches a minimum for a detuning δ = -Γ/2 (where Γ is the transition linewidth), while increasing with |δ| at val- ues |δ| Γ, in contrast with sub-Doppler molasses. In addition, the drift velocities of optical molasses with unbalanced power between counter-propagating beams are far larger than those expected of sub-Doppler molasses [START_REF] Lett | Optical Molasses[END_REF]. We use the 2 3 S 1 → 2 3 P 2 transition of Helium-4, which allows in principle for sub-Doppler cooling, and yet our results show no evidence for such effects. We present a physical argument showing that the special properties of metastable Helium ( 4 He*) strongly inhibit sub-Doppler cooling in the experimental configuration we probe.

II. THEORETICAL CONSIDERATIONS

A. Laser cooling mechanisms

Laser cooling of an atomic gas relies upon the exchange of momentum between the atoms and the nearresonant light field, resulting in a mechanical force F on the atoms. For small velocities, the equilibrium temperature T of these cooling schemes is given by the ratio of a momentum-space diffusion constant D (given by the fluctuations of the force) to the velocity-damping coefficient α. In the following we will first recall the main theoretical results for laser Doppler cooling of a two-level atom, then proceeding to discuss a multi-level atom where sub-Doppler mechanisms may appear.

The mechanical interaction of a near-resonant light beam (frequency ω) with a two-level atom (atomic resonance frequency ω 0 ) is dominated by the radiation pressure effect and its associated force [START_REF] Lett | Optical Molasses[END_REF]. This light-matter interaction is characterised by the laser detuning from resonance δ = ω -ω 0 , the laser intensity I, the natural linewidth of the transition Γ, and the saturation intensity of the transition I sat . A convenient physical quantity is the generalized saturation parameter

s = s 0 1 + 4δ 2 /Γ 2 , (1)
which characterizes the excited state population in a 2level atom, s/2(1 + s). Here s 0 = I/I sat = 2Ω 2 /Γ 2 is the on-resonance saturation parameter, and Ω is the Rabi frequency.

At low saturation of the atomic transition s 1 and small velocities |kv| |δ|, the average force acting on a two-level atom moving at velocity v takes the form

F 2level = -α 2level v with α 2level = -4 k 2 2δ/Γ 2Ω 2 /Γ 2 (1 + 2Ω 2 /Γ 2 + (2δ/Γ) 2 ) 2 . ( 2
)
Here k is the photon wavevector, and the reduced Planck's constant. The force F 2level is plotted in Figure 2 (dashed line) for the typical experimental parameters of optical molasses.

Calculating the diffusion constant D 2level for the twolevel atom, one can obtain the equilibrium temperature k B T 2level = D 2level /α 2level , where k B is Boltzmann's constant. Following the generalization to cooling in 3D [START_REF] Lett | Optical Molasses[END_REF], the temperature of a 3D molasses formed by three orthogonal pairs of counter-propagating laser beams is,

k B T 2level = Γ 2 1 + I tot /I sat + (2δ/Γ) 2 4|δ|/Γ , (3)
where the I tot = 6I is the total intensity resulting from all six laser beams. In the Doppler regime, the minimum possible temperature is achieved for a detuning δ = -Γ/2 and vanishing intensity. This minimum is commonly referred to as the Doppler limit, T D = Γ/2k B . We now turn the discussion to an atom with several ground-state levels where sub-Doppler mechanisms of cooling may be involved [START_REF] Lett | Optical Molasses[END_REF][START_REF] Weiss | Optical Molasses and Multilevel Atoms: Experiment[END_REF][START_REF] Dalibard | Laser Cooling below the Doppler Limit by Polarization Gradients: Simple Theoretical Models[END_REF]. For this discussion we restrict ourselves to the atomic structure of the 4 He* atom on the 2 3 S 1 → 2 3 P 2 transition (see Fig. 2a), and the laser configuration used in the experiment: counterpropagating laser beams with opposite circular polarization (configuration σ + -σ -) along three orthogonal axes.

Consider first a single axis of the system. Following the derivation from [START_REF] Chang | Two-Photon Laser-Cooling Mechanism in Multilevel Interaction Schemes[END_REF] an explicit formulation of the semi-classical force F on a moving atom can be obtained. This approach takes into account atomic coherence effects up to second-order in absorption-emission processes. In Fig. 2 we plot this force F along with that exerted on a two-level atom F 2level . In the low-velocity region (see Fig. 2c) the cooling force on the multi-level atom exhibits a sharp feature with an associated damping coefficient α significantly larger than that of a two-level atom. This feature arises from the presence of atomic coherences and two-photon processes, and is the signature of sub-Doppler cooling mechanisms. Since the momentumspace diffusion coefficient hardly changes, the new damping coefficient would lead one to expect sub-Doppler cooling.

In this configuration, the additional cooling results from a motion-induced population difference between the ground-state levels, and is referred to as σ + -σ -polarization gradient cooling. In a 3D configuration, it is well known that Sisyphus cooling can also occur due to spatially dependent light shifts of the atomic ground-state levels. The electric field from laser beams along orthogonal axes interfere, resulting in a modulation of the intensity, which in turn may lead to Sisyphus cooling as in a one-dimensional lin ⊥ lin configuration. Within the parameter range of interest (|δ| ∼ Γ), both sub-Doppler effects result in similar velocity capture ranges and equilibrium temperatures [START_REF] Lett | Optical Molasses[END_REF][START_REF] Weiss | Optical Molasses and Multilevel Atoms: Experiment[END_REF][START_REF] Dalibard | Laser Cooling below the Doppler Limit by Polarization Gradients: Simple Theoretical Models[END_REF].

The comparison between a 2-level and a multi-level atom presented in Fig. 2 suggests two important conclusions. On the one hand the two-photon structure occurs only in a small range close to zero velocity (see Fig. 2c). In particular, for typical experimental parameters with Helium-4 atoms this velocity range is smaller than the recoil velocity v R = k/m, where m is the particle's mass. This implies that sub-Doppler cooling is highly ineffective in a red molasses of 4 He* since the velocity capture range for such processes is much smaller than the lowest velocity achievable through Doppler cooling. On the Helium atom in a σ + -σ -standing wave in the case of a two-level atom (dashed line, calculated as in [START_REF] Lett | Optical Molasses[END_REF]) and in that of a multi-level atom on a J = 1 → J = 2 transition (solid line, calculated as in [START_REF] Chang | Two-Photon Laser-Cooling Mechanism in Multilevel Interaction Schemes[END_REF]). Detuning and intensity correspond to typical values used in the experiment for optical molasses. (c) Expanded view of the low velocity region of the semi-classical forces, high-lighting the narrow range of the sub-Doppler feature. In both calculations, s = 0.02.

other hand, on a larger velocity range corresponding to Doppler velocities the friction coefficient for the multilevel atom is close to that of an effective two-level atom.

In the following section, we will study the sub-Doppler capture velocity range in greater detail, showing that indeed 4 He* occupies a special place in the study of laser cooling mechanisms.

