Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Deep sub-Doppler cooling of Mg in MOT formed by light waves with elliptical polarization Abstract: We study magneto-optical trap of 24Mg atoms operating on the closed triplet 3P23D3 ( = 383.3 nm) transition formed by the light waves with elliptical polarization (--* configuration). Compare to well-known trap formed by light waves with circular polarization ( +-- configuration) the suggested configuration offer the lower sub-Doppler temperature for trapped Mg atoms, that can’t be reached in conventional +-- MOT. Oleg Prudnikov D.V. Brazhnikov, A.V. Taichenachev, V.I. Yudin, A.N. Goncharov Institute of Laser Physics, Novosibirsk 2016 Basic mechanism of laser cooling 1) Two-level model radiation pressure for moving atom dipole force F k Doppler cooling k BTD / 2 optical potential sub-doppler laser cooling 2) polarization effects lin lin Jg 1 Je 2 J.Dalibard and C.Cohen-Tannoudji, J.Opt.Soc. Am. B (1989). 3) light fields with elliptical polarization "anomalous" sub-dopller cooling effects rectification of dipole force jg 1/ 2 je 3 / 2 F 0 F ( 0 = 0, ) = 0 • O.N. Prudnikov, A. V. Taichenachev, A. M. Tumaikin and V. I. Yudin, Kinetics of atoms in the field produced by elliptically polarized waves, JETP vol.98, pp. 438-454, (2004) • A.V. Bezverbnyi, O.N. Prudnikov, A.V. Taichenachev, A.M. Tumaikin, V.I. Yudin The light pressure force and the friction and diffusion coefficients for atoms in a resonant nonuniformly polarized laser field JETP v.96, pp. 383-401, (2003) • O.N. Prudnikov, A.V. Taichenachev, A.M. Tumaikin, V.I. Yudin Rectification of the dipole force in a monochromatic field created by elliptically polarized waves JETP vol.93, pp.63-70 (2001) Kinetics of atoms in light fields Two coordinate density matrix example: optical transition 12 ˆ ( z1 , z 2 ) ˆ ( p1 , p2 ) ˆ (r , p ) Coordinate representation Momentum representation d 3r ˆ (r q / 2, r q / 2) exp(i p q / ) ( 2 ) 3 d 3u ˆ ( p u / 2, p u / 2) exp(i r u / ) (2 )3 d i ˆ Hˆ , ˆ ˆ ˆ dt d gg i ˆ Hˆ eff , ˆ gg ˆ ˆ gg dt Semiclassical approach Fokker-Plank equation p i i W t i m 2 Fi ( r , p ) W Dij ( r , p ) W i pi ij pi p j ˆ ee ˆ ge ˆ r1 r2 r = (r2 + r1)/2, q = r1 - r2 ˆ eg ˆ gg for S = ||2/(2/2+2) << 1 Quantum approaches / - quantum Monte-Carlo wave-function method [ J. Dalibard,et.al. PRL 68, 580 (1992) ] - Band theory (cooling in optical potential) [Y. Castin,et.al. Europhys. Lett.14, 761 (1991)] - cooling in + - - field [A. Aspect,et.al. PRL 61,826 (1988)]. W (r , p) Tr ˆ (r , p ) p- family approaches Generalized continuous fraction method for density matrix equation Atom-laser interaction part of Hamiltonian can be expressed as sum of two parts from opposite ligth waves. Vˆ Vˆ1eikz Vˆ2e ikz ˆ ( z , q) ˆ ( n ) (q)e ikz n Density matrix equation in coordinate representation takes the following form for spatial harmonics (n): d i ˆ Hˆ , ˆ ˆ ˆ dt d (n) i ( n ) ˆ ( n ) ˆ ( n 1) ˆ ( n 1) ˆ n ˆ L0 ˆ L ˆ L ˆ dt M q 1. We assume that the spatial coherence of density matrix is damped at enough large distance qmax we consider the spatial interval [–qmax..qmax] and make a mesh with discrete points qi (total Nq points). On the mesh we define derivative (n) in standard form: 1 ˆ q i ˆ q ˆ q i 1 q 2q i 1 2. The equation can be written in Liouville representation with Liouvile operators L0, L+, L- and density matrix in Liouville form: . ee e , e/ ;qi eg e , g/ ; qi ˆ qi qi ge / mggg , e ;qi mg , mg/ ;qi . d (n) i n G ( n ) L ( n1) L0 ( n) L ( n 1) dt M Example: For the case of optical transition 1/23/2 the vector contains 18 x Nq elements, for the case of 1 2 it contains 34 x Nq elements. O.N. Prudnikov, A.V. Taichenachev, A.V. Tumaikin, V.I. Yudin, JETP