LLaser Cooling at Resonance
Yaakov Yudkin and Lev Khaykovich
Department of Physics, QUEST Center andInstitute of Nanotechnology and Advanced Materials,Bar-Ilan University, Ramat-Gan 5290002, Israel (Dated: September 26, 2018)We show experimentally that 3-D laser cooling of lithium atoms is achieved when the laser light istuned exactly to resonance with the atomic transition. For a theoretical description of this surprisingphenomenon we resolve to a full model which takes into account both the entire atomic structureand the laser light polarization. Here we build such a model for Li atoms cooled on the D -linein a σ + − σ − laser configuration. We take all 24 Zeeman sub-levels into account and obtain goodagreement with the experimental data. Moreover, by means of Monte-Carlo simulations we showthat coherent processes play an important role in showing consistency between the theory and theexperimental results. I. INTRODUCTION
The mechanism of Doppler cooling of neutral atomsby laser light was first suggested in the mid 70s [1, 2].A simplified theoretical analysis utilizes the fluctuation-dissipation theorem (FDT) and is based on finding thesteady state between the friction force and the diffusion.Applied for a two-level atom it shows that a minimaltemperature of k B T D = (cid:126) Γ / δ = − Γ /
2, where Γ is the natural linewidth of theexcited state. This temperature is known as the Dopplerlimit [3, 4].A decade later these theoretical ideas were imple-mented experimentally on alkali atoms [5]. But soon, toeveryone’s surprise, temperatures far bellow the Dopplerlimit were observed [6]. This phenomenon was soon ex-plained by the sub-Doppler cooling mechanism due tothe degenerate ground states of a multilevel atom [7],which is what real atoms are. The overwhelming successof sub-Doppler cooling triggered an enormous activity inthe field in the following decades paving the way for sev-eral major new research directions [8].Nevertheless, it is Doppler cooling that remains a work-horse of all laser cooling experiments with atoms and,more recently, with more and more complex molecules [9–11]. This is because sub-Doppler cooling has a signif-icantly lower velocity capture range than the Dopplercooling and it is less universal, being dependent on theground state level degeneracy. For example, alkaline-earth and rare-earth atoms have a non-degenerate groundstate and can be expected to satisfy the conditions forDoppler cooling without a sub-Doppler mechanism set-ting in. Indeed, general agreement with Doppler coolingtheory has been shown in experiments with bosonic iso-topes of Hg atoms [12]. However, in this experiment aswell as in other experiments with these types of atoms,observation of the Doppler limit remained elusive [13–16].Open shell lanthanide atoms, which were introduced tolaser cooling more recently, surprisingly can satisfy sim-ple two-level atom conditions [17–20]. Doppler theoryapplies well there, but again the Doppler limit was notachieved. Thus, during the fast evolution of laser cooling, it isits most basic mechanism that has been waiting a longtime for its theory to be confirmed. Only recently it hasbeen noticed that the special properties of metastable He atoms allow a nearly ideal realization of the Dopplercooling regime. Thus, confirmation of several predictionsof the Doppler theory in 3D were reported [21]. Lithiumatoms share some of these special properties with heliumatoms. Indeed, it is well known that the sub-Dopplercooling mechanism fails in lithium as it fails in metastablehelium [22]. However, in contrast to helium, it is also wellknown that experimental results in lithium show onlyqualitative agreement with the Doppler theory [23, 24].Here we report one of the most intriguing and surpris-ing deviations from the simple Doppler cooling theoryin 3-D cooling of Li atoms: we observe a steady statetemperature when the laser is tuned exactly to resonancewith the atomic transition. We then develop a realisticmulti-level theory in a 1-D laser configuration and showgood agreement with experimental results under ”nor-mal” laser cooling conditions. In addition, we constructMonte-Carlo simulations for a better understanding ofthe role of