B. Capture range of sub-Doppler cooling

The velocity capture range for Doppler cooling is v D c |δ|/k, directly reflecting the largest Doppler effect that can be compensated with a detuning δ from resonance. Similarly, the capture range for sub-Doppler cooling reflects the new physical mechanisms involved. For Sisyphus cooling (lin ⊥ lin) to be effective, a ground-state atom should have a high probability of being optically pumped into a different ground-state atom on a wavelength distance [START_REF] Cohen-Tannoudji | Fundamental Systems in Quantum Optics[END_REF]. In the semi-classical regime, this yields a condition on the atomic velocity v Γ /k, where Γ is the optical pumping rate from one ground-state level to another ground-state level. The capture velocity is thus estimated to be v ⊥ c Γ /k. In the case of σ + -σ -polarization gradient cooling, the capture velocity is given by the pumping rate at small detunings δ ∼ -Γ, v σ c Γ /k [START_REF] Chang | Two-Photon Laser-Cooling Mechanism in Multilevel Interaction Schemes[END_REF], while it is set by the light shift δ of the ground-state levels at large detuning |δ| Γ, and is defined in the same way to be v σ c |δ |/k, [START_REF] Cohen-Tannoudji | Fundamental Systems in Quantum Optics[END_REF]. The optical pumping rate and the light shift can be expressed conveniently as [START_REF] Cohen-Tannoudji | Fundamental Systems in Quantum Optics[END_REF]

Γ = Γ 2 2Ω 2 /Γ 2 1 + 2Ω 2 /Γ 2 + (2δ/Γ) 2 = Γ 2 s 1 + s δ = δ 2 2Ω 2 /Γ 2 1 + 2Ω 2 /Γ 2 + (2δ/Γ) 2 = δ 2 s 1 + s (4)
A large capture velocity range for sub-Doppler cooling requires a large Γ and/or δ . These two quantities are decreasing functions of the detuning δ when |δ| Γ and δ is maximum for |δ| = Γ/2. The Doppler temperature (see Eq. 7) is minimum in the same range of detuning δ. In the following, we will therefore concentrate on laser detuning |δ| of the order of Γ, the most favorable situation to observe signatures of sub-Doppler cooling (although not the lowest temperature). The capture velocities for both sub-Doppler mechanisms are then

v ⊥ c v σ c = v c = Γ 2k s 1 + s . ( 5
)
For the parameters represented in Fig. 2 (s = 0.02) and using Eq. 5, we find v c 0.01Γ/k in agreement with the numerical calculation of the force shown in Fig. 2. Identifying a root-mean-square (RMS) velocity v D of a 3D gas at the Doppler temperature

T D , 1 2 mv 2 D = 3 2 k B T D , we find v c v D = 1 6
T D T R s 1 + s , (6)
where T R is the temperature associated with the recoil energy small in the case of 4 He* as compared to other species. In the Table I we present a comparison of the ratio T D /T R for several atomic species commonly used in laser-cooling experiments.

k B T R = 2 k 2 /2m.
Eq. 6 indicates that the capture velocity v c /v D increases with saturation parameter s, approaching the limit T D /6T R . However, s can not be increased arbitrarily. For red-detuned optical molasses, the friction coefficient α is known to change sign at large saturation parameter [START_REF] Minogin | Resonant Light Pressure Forces in a Strong Standing Laser Wave[END_REF][START_REF] Aspect | Cooling Atoms with Stimulated Emission[END_REF]. This change of sign occurs when the contribution of multi-photon processes to the force becomes large enough to modify the excited state decay rate, and typically occurs at s ∼ 1. Supposing a maximum saturation parameter s max = 1/2, we estimate an upper limit on the capture velocity v up c /v D . The values for v up c /v D are given in Table I. It is clear that Helium, and to a lesser extent Lithium, is special with respect to other alkali atoms since the capture range for sub-Doppler cooling is smaller than the range of velocities achievable with Doppler cooling. This statement applies both to the σ + -σ -and lin ⊥ lin configurations. Thus we conclude that sub-Doppler cooling is not expected to play a role on the 2 3 S 1 → 2 3 P 2 transition of Helium-4. Indeed, in the experiment we do not observe any signatures of sub-Doppler cooling. The semi-classical arguments we have presented above highlight the special place that Helium occupies among laser cooled species. A precise and quantitative condition for the appearance of efficient sub-Doppler cooling would require a fully quantum, three-dimensional computation and is beyond the scope of this paper.

C. Equilibrium temperature for a multi-level atom in the Doppler regime

From the above arguments, sub-Doppler effects on metastable Helium on the 2 3 S 1 → 2 3 P 2 transition are expected to be negligible. We will therefore compare the temperature measurements in 3D gases to the predictions of Doppler theory [START_REF] Lett | Optical Molasses[END_REF]. To account for the multilevel atomic structure in the 3D Doppler theory we take a weighted sum over all possible one-photon transitions, where the weights are given by the square of the Clebsch-Gordan coefficients (see Fig. 2a). This leads to a rescaling of the saturation intensity I 0 = 9/5 I sat , and an equilbrium temperature

k B T = Γ 2 1 + I tot /I 0 + (2δ/Γ) 2 4|δ|/Γ . ( 7
)
where I tot is the total intensity in the 6 beams. A similar approach has been used in [START_REF] Townsend | Phase-Space Density in the Magneto-Optical Trap[END_REF].

III. DESCRIPTION OF THE EXPERIMENTAL APPARATUS

A. 4 He* Magneto-Optical traps

Our measurements are performed with an apparatus which cools and traps metastable Helium atoms in a MOT. 4 He* atoms are produced in a hot plasma (dcdischarge) and slowed down to trappable velocities on the order of several tens of meters per second with a 2.5 m long Zeeman slower. The slowed atoms enter the science chamber where three orthogonal pairs of counterpropagating laser beams are shone onto the atoms in the presence of a quadrupole magnetic field. The cooling light, which addresses the 2 3 S 1 → 2 3 P 2 transition, is derived from a Koheras AdjustiK Y10 fiber laser from NKT Photonics with a manufacturer stated linewidth < 10 kHz. During the MOT phase, the typical intensity per beam is ∼ 20I sat at a detuning δ M OT = -2π × 50 MHz -31Γ from the atomic transition, where the transition linewidth is Γ = 2π × 1.6 MHz. The magnetic field gradient along the coil axis is B x = 24 G/cm. Under these conditions, 8 × 10 8 atoms at a temperature of 1.5(1) mK are loaded within 2 seconds.

Detection of the gas is performed using a thermoelectrically cooled InGaAs camera (XEVA type from Xenics company -256 × 320 pixels with a pixel size of 30 × 30 µm). This technology is suited to image metastable Helium atoms with a quantum efficiency of ∼ 80 % at 1083 nm. The camera collects the fluorescence of the atoms from the probe beams. The latter are made of the six beams we use to make a MOT which are tuned onto resonance of the atomic transition during the imaging pulse. The duration of the imaging pulse is 100 µs and the total intensity is about 175 I sat , where I sat = πhcΓ/3λ 3 0.165 mW/cm 2 is the saturation intensity of the J = 1 → J = 2 cycling transition. Sizes and temperature of the 4 He* clouds are extracted by monitoring the time-of-flight expansion of the initially trapped gases and fitting the imaged 2D density profiles with a Gaussian function.

