0 Continuous - wave Doppler - cooling of hydrogen atoms with two - photon transitions

We propose and analyze the possibility of performing two-photon continuouswave Doppler-cooling of hydrogen atoms using the 1S − 2S transition. “Quenching” of the 2S level (by coupling with the 2P state) is used to increase the cycling frequency, and to control the equilibrium temperature. Theoretical and numerical studies of the heating effect due to Doppler-free two-photon transitions evidence an increase of the temperature by a factor of two. The equilibrium temperature decreases with the effective (quenching dependent) width of the excited state and can thus be adjusted up to values close to the recoil temperature. Typeset using REVTEX 1 Laser cooling of neutral atoms has been a most active research field for many years now, producing a great deal of new physics. Still, the hydrogen atom, whose “simple” structure has lead to fundamental steps in the understanding of quantum mechanics, has not yet been laser-cooled. The recent experimental demonstration of the Bose-Einstein condensation of H adds even more interest on laser cooling of hydrogen [1]. One of the main difficulties encountered in doing so is that all transitions starting from the ground state of H fall in the vacuum ultraviolet (VUV) range (121 nm for the 1S − 2P transition), a spectral domain where coherent radiation is difficult to generate. In 1993, M. Allegrini and E. Arimondo have suggested the laser cooling of hydrogen by two-photon π pulses on the 1S − 3S transition (wavelength of 200 nm for two-photon transitions) [2]. Since then, methods for generation of CW, VUV, laser radiation have considerably improved, and have been extensively used in metrological experiments [3]. This technical progress allows one to realistically envisage the two-photon Doppler cooling (TPDC) of hydrogen in the continuous wave regime, in particular for the 1S − 2S two-photon transition. Laser cooling relies on the ability of the atom to perform a great number of fluorescence cycles in which momentum is exchanged with the radiation field. It is well known that 2S is a long-lived metastable state, with a lifetime approaching one second. From this point of view, the 1S − 2S two-photon transition is not suitable for cooling. On the other hand, the minimum temperature achieved via Doppler cooling is proportional to the linewidth of the excited level involved on the process [4], a result that will be shown to be also valid for TPDC. From this point of view 2S is an interesting state. In order to conciliate these antagonistic properties of the 1S−2P transition, we consider in the present work the possibility of using the “quenching” [5] of the 2S state to control the cycling frequency of the TPDC process. For the sake of simplicity, we work with a onedimensional model. We write rate equations describing TPDC on the 1S − 2S transition in presence of quenching. The quenching ratio is considered as a free parameter, allowing control of the equilibrium temperature. The cooling method is then in principle limited only by photon recoil effects. 2 We also develop analytical approaches to the problem. A Fokker-Planck equation is derived, describing the dynamics of the process for temperatures well above the recoil temperature Tr (corresponding to the kinetic energy acquired by an atom in emitting a photon). A numerical analysis of the dynamics of the cooling process completes our study. Let us consider a hydrogen atom of mass M and velocity v parallel to the z-axis (Fig. 1) interacting with two counterpropagating waves of angular frequency ωL with 2ωL = ω0 + δ, where ω0/2π = 2.5 × 10 14 Hz is the frequency corresponding to the transition 1S → 2S, and also define the quantity k ≃ 2kL = 2ωL/c. The shift of velocity corresponding to the absorption of two-photons in the same laser wave is ∆ = h̄k/M = 3.1 m/s. We will neglect the frequency separation between 2S and 2P states (the Lamb shift – which is of order of 1.04 GHz) and consider that the one-photon spontaneous desexcitation from the 2P states also shifts the atomic velocity of ∆ randomly in the +z or −z direction. Note that Tr = M∆ 2/kB ≈ 1.2 mK for the considered transition (kB is the Boltzmann constant). We neglect the photo-ionization process connecting the excited states to the continuum. This is justified by the 1/E decreasing of the continuum density of states as a function of their energy E and by the fact that a monochromatic laser couples the excited levels