coherent processes in the original theory andfor simulation of 3-D cooling conditions. To the bestof our knowledge, a realistic and quantitative theory forlaser cooling of lithium has never been presented before.Laser cooling at resonance, apart from being an in-teresting subject by its own rights, can be of potentialinterest in applications where the combination of cool-ing and high photon scattering rate is required. This isthe case in a recently developed advanced technique foraccurate atom counting at the level of single atom res-olution [25, 26]. Evidently, cooling atoms at resonance,where scattering of photons is maximal, provides the bestconditions for atom number counting with MOT beams.In this paper we first describe the experiment (sec. II)and demonstrate the accuracy with which the laser fre-quency is determined. In sec. III a realistic multi-levelmodel is developed. In sec. IV we compare experimentto theory and discuss the region of agreement and de-viations from it, and in sec. V we describe Monte-Carlosimulations of 3-D cooling conditions. a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a r II. EXPERIMENTA. Locking of the MOT Lasers
In the experiment we collect and cool Li atoms in amagneto-optical trap (MOT) in the apparatus describedelsewhere [27]. A significant improvement in the laserlocking scheme has been introduced to allow precise de-termination of the detuning δ = ω L − ω A , where ω L is thelaser frequency and ω A is the resonance frequency of theatom. The previous locking scheme involved a feedbackloop wired to the piezoelectric element on which the grat-ing of the external cavity semiconductor laser is mounted.This method is limited to a locking bandwidth of ∼ ∼
300 kHz.This is not good enough for investigation of the expectedsharp heating of atoms in the vicinity of resonance. Topush the laser width below 100 kHz we wire the feedbackloop to the current that runs the lasing diode. This waythere is no (slow) mechanical response involved and alarger locking bandwidth is achieved. After implement-ing this in our system the locking bandwidth was boostedby more than a factor of 20 ( ∼
50 kHz) and the laserlinewidth was pushed down to ∼
60 kHz. This providesus with an accuracy of laser frequency determination of0 .
01 Γ, where Γ / π = 5 . D -line of Li atoms. B. Locating the Resonance
After having obtained an improved laser frequency sta-bility we turn to precise determination of the absolutefrequency. For this purpose, the fluorescence signal ofatoms loaded into the MOT is measured as a function ofthe detection beam detuning. After being released fromthe optical molasses and followed by a short time of flight(TOF), the atom cloud is illuminated with a short andintense pulse of the detection light provided by the MOTpump laser only whose frequency is marked as a red ar-row in Fig. 1. The pump laser intensity of the pulse perbeam is I/ . I sat , where I sat = 2 .
54 mW/cm isthe saturation intensity and the pulse duration is 100 µ s.Applying only the pump laser allows us to isolate the | F = 2 (cid:105) → | F (cid:48) = 3 (cid:105) closed transition from other adja-cent blue detuned excited states (see Fig. 1) which areopen because they induce efficient optical pumping tothe | F = 1 (cid:105) state. The resulting fluorescence light is im-aged onto a pco.pixelfly camera. The fluorescence signalas a function of the detection pulse detuning is shownin Fig. 2, where we define the position of the maximumas δ = 0. A perfectly isolated transition is expected tofollow a symmetric Lorentzian curve. However, the ex-perimental data is not symmetric which is related to theoptical pumping already mentioned earlier. Above of theresonance ( δ >
0) the laser frequency approaches the res-onance transition frequency of the next hyperfine level of | − 1〉 | + 3〉 | − 3〉| − 2〉 |0〉 | + 1〉 | + 2〉 |𝐹 ′ = 3〉 | − 1〉| − 2〉 |0〉 | + 1〉| + 2〉 |𝐹 ′ = 2〉 | − 1〉 |0〉 | + 1〉 |𝐹 ′ = 1〉 |0〉 |𝐹 ′ = 0〉 | − 2〉 |0〉 | − 1〉 | + 1〉| + 2〉 |𝐹 = 2〉 |0〉| − 1〉 | + 1〉 |𝐹 = 1〉 803.5 𝑀𝐻𝑧1.56 Γ