B. Optical molasses

After the MOT phase we implement an optical molasses on the 2 3 S 1 → 2 3 P 2 transition, as we shall now describe. At the end of the MOT phase we ramp the magnetic field gradient to zero and ramp both the detuning and intensity of the laser beams from the MOT values to those of the molasses within 20 ms (see Fig. 3a). This ramp of the parameters allows us to capture and cool half of the atoms (N = 4 × 10 8 ) in the molasses. The polarization of the light beams during the molasses stage is identical to that of the MOT. We then wait for a variable time t mol at fixed final intensity and detuning of the laser beams to reach thermal equilibrium. We monitor the time-of-flight expansion of the optical molasses cloud to extract its temperature.

IV. RESULTS

A. Temperature

In Fig. 1 we present the results of the temperature measurements performed on the MOT and on the optical molasses by recording the TOF expansion of the atomic clouds. Temperature is plotted as a function of the laser detuning δ. We observe a minimum temperature at a detuning δ = -Γ/2 as predicted by Doppler cooling theory. The minimum measured temperature T = 1.3(1) T D is close to the expected Doppler limit, which occurs for vanishingly small light intensity. In addition we compare our temperature measurements to the Doppler prediction of Eq. 7 and we find excellent agreement over the entire range of detunings studied in the experiment. For the high-temperature measurements, the confining potential provided by the MOT is necessary to reach thermal equilibrium on the timescale of the experiment. On the other hand, the low-temperature measurements are performed on an optical molasses (see Figs. 1 and3c), since otherwise density-dependent Penning collisions would severely reduce the number of trapped atoms [START_REF] Shlyapnikov | Decay Kinetics and Bose Condensation in a Gas of Spin-Polarized Triplet Helium[END_REF].

B. Equilibration time for 3D optical molasses

In Fig. 3b we plot the time evolution of the atom cloud temperature during the molasses phase. On a short timescale of t mol ∼ 1 ms, we observe a rapid decrease of the temperature. However, reaching the equilibrium temperature requires durations even longer. Close to the Doppler limit T D (δ = -Γ/2 and I tot /I sat = 1/10), the molasses temperature reaches equilibrium on a timescale t mol 10 ms. We have observed that the equilibration time can vary drastically in the presence of uncompensated bias magnetic fields and power imbalances between counter-propagating beams. As we will see in Section IV D, the equilibirum molasses state is also very sensitive to such technical issues.

In Doppler theory, the timescale to reach the equilibrium temperature is directly related to the velocity damping coefficient, predicting an expected cooling time of τ cool = m/2α [START_REF] Lett | Optical Molasses[END_REF]. For the low value of the saturation parameter s 0.05 and detuning |δ| ∼ Γ used in the data presented in Fig. 3, the expected Doppler cooling time is t cool 0.5 ms. Although this timescale is similar to that of the observed initial rapid decrease in temperature, for the above stated reasons we can not use this measurements to estimate α.

C. MOT sizes

The Penning collision rate in non-polarized 4 He* is relatively high [START_REF] Shlyapnikov | Decay Kinetics and Bose Condensation in a Gas of Spin-Polarized Triplet Helium[END_REF], limiting the atomic density of lasercooled Helium clouds to ∼ 10 9 cm -3 . As a result, photon re-scattering effects, which can result in heating of the atom cloud in other species, are typically negligible for Helium [36]. As a consequence, the equilibrium of metastable Helium MOTs is reached in the absence of both multiple scattering and sub-Doppler cooling. In this simple regime, the equilibrium temperature is given by Eq. 7 and the MOT sizes derive from the equipartion theorem at the Doppler temperature. The force acting onto the MOT can be written F = -α v -κ x x -κ y y -κ z z at low velocities and close to the trap center. The expected RMS cloud size σ i of MOT is then given by

1 2 κ i σ 2 i = 1 2 k B T, (8)
with the three-dimensional equilibrium temperature T calculated from Eq. 7 and index i referring to the different coordinate axis, i = {x, y, z}. The one-dimensional spring constants κ i are given by

κ i = -α 2level µB i k (9)
where B i is the magnetic field gradient along direction i and µ = 3µ B /2 is an average magnetic moment for the multi-level atom on the the 2 3 S 1 → 2 3 P 2 transition (Landé factors are g J (2 3 S 1 ) = 2 and g J (2 3 P 2 ) = 3/2).

From this model we calculate the expected MOT sizes from the known experimental parameters. We have measured the RMS MOT sizes as a function of both the axial magnetic-field gradient B x and the detuning δ of the MOT beams. We plot in Fig. 4 the experimental measurements along with the theoretical predictions in the Doppler cooling regime. The behavior of the MOT sizes with both the magnetic field gradient and the laser detuning are in good agreement with the prediction from Eq. 8 with no adjustable parameters, further demonstrating the validity of the Doppler model. We emphasize that the scaling of the MOT size with the detuning is different in the sub-Doppler regime where the temperature scales inversely proportional to |δ| [START_REF] Townsend | Phase-Space Density in the Magneto-Optical Trap[END_REF]. As expected the MOT temperature is independent of B , as shown in the inset of Fig. 4a. Previous studies of MOT sizes were either conducted in sub-Doppler regime of cooling [START_REF] Wallace | Measurements of Temperature and Spring Constant in a Magneto-Optical Trap[END_REF][START_REF] Kohns | On-Line Measurement of Sub-Doppler Temperatures in a Rb Magneto-optical Trap-by-Trap Centre Oscillations[END_REF][START_REF] Höpe | Neutral Cesium Atoms in Strong Magnetic-Quadrupole Fields at Sub-Doppler Temperatures[END_REF] or with alkaline-earth-metal atoms where high diffusion coefficients and multiple scattering prevent observations similar to ours [START_REF] Xu | Dynamics in a Two-Level Atom Magneto-Optical Trap[END_REF][START_REF] Loo | Investigations of a Two-Level Atom in a Magneto-Optical Trap Using Magnesium[END_REF].

D. Stability of Doppler molasses and drift velocity

One experimental observation that puzzled researchers in the very first experiments with laser cooling was the long lifetime of the optical molasses [START_REF] Chu | Three-Dimensional Viscous Confinement and Cooling of Atoms by Resonance Radiation Pressure[END_REF][START_REF] Lett | Optical Molasses[END_REF]. These measurements were associated with an unexpected robustness of the laser-cooling against an intensity imbalance between counter-propagating laser beams. The additional decay rate induced by this imbalance was measured to be 10 to 20 times lower than that predicted for Doppler molasses. This discrepancy was soon after explained by the presence of sub-Doppler cooling mechanisms [START_REF] Lett | Optical Molasses[END_REF][START_REF] Dalibard | Laser Cooling below the Doppler Limit by Polarization Gradients: Simple Theoretical Models[END_REF].

We have investigated the stability of the Helium molasses against intensity imbalance between counterpropagating laser beams. Following [START_REF] Lett | Optical Molasses[END_REF], we define the imbalance parameter by P 1 = (1 + )P 2 , where P 1 (P 2 ) is the power in the forward-propagating (counterpropagating) beam along one given axis. We measure the drift velocity v d of the molasses cloud along this axis in the following way. We repeat experimental cycle with varying molasses time t mol , taking fluorescence pictures after a fixed and short time-of-flight t T OF = 0.1 ms.