only to a very small range of continuum levels. The atom is subjected to a controllable quenching process that couples the 2S state to the 2P state (linewidth Γ2P = 6.3 × 10 s). The adjustable quenching rate is Γq. Four two-photon absorption process are allowed: i) absorption of two photons from the +z-propagating wave (named wave “+” in what follows), with a rate Γ1 and corresponding to the a total atomic velocity shift of +∆; ii) absorption of two photons from the −zpropagating wave (wave “−”), with a rate Γ−1 and atomic velocity shift of −∆; iii) the absorption of a photon in the wave “+” followed by the absorption of a photon in the wave “−”, with no velocity shift and iv) the absorption of a photon in the wave “−” followed by the absorption of a photon in the wave “+”, with no velocity shift. The two latter process are indistinguishable, and the only relevant transition rate is that obtained by squaring the sum of the amplitudes of these process (called Γ0). Also, these process are “Doppler-free” 3 (DF) as they are insensitive to the atomic velocity (to the first order in v/c) and do not shift the atomic velocity. Thus, they cannot contribute to the cooling process. As atoms excited by the DF process must eventually spontaneously decay to the ground state, this process heats the atoms. In the limit of low velocities, the transition amplitude for each of the four processes is the same. One thus expects the DF transitions to increase the equilibrium temperature by a factor of two. We can easily account for the presence of the quenching by introducing an effective linewidth of the excited level (which, due to the quenching process, is a mixing of the 2S and 2P levels) given by Γe = Γ2P Γq Γq + Γ2P = gΓ2P (1) with g ≡ Γq/(Γq + Γ2P ). This approximation is true as far as the quenching ratio is much greater than the width of the 2S state (note that this range is very large, as the width of the 2S state is about 10 times that of the 2P state). The two-photon transition rates [6] are given by: Γn = Γ2P g 2 (1 + 3δn0)Ī 2 (δ̄ − nKV ) + g/4 (2) where n = {−1, 0, 1} describes, respectively, the absorption from the “−” wave, DF transitions, and the absorption from the “+” wave. Ī ≡ I/Is where Is is the two-photon saturation intensity, δ̄ is the two-photon detuning divided by Γ2P , K ≡ k∆/Γ2P = 0.26 and V ≡ v/∆. The rate equations describing the evolution of the velocity distribution n(V, t) and n(V, t) for, respectively, atoms in the ground and in the excited level are thus ∂n(V, t) ∂t = − [Γ−1(V ) + Γ0 + Γ1(V )]n(V, t) + Γe 2 [n∗(V − 1) + n∗(V + 1)] (3a) ∂n(V, t) ∂t = Γ−1(V − 1)n(V − 1, t) + Γ0n(V, t) + Γ1(V + 1)n(V + 1, t)− Γen ∗(V, t). (3b) The deduction of the above equations is quite straightforward (cf. Fig 1). The first term in the right-hand side of Eq. (3a) describes the depopulation of the ground-state velocity 4 class V by two-photon transitions, whereas the second term describes the repopulation of the same velocity class by spontaneous decay from the excited level. In the same way, the three first terms in the right-hand side of Eq. (3b) describe the repopulation of the excited state velocity class V by two-photon transition, and the last term the depopulation of this velocity class by spontaneous transitions. For each term, we took into account the velocity shift (V → V ± 1) associated with each transition and supposed that spontaneous emission is symmetric under spatial inversion. For moderate laser intensities, one can adiabatically eliminate the population of excited level. This is valid far from the saturation of the two-photon transitions and reduces the Eqs. (3a-3b) to one equation describing the evolution of the ground-state population: dn(V, t) dt = − [ Γ0 + Γ−1(V ) 2 + Γ1(V ) 2 ] n(V, t) + 1 2 {Γ0 [n(V − 1, t) + n(V + 1, t)] + Γ−1(V − 2)n(V − 2, t) + Γ1(V + 2)n(V + 2, t)} (4) Eq.(4) is in fact a set of linear ordinary differential equations coupling the populations of velocity classes separated by an integer: V, V ± 1, V ± 2, · · ·. This discretization exists only in the 1-D approach considered here, but it does not significantly affect the conclusions of our study, while greatly simplifying the numerical approach. Eqs.(4) can be recast as dn/dt = Cn, where C is a square matrix and n is the vector (· · ·n(−i, t), · · ·n(0, t), n(1, t), · · ·). Numerically, the equilibrium distribution is obtained in a simple way as the eigenvector neq of C with zero eigenvalue. In this way, the asymptotic temperature is obtained as :