v. 104, p839, (2007) O. N. Prudnikov, R.Ya. Ilenkov, A. V. Taichenachev, A. M. Tumikin, and V. I. Yudin, JETP v.112, p.939 (2011) Solution of density matrix equation Example 2 Example 1 spatial distribution E Tr Ĥ ̂ (b) density matrix spatial harmonics (c) Energy (a) momentum distribution spatial coherence (d) Energy of 85Rb atoms in linlin field (5S1/2 (F = 3)5P3/2 (F'=4)). The black and white dots represents the temperature measurements results [P. S. Jessen, et.al., Phys. Rev. Lett. 69, 49 (1992)] Density matrix spatial (а) and momentum (b) distributions for atoms with Jg=1/2Je=3/2 optical transition in standing wave with linear polarization (= - /2, = , R = 0.1). Density matrix spatial harmonics of the ground R{g (n)} and excited state R{e (n)} (c). Tr ˆ (q 0) R e ( n ) Tr ˆ ee ( n ) (q 0) R g (n) gg ( n ) Spatial coherence functions for the ground Qg and excited state Qe (d). ( q ) Tr ˆ ( q ) Q e (q ) Tr ˆ ee ( n 0 ) ( q) Qg gg ( n 0 ) max. of U min. of U Momentum distribution (solid line), and Gaussian approximation (dashed line). p2 I/Is = 1.4, = -8 F ( p ) e 2 mT Sub-Doppler cooling with taking into accounts all quantum recoil effects Sub-Doppler laser cooling takes place at low saturation S = ||2/(2 + 2/4) << 1. In this limit authors use reduced form of density matrix equation that allows to reduce the numbers of elements matrix gg is function of two parameters only! reduced equation 1) detuning / 2) optical potential depth u = S/R gg i ˆ Hˆ eff , ˆ gg ˆ ˆ gg t ( or intensity I ) • In band theory gg is function of only one parameter (U0/ER) [Y. Castin Europhysics Lett.v.14, 761, (1991) ] pˆ 2 Hˆ eff S Vˆ Vˆ 2M Example 1: atom with jg=1/2je=3/2 in lin lin Band theory approximation: U0 ER 2 Ek prms / 2M Ek/ pe2 / 2 M Fig. Kinetic energy for different detunings [Y. Castin, et al. Europhysics Lett. v.14, 761, (1991)] approximation assumes vibration levels splitting large the levels width caused by spontaneous relaxation and tunneling effects - it do not takes into account above barier atoms - is not valid for deep optical potential (small detunings) We see significant difference for field parameters out of secular approximation. O. N. Prudnikov, et.al. "Steady state of a low-density ensemble of atoms in a monochromatic field taking into account recoil effects", JETP v.112, pp.939-945 (2011) Sub-Doppler cooling with taking into accounts all quantum recoil effects reduced equation VS gg i ˆ Hˆ eff , ˆ gg ˆ ˆ gg t base equation d i ˆ Hˆ , ˆ ˆ ˆ dt ˆ ee ˆ ge ˆ ˆ eg ˆ gg pˆ 2 Hˆ eff S Vˆ Vˆ 2M matrix gg is function of two parameters only! 1) detuning / 2) optical potential depth u = S/R matrix gg is function of 3 parameters! ( or intensity I ) 3) semiclassical parameters R. R k 2 2 M Example 2: Solution on a base of general equation R = 0.05 Results: 1) density matrix is function of 3 parameters (additional parameter is R) 2) reduced equation is valid for only extremely low R , i.e. in conditions of semiclassical approach! R = 0.01 reduced equation R R = 0.005 k 2 2 M i.e. when we restrict consideration by reduced equation with effective Hamiltonian we stay in conditions of semiclassical approach. Fig. Kinetic energy of atoms with jg=1/2 je=3/2 optical transition in linlin field for = -5. O. N. Prudnikov, et.al. "Steady state of a low-density ensemble of atoms in a monochromatic field taking into account recoil effects", JETP v.112, pp.939-945 (2011) Sub-Doppler cooling of 24Mg in +-- field: semiclassical and quantum approaches 33P2 → 33D3 [*] M. Riedmann, et. al. PRA v.86, 043416 (2012). (T 1 mK) M 238 k 2 / 476 Erec 1.28 mK Semiclassical approach: 2 26.3 MHz k / p 1 ultracold fraction with |p|<3hk 5 5 momentum distribution 5 I 20 mW / cm 2 T p 2 M