Recording the time evolution of the center of the molasses cloud, we extract a drift velocity v d . In Fig. 5 we plot the drift velocity as a function of the power imbalance between the two laser beams. Following the derivation of the Doppler force valid in the limit of low intensity s 0 1 [START_REF] Lett | Optical Molasses[END_REF], we derive the net force in the case of an unbalanced beam intensity. In this case, the scattering forces from counter-propagating beams balance at a non-zero velocity v d . Although we obtain an analytical formula for v d under those circumstances, we express the result in orders of the imbalance to elucidate the physical content,

v d = Γ 8k 1 + 4δ 2 /Γ 2 2|δ|/Γ - δ 2 8k 1 + 4δ 2 /Γ 2 4δ 2 /Γ 2 + . . . (10)
In particular, the first order term of Eq. 10 is found to be consistent with that reported in [START_REF] Lett | Optical Molasses[END_REF] and results from writing the force F = -α 2level v with α 2level being expressed from Eq. 2.

We plot in Fig. 5 the results of the predictions for the intensity and detuning used in the experiment. The good agreement of the measurements with the Doppler prediction without adjustable parameters is an additional confirmation that the red optical molasses we manipulate are indeed in the regime of Doppler cooling, and that sub-Doppler cooling mechanisms are ineffective on the 2 3 S 1 → 2 3 P 2 transition of 4 He*. Lastly, we note that similar behavior is observed for an imposed bias magnetic field.

V. CONCLUSION

In this paper we have reported on the first experimental observation of three-dimensional laser cooling at the Doppler limit. We have found quantitative agreement with the Doppler theory of (i) the temperature dependence as a function of the laser detuning, (ii) the MOT sizes and (iii) the velocity drift in optical molasses with unbalanced power between counterpropagating light beams. In particular we find a minimum temperature close to the celebrated Doppler temperature of T D = Γ/2k B at a laser detuning of δ = -Γ/2, as predicted in the model of Doppler laser cooling. We have shown that this behavior arises due to the special properties of metastable Helium atoms which render the capture range of sub-Doppler cooling inaccessible. Our discussion and criteria regarding the efficiency of sub-Doppler effects are not restricted to the specific experimental results presented here.

The capture range for sub Doppler mechanisms is limited by the low intensity required for the red-detuned molasses stability. But our work suggests that, since in molasses higher intensities can be used [START_REF] Aspect | Cooling Atoms with Stimulated Emission[END_REF][START_REF] Grier | Λ-Enhanced Sub-Doppler Cooling of Lithium Atoms in D1 Gray Molasses[END_REF], sub Doppler cooling on the blue side of resonance may be possible. At the temperatures achieved here through Doppler cooling, we expect to efficiently load blue-detuned 2 3 S 1 → 2 3 P 1 molasses and reach sub Doppler temperatures. The combination of these techniques may then permit all-optical production of degenerate Helium-3 and Helium-4 gases, which would constitute a significant experimental advance.

More generally, the results presented here help to make clear the capabilities and limitations of the various laser cooling techniques. Such an understanding is important as laser cooling is extended to an ever widening array of atoms [START_REF] Donnan | A Proposal for Laser Cooling Antihydrogen Atoms[END_REF] and molecules [START_REF] Barry | Magneto-optical trapping of a diatomic molecule[END_REF]. We speculate, for example that our results may be of relevance to the problem of cooling H and anti-H [START_REF] Donnan | A Proposal for Laser Cooling Antihydrogen Atoms[END_REF]. The proposed cooling transition 1S 1/2 → 2P 3/2 has a ratio T D /T R = 3.70, and thus a situation resembling that presented above may occur.

FIG. 1 .

 1 FIG. 1. Temperature in magneto-optical trap and optical molasses as a function of the detuning of the laser cooling beams. Temperatures are extracted from monitoring time-offlight expansion of the gas. Error bars (one standard deviation) reflect the error from fitting. The total intensity (sum of the six beams) used in the experiment are respectively IMOT = 100 Isat and IMOL = Isat/10. The lines are the results of Doppler theory of Eq. 7, for intensities IMOT (solid line) and IMOL (dashed line).

FIG. 2 .

 2 FIG. 2. (a)Atomic structure of a multi-level atom on a J = 1 → J = 2 transition. (b) Semi-classical force for a moving Helium atom in a σ + -σ -standing wave in the case of a two-level atom (dashed line, calculated as in[START_REF] Lett | Optical Molasses[END_REF]) and in that of a multi-level atom on a J = 1 → J = 2 transition (solid line, calculated as in[START_REF] Chang | Two-Photon Laser-Cooling Mechanism in Multilevel Interaction Schemes[END_REF]). Detuning and intensity correspond to typical values used in the experiment for optical molasses. (c) Expanded view of the low velocity region of the semi-classical forces, high-lighting the narrow range of the sub-Doppler feature. In both calculations, s = 0.02.

FIG. 3 .

 3 FIG. 3. (a) Sketch of the experimental cycle to probe optical molasses. After the MOT phase, the magnetic field gradient B , the beam intensity Itot and the detuning δ are ramped over 20 ms. A molasses phase at constant parameters lasts t mol . After switching off the molasses beams, fluorescence pictures are taken after a time-of-flight tT OF . (b) Temperature of optical molasses as a function of the duration t mol of the second stage of molasses. Solid lines are guide to the eye. (c) Temperature of optical molasses as a function of the laser detuning. Comparison with Doppler theory for laser cooling multi-level atoms of Eq. 7 (solid lines). The different sets correspond to different intensity in the cooling beams.

FIG. 4 .

 4 FIG. 4. RMS sizes of the MOT clouds σx, σy (blue, red) as a function of the laser detuning (a) and as a function of the magnetic field gradient (b). The lines result from numerical calculation (no adjustable parameter) of the expected sizes in the regime of Doppler cooling (see text), accounting for a 5% error on the calibration of the light intensity. Inset in (b) : temperature of the MOT as a function of the magnetic field gradient B .

4 FIG. 5 .

 45 FIG.5. Drift velocity v d of the molasses cloud as a function of the power imbalance between two-counter propagating laser beams. For this measurement, we use δ = -Γ and I/Isat 1/40. The dashed line is the prediction of[START_REF] Lett | Optical Molasses[END_REF] which is first order in . The solid line is the full prediction of Doppler theory, valid in the low-intensity limit.

ACKNOWLEDGMENTS

We acknowledge fruitful discussions with D. Boiron, T. Bourdel, C. Cohen-Tanoudji and L. Sanchez-Palencia. We thank A. Guilbaud, F. Moron, F. Nogrette and A. Villing, along with all members of the Atom Optic group at LCF, for technical help during the building of the experimental apparatus. We acknowledge financial support from the Région Ile-de-France (DIM Daisy), the RTRA Triangle de la Physique (Junior Chair to D. C.), the European Research Council (senior grant Quantatop) and the Institut Francilien de Recherche sur les Atomes Froids (IFRAF). LCF is a member of IFRAF.


Emma Hooper
email: emmahooper@hotmail.fr

Sustainable growth and financial markets in a natural resource rich country

Keywords: Exhaustible natural resources, exogenous growth, financial markets JEL Classification: E20, O40, Q32, E44

We study the optimal growth path of a natural resource rich country, which can borrow from international financial markets. More precisely, we explore to what extent international borrowing can overcome resource scarcity in a small open economy, in order to have sustainable growth. First, this paper presents a benchmark model with a constant interest rate. We then introduce technical progress to see if the economy's growth can be sustainable in the long-run. Secondly, we analyse the case of a debt elastic interest rate, with a constant price of natural resources and then with increasing prices. The main finding of this paper is that borrowing on international capital markets does not permit sustainable growth for a country with exhaustible natural resources, when the interest rate is constant. Nevertheless, when we endogenize the interest rate the consumption growth rate can be positive before declining.

Introduction

Surprisingly, oil or gaz producers, even though they have important revenues due to their natural resources, are usually facing high level of debt. For instance, Angola, which is one of the first oil producer of Africa has to deal with an external debt of 22,170 billion of US dollar, and Brazil had a debt of 440,478 billion of US dollar in 2012. In this paper, we shall investigate whether a natural resource dependent country with an external debt can achieve sustainable output and consumption growth rates, thanks to its access to international financial markets.

Since the pioneering work of [START_REF] Hotelling | The Economics of Exhaustible Resources[END_REF], the natural resource literature flourished in the 1970s-1980s, in the aftermath of the 1973 and 1979 oil crises. A significant part of those studies were based on extraction models, where the producer maximizes its profits by lowering its exploration and production costs. For instance, [START_REF] Gilbert | Optimal Depletion of an Uncertain Stock[END_REF], [START_REF] Heal | Uncertainty and the Optimal Supply Policy for an Exhaustible Resource[END_REF], and [START_REF] Loury | The Optimal Exploitation of an Unknown Reserve[END_REF] examined optimal extraction paths under reserve uncertainty. Pindick (1978) added price volatility to this kind of models. Our paper is at the crossroads of two literatures.The first stream of literature deals with sustainable growth with natural resources. The cornerstone of those studies is [START_REF] Stiglitz | Growth with exhaustible natural resources: Efficient and optimal growth paths[END_REF] neoclassical optimal growth model, where natural resources are essential for production. He points out that optimal growth and a sustainable per capita consumption is feasible with exhaustible natural resource in limited supply, if the ratio of the rate of technical change to the the rate of population growth is greater than or equal to the share of natural resources. [START_REF] Solow | Intergenerational equity and exhaustible resources[END_REF] also uses a neoclassical growth model, where he applies the maxmin principle to the intergenerational problem of optimal capital accumulation, in general and then in particular with scarce natural resources. [START_REF] Dasgupta | The optimal depletion of exhaustible resources[END_REF] have a different approach of technical change, they do not view it as a smooth gradual process, but as technological breaktrough. They thus introduce a "backstop technology", which represents a major discovery, a new substitute for the depletable resource. An example might be the substitution of fossil fuel energy by solar or wind energy. It should be emphasized that this date of discovering is completely unknown. Nevertheless, this substitution process is a way to overcome resource scarcity and avoid a falling level of per capita conumption. Dasgupta and Stiglitz (1976) and Heal (1978) follow this approach. More recently, [START_REF] Benchekroun | The optimal depletion of exhaustible resources: A complete characterization[END_REF] show that in a Dasgupta-Heal-Solow-Stiglitz based model, the initial consumption under a utilitarian criterion starts below the maximin rate of consumption if and only the resource is abundant enough. They underline the fact that the present generation does not necessarily benefits most from a windfall of resources. In addition, some articles decided to introduce endogenous growth with non renewable natural resources in this literature, as [START_REF] Schou | A Growth Model with Technological Progress and Nonrenewable Resources[END_REF]. In particular, [START_REF] Barbier | Endogenous Growth and Natural Resource Scarcity[END_REF] proposed a "Romer-Stiglitz" model, where endogenous technical change can overcome resource scarcity. [START_REF] Grimaud | Non-renewable resources and growth with vertical innovations: optimum, equilibrium and economic policies[END_REF] analyses the effect of intervention instruments on the rate of resource extraction and growth rate in a Schumpeterian model. Furthermore, [START_REF] Groth | Can non-renewable resources alleviate the knife-edge character of endogenous growth?[END_REF] show in a one-sector optimal endogenous growth model, growth per capital consumption is unstable unless there is population growth. They adopt a "semi-endogenous" framework to allow a stable long-run growth. The second body of literature embrace extensions of the Ramsey model to an open economy. It is shown that the open economy version of the Ramsey model leads to a number of paradoxical conclusions. Indeed, Barro and Salai-Martin(2003) display that the speed of convergence for capital and output is infinite, ie the economy jumps into the steady state. Moreover, consumtions tends to zero, except for the most patient country. Some attempts have been made to improve those counterfactual resluts, by introducing a constraint on international credit or adjustment costs. The infinite speed of convergence for capital and output does not apply to countries that are effectively constrained to borrow, and consumption does not decrease to zero. With adjustment costs for investments, capital and output are less instantaneous, even if capital markets are perfect.

Our main contribution is to introduce international borrowing in an exogenous Ramsey growth model, with exhaustible natural resources, in absence and then in presence of technical change. As a matter of fact, very few studies examine optimal exogenous growth models with resource exraction in an open economy. There are some exceptions, as [START_REF] Poelhekke | Volatility, Financial Development and the Natural Resource Curse[END_REF], [START_REF] Van Der Ploeg | Harnessing Windfall Revenues: Optimal Policies for ResourceRich Developing Economies[END_REF] who have introduced international finance in their models of resource-rich countries, which focus more on managing resource windfalls. This paper analyses whether a resource dependent economy can have a sustainable growth rate thanks to international borrowing or technical progress, while its main source of growth is bound to disappear. The resources here are essential for production, non renewable, and can apply to oil, gaz, shale gaz or to different minerals as copper, gold, diamonds. In other words, we explore whether indebtedness or technical change can help overcome resource scarcity.

In the next section, we set forth a general Ramsey model in a small open economy, with exhaustible natural resources. In section 3, we focus on a particular case where the interest rate r is constant. We then introduce a resourceaugmenting technical progress that helps raise the efficiency of resource use. In section 4, we generalize our model with an interest rate which is increasing in the level of debt. We then try to see if our results are different when resource prices are increasing.

Theorethical framework

We consider a simple neoclassical growth model with natural resources. It is a small open economy with borrowing capacities.

Output is produced with a constant-returns-to-scale technology, using a Cobb-Douglas production function:

Y = F (K, R) = K α ((1 -γ)R) 1-α , 0 < α < 1 (1)
where K is the stock of man-made capital, R a nonrenewable natural resource, and γ the share of the country's natural resources that are exported abroad.

Labor is supposed to be constant.

According to diminishing returns to the accumulation of capital,

F (K) = F K > 0 and F (K) < 0.
The man-made capital depreciates at rate δ:

K(t) = I(t) -δK(t), δ ∈ [0, 1] (2)
with I(t) the investment.

Natural resources

Let S(t) be the stock of non-renewable resources available at time t, and R(t) the rate of extraction of this resource at time t. The economy has an initial stock of natural resources S(0).

The stock decreases over time with the rate of extraction:

Ṡ(t) = -R(t) (3)
with R(t) a continuous function of time, K(0) > 0 and S(0) > 0.