k 2 R 1 2 M / 2 Here the solid lines represent quantum simulation results, and the dashed lines - semiclassical approach results O. N. Prudnikov, D. V. Brazhnikov, A. V. Taichenachev, V. I. Yudin, A. E. Bonert, R. Ya. Il'enkov, and A. N. Goncharov, Quantum treatment of two-stage sub-Doppler laser cooling of magnesium atoms, Phys. Rev. A 92, 063413 (2015) Sub-Doppler cooling conditions for S = ||2/(2/2+2) << 1 Sub-Doppler temperature T p2 M D ( 0) s z k 2 ( / ) S M ( / ) S z slow atom approach: p M S/k negligible in semiclassical approach k 2 R 2M What happens for non negligible R ? optical transition Is [mW/cm2] (/) function for atoms with 23 optical transition R 1 ~ 2M [nm] p 5 hk . 24Mg 33P2 → 33D3 61.7 1) violation of slow atom approach 2) range of sub-Doppler force might be few recoil momentum. 383.9 238 atom may not "see" sub-Doppler force! Sub-Doppler cooling of 24Mg: quantum approach laser cooling in +-- field T 1.28 mK I (mW/cm2) Doppler temperature . I (mW/cm2) I (mW/cm2) Fig. Optimal parameters of intensity and minimum temperature for cooling in +-- field laser cooling in linlin field I (mW/cm2) lin lin Doppler temperature I (mW/cm2) I (mW/cm2) Fig. Optimal parameters of intensity and minimum temperature for cooling in linlin field Sub-Doppler cooling of 24Mg in MOT No trapping force in linlin filed! 1.28 mK [deg.] [deg.] Fig. Temperature of laser cooling of 24 Mg in lin--lin field ( = -/4) [deg.] [deg.] -- * field I (mW/cm2) Fig. Temperature of laser cooling of 24Mg in --* field ( = -/4, = -) and fraction of atoms with |p|<3hk. I [mW/cm2] 100 200 300 0 [deg.] -0.28 -0.86 -1.15 T [h] 0.097 0.118 0.157 -2 -5.16 -5.16 53.7 48.2 43.4 0 [deg.] N|p|<3hk [%] Fig. Optimal intensity and minimum temperature in lin--lin field ( = -/4) Fig. Optimal ellipticity and minimum temperature for different intensity in -- field ( = -/4). Magneto-optical potential for 24Mg in in --* MOT assuming linear growth of magnetic field in trapping zone U (H ) RW H ( RW ) H/ = 1 ( H = 12.7 Gs ) H ( RW ) F ( H ) (v 0, H ) d H 0 Fig. Force in h units and magneto-optical potential in hRW/ units (RW is beam radius) as function of Zeeman splitting on MOT edge in +-- MOT. (I = 100mW/cm2, = -) Rw is radius of light beams forming the MOT --* Fig. Force in h units and magneto-optical potential in hRW/ units (RW is beam radius) as function of Zeeman splitting on MOT edge in --* MOT. (I = 100mW/cm2, = -) Estimation: RW = 0.5 cm, for magnetic field gradient z H = 12.7 Gs/cm ( H/ = 0.5 Gs at edge), the MOT depth U(H) = 0.094 hRW/ 1.56K that much exceed sub-Doppler cooling temperature T125 K. Magneto-optical force for trapping in --* MOT Trap should be stable for moving atom as well. Magneto-optical force as function of atoms velocity and magnetic field. (lin--lin field configuration with = -/4, = -, I = 100 mW/cm2) here force push the atoms out of the trap H/ = 1 ( H = 12.7 Gs ) Magneto-optical force trapping zone for --* MOT ( = -/4, = -, I = 100 mW/cm2) Number of trapped atom Ncvc4 [C. Monroe, et.al. PRL. 65, 1571 (1990)] vc> 3.5k/ results Nc107 - 108 at. Critical magnetic field for stable --* MOT Results 1. Quantum recoil effects make sub-Doppler cooling of 24Mg on 3P2-3D3 not efficient in +-- field configuration. 2. For deep sub-Doppler cooling linlin configuration can be used. (Can't be used for MOT!) 3. For deep sup-Doppler cooling in MOT --* light field configuration can be used. - enough deep magneto-optical potential - for stable MOT magnetic field should not exceed critical values in trapping zone. - positive small ellipticities ( < 5 deg.) of light waves forming the MOT do not much increases the cooling temperatures but increase much the critical values for magnetic field.