The government

We consider an economy with a government that has constant relative risk aversion prefereneces. The intertemporal utility function is:

∞ 0 e -ρt U (C(t))dt (4) with U(C(t)) = C 1-η -1 1-η for η = 1, η > 0 and U(C(t)) = ln(C(t)) for η = 1
The variable C(t) denotes consumption at time t, and η is the coefficient of relative risk aversion. The discount rate ρ is assumed to be strictly positive.

The government has access to international capital markets, and can borrow or lend at an interest rate r, which is exogenous and which depends on the level of tne country's debt B(t) at time t. One part of the country's resources γR is exported abroad, and the rest (1-γ)R is sold on the domestic market. The government's dynamic budget constraint is:

Ḃ(t) = C(t) + r(B(t))B(t) + I(t) -Y (t) -γpR(t) ( 5
)
where p is the price of the natural resources sold aborad, and thus γpR(t) is the natural resources revenu received by the government from exports.

The government maximises its lifetime utility on an infinite horizon, subject to the capital accumulation equation ( 2), to the natural resources stock constraint equation ( 3) and the budget consraint equation ( 5). The problem to be solved is thus:

max {C,I,R} ∞ 0 e -ρt U (C(t))dt s.t. K(t) = I(t) -δK(t) Ṡ(t) = -R(t) Ḃ(t) = C(t) + r(B(t))B(t) + I(t) -Y (t) -γpR(t) K(0) > 0, S(0) > 0
There are three state variables S(t), B(t), and K(t). Three control variables C(t), I(t) and R(t) to be chosen.

The current value Hamiltonian is :

H (C, R, I, λ 1 , λ 2 , λ 3 , t) = U(C(t)) + λ 1 (t)(I(t) -δK(t)) -λ 2 (t)R(t) + λ 3 (C(t) + r(B(t))B(t) + I(t) -Y (t) -γp(t)R(t))
The co-states λ 1 (t), λ 2 (t), λ 3 (t) represent respectively the shadow price of accumulated capital, of the resource stock and the shadow price of debt.

The optimality conditions are given by:

U (C(t)) = -λ 3 (t) (6) -λ 3 (t)(F R + γp) = λ 2 (t) (7) -λ 3 (t) = λ 1 (t) (8) λ1 (t) = λ 1 (t)(δ + ρ) + λ 3 (t)F K (9) λ2 (t) = ρλ 2 (t) (10) λ3 (t) = λ 3 (t)(ρ -r (B(t))B(t) + r(B(t))) (11)
At the optimum rate, the rate of return is the same at all points in time, being equal to the social discount rate (equation 10): λ2(t) λ2(t) = ρ As the Hotelling rule goes, the shadow price of the resource in the ground, also called the scarcity rent, grows at the discount rate.

The transversality conditions are:

lim t→∞ e -ρt λ 1 (t)K(t) = 0 lim t→∞ e -ρt λ 2 (t)S(t) = 0 lim t→∞ e -ρt λ 3 (t)B(t) = 0
It is not possible to accumulate capital and debt indefinitely. As in the long run, the natural resource will be exhausted, the country cannot extract those resources indefinitely.

Model with a constant interest rate

The economy has unrestricted access to a perfect world capital market.

Model without technical progress

We now assume that the interest rate equals a constant r. We also assume that r ≤ ρ applies, because if not the economy would eventually accumulate enough assets to violate the small-country assumption that we made.

With an interest rate r constant, the government's dynamic budget constraint becomes:

Ḃ(t) = C(t) + rB(t) + I(t) -Y (t) -γpR(t) (12)
The optimality conditions remain the same as in the general model, except for equation ( 11) that becomes:

λ3 (t) = λ 3 (t)(ρ -r) (13)
Using ( 8) and ( 11) yields that the marginal productivity of capital is equal to the depreciation rate plus the interest rate:

F K = δ + r (14)
Besides, using the production function and equation ( 14), the marginal productivity of capital can be reexpressed as:

F K = α( K (1 -γ)R ) α-1 = δ + r
It leads to a ratio capital natural resource K (1-γ)R which is constant, when the interest rate r is constant:

K (1 -γ)R = ( δ + r α ) 1 α-1
From ( 9) and ( 11), the marginal productivity of natural resources depends on the price of those resources and on the interest rate:

F R + γp = - λ 2 (0) λ 3 (0) e rt (15)
As one can note, there is an incompatibility between equation ( 15), which grows exponentially, and the marginal productivity of the natural resources derived from the production function,

F R + γp = (1 -α)( K (1-γ)R ) α + γp, which is constant.
We have to distinguish two cases: -If the country does not export its natural resources, then γ = 0, the model does not work. Equation (15) becomes:

F R = - λ 2 (0) λ 3 (0) e rt ,
which is not possible as F R is constant.

-If the country exports its natural resources, equation ( 15) holds if and only if prices increase at a rate r. Therefore, prices can be reexpressed as:

p(t) = p(0)e rt
Using equations ( 14) and ( 15) boils down to a new Solow-Stiglitz condition:

ḞR + γ ṗ F R + γp = F K -δ = r (16)
We can see that along the optimal path, the growth rate of the marginal productivity of the natural resources plus the growth rate of the prices have to be equal to the interest rate.

Proposition: The optimal rate of consumption is given by

g C = Ċ C = r -ρ η , η > 1 (17) If η = 1, Ċ C = r -ρ
Proof : This is straightforward from equations ( 6) and ( 11), which gives the Keynes-Ramsey rule.

As r ≤ ρ, the rate of consumtion is negative, therefore consumption decreases asymptotically towards zero. This result confirms what has been shown in the literature extending the Ramsey model to an open economy with international borrowing, where consumption also tended to zero.

We shall see now what are the implications of a constant ratio capital natural resource on output.

Proposition: The optimal path of output and stock of capital approach zero.

Proof : We know that the natural resources are exhaustible, so that the rate of extraction of those resources tends towards zero: lim t→+∞ R = 0 Since the ratio K R is constant, this implies that the accumulation of capital also approaches zero lim

t→+∞ K = 0
As K and R are falling to zero, therefore output also decreases towards zero:

lim t→+∞ Y = 0
In other words, not just the growth rate but even the level of output will vanish.

In addition, from equation ( 2), we also have a level of investment approaching zero: lim

t→+∞ I = 0
Those no-output and no-growth results appear to be counterfactual. Especially, the fact that production also falls towards zero leads to an economy on the way to extinction.

Adding exhaustible natural resources to an open economy version of the

Ramsey model leads thus to different results from those models with no natural resources, where the speed of convergence for capital and output is infinite.

A balanced growth path (BGP) is defined as a path along which the quantities Y,C and K change at constant proportionate rates. Let g x denote the growth rate of the variable x > 0, that is g x = ẋ

x .

Proposition: Along a BGP, g Y = g K = g I = g C = g R Proof : Let reexpress the production function in growth rate, such as g x = ẋ

x :

Y = ( K (1 -γ)R ) α R Then, g Y = αg K (1-γ)R + g R As the ratio K (1-γ)R is constant, its growth rate equals zero. Therefore, g K = g R and g Y = g R Then, g Y = g K = g I = g R = g C = r-ρ η , η > 1
Concerning the debt path, we need to consider the constraint budget:

Ḃ(t) = C(t) + rB(t) + I(t) -Y (t) -γpR(t): B(t) = b 1 e rt + b 2 e r-ρ η t + b 3 e ( r-ρ η +r)t
Solving this, with the transversality condition lim t→∞ e -ρt λ 3 (t)B(t) = 0 and the corresponding solution λ 3 (t) = λ 3 (0)e (ρ-r)t yields: From the transversality condition, we can see that b 1 = 0 and as r ≤ ρ, lim t→+∞ B(t) = 0. This induces that maxg B = r -ρ η

In conclusion, in a small open economy with exhaustible natural resources when there is no technical progress and a constant interest rate, all the variables output, capital, consumption, natural resources, the level of debt grow at the same rate along the BGP and decline asymptotically towards zero.

Therefore, as in the literature of sustainable growth with natural resources, such as [START_REF] Stiglitz | Growth with exhaustible natural resources: Efficient and optimal growth paths[END_REF] or [START_REF] Dasgupta | The optimal depletion of exhaustible resources[END_REF], sustainability is not feasible without technical progress.

Model with technical progress

We now introduce a resource-augmenting technical progress in the model. Therfore, even though natural resources are still exhaustible, the rate of extraction does no longer fall asymptotically to zero as it used to do in the precedent sections. This assumption can be supported by the fact that technological change can lead to the discovery of new fields, like in Brazil, or allow the exploitation of resources previously thought not to be economically accessible, as offshore high pressure -high temperature wells for

The new Cobb-Douglas production function is:

Y = K α (A(1 -γ)R) 1-α , 0 < α < 1
with A the resource-saving technological change. A is growing at some constant exogenous rate τ > 0, so that A = e τ t Prices and the interest rate are assumed to be constant.

We have the same optimality conditions as in the benchmark model, except that this time the marginal productivity of capital and the marginal productivity of the resource can be rexpressed as:

F K = α( K A(1 -γ)R ) α-1 (18) F R = (1 -α)( K A(1 -γ)R ) α A (19)
From equation ( 14), yields:

F K = δ + r
This implies that the ratio

K A(1-γ)R is constant: K A(1 -γ)R = ( δ + r α ) 1 α-1 (20)
Moreover, the modified Solow-Stiglitz efficiency condition still holds:

ḞR + γp F R + γp = F K -δ = r
In other words, along the optimal path the the interest has to be equal to the growth rate of the marginal productivity of the resource plus the government's revenue of exported resources.

Using this last efficiency condition and equation ( 23) yields:

ḞR F R = α( K A(1-γ)R K A(1-γ)R ) + Ȧ A (21) ḞR + γp F R + γp = (α( K A(1-γ)R K A(1-γ)R ) + Ȧ A )F R F R + γp As the ratio K A(1-γ)R is constant, therefore its growth rate g K AR = K A(1-γ)R K A(1-γ)R
is equal to zero. Equation ( 25) can thus be reexpressed as:

ḞR + γp F R + γp = τ F R (1 -α)( K A(1-γ)R ) α A + γp Proposition:
The optimal growth rate of consumption is given by:

g C = Ċ C = r -ρ η
Proof : This is straightforward from equations ( 6) and ( 11), which gives the Keynes-Ramsey rule. Therefore, the consumption growth rate is negative, as r ≤ ρ. Consumption thus cannot be sustainable in the long run.

Our result is thus different from [START_REF] Stiglitz | Growth with exhaustible natural resources: Efficient and optimal growth paths[END_REF], where thanks to techincal progress there is sustainability in a closed economy with exhaustible natural resources, if the ratio of the rate of technical change to population gowth rate is higher or equal than the share of natural resource in production.

Proposition: Along a BGP, g Y = g K = g I = g C and g R < 0

Proof :

g A(1-γ)R = g A + g (1-γ)R = g A We thus assume g R < 0
As the ratio K A(1-γ)R is constant, by differenciating logarithmically we have:

g K A(1-γ)R = g K -g A(1-γ)R = 0 and g A(1-γ)R = g A + g (1-γ)R = g A + g R , as γ constant
This implies that:

g K = g A + g R Furthermore, as g K = g C , we have: g R = g C -g A Then, g R = r -ρ η -τ As g R < 0, therefore g C = r-ρ η < τ
Concerning the debt path, we need to consider the constraint budget, knowing that Y, I and R tend asymptotically towards zero, with constant prices:

B(t) = b 1 e rt + b 2 e r-ρ η t
Solving this, with the transversality condition lim t→∞ e -ρt λ 3 (t)B(t) = 0 and the corresponding solution λ 3 (t) = λ 3 (0)e (ρ-r)t yields:

lim t→+∞ λ 3 (0)(b 1 e rt + b 2 e r-ρ η t )e -rt = 0 λ 3 (0) lim t→+∞ (b 1 + b 2 e ( r-ρ η -r)t ) = 0
From the transversality condition, we can see that b 1 = 0 and as r ≤ ρ, lim t→+∞ B(t) = 0. This implies that

maxg B = r -ρ η -r
In conclusion, in a small open economy with exhaustible natural resources when there is technical progress and a constant interest rate, all the variables output, capital, consumption, natural resources, level of debt decline towards zero. Moreover, along the BGP those variables do not grow at the same rate.

Surprisingly, technical progress cannot overcome resource scarcity, as the interest rate is exogenous and constant Therefore, contrary to [START_REF] Stiglitz | Growth with exhaustible natural resources: Efficient and optimal growth paths[END_REF] positive growth rates for output and consumption cannot be sustained in the long run.

Model with a debt-elastic interest rate

The economy faces now limitations in its access to the world financial markets. It is indeed more realistic than a country that can borrow as much as it wants at a fixed rate.

Case with constant prices

The interest rate r(B) depends now on the level of debt. We thus assume that r(B) is increasing in the aggregate level of foreign debt.

Proposition: The optimal level of debt decreases and output falls to zero in the long-run.

Proof : Using the optimality conditions of the general model, especially equations ( 8), ( 9) and ( 11), we have the marginal productivity of the capital that depends on the interest rate:

F K = δ + r (B).B + r(B) (22)
As

F K = α( K (1-γ)R ) α-1 , we can find the ratio K (1-γ)R : K (1 -γ)R = ( δ + r (B).B + r(B) α ) 1 α-1 (23)
From equations ( 7), ( 10), ( 11) and ( 17), the marginal productivity of the natural resources is given by:

+ γp F R + γp = F K -δ = r (B).B + r(B) (24)
This relation corresponds to the new Solow-Stiglitz efficiency condition in an open economy with a debt-elastic interest rate.

But F R can be also written as:

F R = (1 -α)( K (1-γ)R ) α
Therefore, ḞR

F R = α( K (1-γ)R K (1-γ)R
) Using (18), yields:

ḞR + γp F R + γp = 1 α * G(B).r (B) * Ḃ with G(B) = α 1 α-1 (r (B).B + r(B) + δ) ( 1 α-1 -1) (r (B).B + r(B) + δ) ( 1
α-1 ) + γp And thus with (20), we can have a function of B, r(B), r'(B), G(B) < 0:

1 α * G(B).r (B) * Ḃ = r (B).B + r(B)
We reexpress this condition as the following autonomous differential equation:

Ḃ = H(B)
with

H(B) = (r (B).B + r(B)) * α G(B) * r (B)
As B > 0, r(B) > 0 and r (B) > 0, and G(B) < 0, H(B) = Ḃ is thus negative.

Therefore, the optimal level of debt B is decreasing with time. This is quite a paradoxical result, as the economic policy recommanded here is to pay off its debt from the beginning and thus not to take any risks.

As in the literature it is more common to use an exponential interest rate, we will use this last formulation to see if our result is still robust.

We refer to Schmitt-Grohe and Uribe ( 2003 Proposition: When the interest rate is exponential, the optimal level of debt decreases and output falls to zero in the long-run.

Proof : The formulation of the interest rate implies r(B) > 0 and r (B) = ψe B-D > 0. G(B) is still negative, thus Ḃ < Debt is thus still decreasing with time.

As the level of debt is falling to zero, the ratio K

(1-γ)R also approaches asymptotically a constant:

K (1 -γ)R = ( ψe B-D .B + r * + ψ(e B-D -1) + δ α ) 1 α-1
We provide a numerical example that illustrates the debt pattern in function of time when the interest rate is exponential (Figure 2). More precisely, we set α = 0, 32, δ = 0, 1, p = 1, γ = 0, 5, r * = 0, 04, D = 0, 7442 and ψ = 0, 8, and we plot the optimal debt path in function of time.

In fact, despite the form of interest rate chosen, the level of debt B is still decreasing towards zero and the ratio K (1-γ)R tends asymptotically towards a constant.

We thus find the same counterfactual results from the benchmark model with a constant interest rate r concerning the accumulation of capital, the level of investment, and output that decrease asymptotically towards zero.

Proposition: The optimal rate of consumption is given by:

Ċ C = F K -δ -ρ η = r (B).B + r(B) -ρ η , η > 1 (25)
Proof : This is straightforward from equations ( 6) and (11). Consumption accumulates at a rate equal to the difference between the net marginal product of capital F K -δ and the discount rate ρ. Nevertheless this time the consumption growth rate g C is depending on the level of debt B t , contrary to the benchmark model. In fact during the transitional dynamics, the consumption growth rate is first positive: we have g C ≥ 0 when r (B).B + r(B) ≥ ρ In the long-run, as the level of debt B tends asymptotically towards zero, this growth rate becomes negative, thus g C ≤ 0, and consumption decreases. To illustrate the consumption growth rate in function of time when the interest rate is exponential (Figure 3), we set α = 0, 32, δ = 0, 1, p = 1, γ = 0, 5, r * = 0, 04, D = 0, 7442 and ψ = 0, 8. We can see that the consumption growth rate is first positive, and then declines in the long-run. Therefore, when we endogenize the interest rate, consumption can grow for a while before decreasing.

Case with increasing prices

From now on, prices have been taken as constant. We now assume prices to increase at a rate θ, with: p(t) = p(0)e θt . We shall see if increasing prices, like in 2002-2008 where the oil price went from 20$ to 140$, can help have a sustainable growth.

The equation ( 20) becomes:

ḞR + γ ṗ F R + γp = G(B).r (B) * Ḃ + γ ṗ α G(B).r (B) * Ḃ + γ ṗ α = r (B).B + r(B)
We reexpress this condition as the following non autonomous differential equation, as p depends now on time:

Ḃ = H 2 (B) with H 2 (B) = α(r (B).B+r(B))-γ ṗ G(B) * r (B)
As we know from below r(B) > 0, r (B) > 0, and G(B) < 0. Moreover, γ > 0 and ṗ > 0. Therefore, when α(r (B).B + r(B)) < γ ṗ, then Ḃ > 0. Let us apply this condition in the case where B(0) takes the value of zero and when the interest rate is exponential, this condition becomes: α(r * + ψ(e -D -1)) < γ ṗ(0). As p(0) is assumed to be equal to one, then ṗ(0) = θ. This inequality is thus verified when ψ(e -D -1) < γθ α -r * . In that case, the country increases its level of debt. In the short-run, it is interesting to see that as debt is growing, consumption grows.

But when α(r (B).B + r(B)) > γ ṗ, then H 2 (B) = Ḃ < 0, the optimal level of debt is still decreasing, even though prices are increasing. Indeed when B(0) takes the value of zero and when the interest rate is exponential, this condition becomes: ψ(e -D -1) > γθ α -r * . This means that even in a period of relative prosperity, when the price of the natural resource is increasing, thus when the country earns more money then it used to do, the government ought to pay off its debt in priority. In the long-run, it confirms the results from the case with a debt-elastic interest rate and constant prices, which is quite similar to what we found in the benchmark model, where consumption declines.

Conclusion

In this paper, we have developed an exogenous optimal growth model of a resource-rich open economy, where natural resources are essential for production and non-renewable.

The main finding of this paper is that sustainability is not feasible when the interest rate is constant, even if there is technical progress, but when the economy faces a debt-elastic interest rate, consumption can grow for a while before decreasing in the long-run.

First, we show that when the interest rate is assumed constant, this economy is faced to decreasing consumption and production. This leads us to some counterfactual no-growth results, as all our variables approach asymptotically zero and grow at the same rate on a balanced growth path. When we add technical progress, results are quite similar to this first case except that natural resources and output, consumption do no longer grow at the same rate. Contrary to [START_REF] Stiglitz | Growth with exhaustible natural resources: Efficient and optimal growth paths[END_REF], resource augmenting technical progress does not overcome resource scarcity. In the literature extending Ramsey models to an open economy, borrowing constraints with a collateral or adjustment costs are usually added in order to found less paradoxical results, but our approach is different. We introduce an interest rate that increases in the aggregate level of debt. It is a more realistic assumption, as countries usually face a risk premium. Second, we show that when we endogenize the interest rate, the optimal level of debt asymptotically declines, and so do the output, capital and investment. Therefore, the country ought to reduce its debt. Nevertheless, the growth rate of consumption is positive during the transitional dynamics before declining.

This paper can be considered as a contribution to the optimal growth models with exhaustible natural resources, that focused mainly on closed economies. One way to extend this paper would be to optimize the model according to γ the share of the country's natural resources exported abroad. Another extension would be to introduce a constraint on international credit, where natural resources could be used as collateral.

  lim t→+∞ λ 3 (0)(b 1 e rt + b 2 e r-ρ η t + b 3 e ( r-ρ η +r)t )e -rt = 0 λ 3 (0) lim t→+∞ (b 1 + b 2 e ( r-ρ η -r)t + b 3 e

  ) debt-elastic interest-rate premium. It can be expressed as:r(B) = r * + ψ(e B-D -1)where ψ(e B-D -1) is the country-specific interest rate premium, r * is the world interest rate, D is the steady-stae level of foreign debt. r * , D and ψ are constant and positive parameters.

Figure 1 :

 1 Figure 1: Exponential interest rate r(B) in function of the level of debt

Figure 2 :

 2 Figure 2: Debt path in function of time With an exponential interest rate r(B)

Figure 3 :

 3 Figure 3: Consumption growth rate in function of time With an exponential interest rate r(B)