671 𝑛𝑚 pu m p r e pu m p FIG. 1. (Color online) The D -line of Li. The ground statehas two hyperfine energy levels and hence both a pump (redarrow) and a repump (blue arrow) laser are needed. The ex-cited state has four hyperfine levels. Note that their order isinverted, i.e. the lowest energy state is | F (cid:48) = 3 (cid:105) . Their energydifference is on the order of the natural linewidth Γ makingthem all strongly overlapping. Each hyperfine state has de-generate Zeeman sub-levels. All allowed dipole transitions aredepicted as dotted lines. - - - - ( Γ ) f l uo r e s en c e s i gna l ( a r b . ) FIG. 2. (Color online) Experimentally locating the positionof the resonance. Since the data is not symmetric it cannotbe fitted to a Lorentzian on the entire range (see text forexplanation). We use only the central points and find a verylow dependence of the position of the resonance on the rangechosen. The red solid line is the fit to 21 central points whilethe blue dashed line uses 31 points. The shift in position of thefit maximum defines the uncertainty in resonance position. the excited state | F (cid:48) = 2 (cid:105) (see fig.1). From there theatoms can decay into the lower hyperfine ground state | F = 1 (cid:105) . Since the picture is taken using the pump laseronly they are now in a dark state and no longer contributeto the fluorescence signal. Below the resonance ( δ < δ .As mentioned, the position of the maximum is definedas δ = 0 with respect to the | F = 2 (cid:105) → | F (cid:48) = 3 (cid:105) tran-sition (see Fig.1) and is used as our reference point forthe detuning. In order to find it the data is fitted to aLorentzian curve while including only the central datapoints and imposing zero overall offset. The range ofpoints used is varied and we see only small fluctuations(standard deviation of 0 .
02 Γ) in the position of the res-onance as a function of the number of points used. Inaddition, the error in the fit is typically below 0 .
03 Γ sowe estimate our statistical error in the determination ofthe resonance to be ∼ .
05 Γ.Such precise determination of the resonance positionrequires careful considerations of possible systematic er-rors. Since we calibrate the position of the resonance witha higher laser intensity compared to the optical molassesconditions (see next section), we consider the residualAC Stark shift at resonance caused by two main effects.The first one is related to the final velocity distributionof the atoms in the could and the second one is relatedto the shift of the | F = 2 (cid:105) state due to the nearby al-lowed | F = 2 (cid:105) → | F (cid:48) = 2 (cid:105) transition at a detuning of δ = − . V pAL (see Eq. (7)) for all Zeemansub-levels of the | F = 2 (cid:105) and | F (cid:48) = 3 (cid:105) states. This isdone by taking respective partial traces over the interac-tion energy multiplied by the atomic density matrix (seeSec. III A for definitions). Our most conservative esti-mates taken at T = 2 T D and above mentioned laser in-tensity, predicts the upper bound of the systematic shiftto be − . − .
05 Γ. In addition, residualmagnetic field offsets are compensated to below 10 mG.
C. Experimental Sequence and Results
First the MOT is loaded with ∼ × atoms. The de-tuning of the pump and repump are switched to matchthe conditions for minimal temperature and the atomscooled to ∼ T D inside the MOT [27]. This is the ini-tial condition for the experiment. Then the magneticfield is turned off and the repump detuning is brought toresonance ( | F = 1 (cid:105) → | F (cid:48) = 2 (cid:105) transition) to optimizerepumping of atoms to the | F = 2 (cid:105) ground state. Simul-taneously the detuning of the pump is tuned to the targetvalue δ and the intensities of the pump and the repumpare set to I p / . I sat and I r / . I sat respec-tively. The atoms are then subject to an optical molassesfor a variable time t mol after which a TOF measurementof the temperature is conducted. As is clear from this de-scription we utilize the same laser beams as for the MOT.Therefore, the laser polarization in the molasses remains ■ ■ ■ ■ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● ● ● ● ● ●● ◆ ◆ ◆ ◆ ◆ ◆ ◆▼ ▼ ▼ ▼ ▼ ▼ ▼ ( ms ) t e m pe r a t u r e ( T D ) FIG. 3. (Color online) Experimental points and fit for thetemperature as a function of the duration of the molasses,after the atoms are cooled to T ≈ T D in the MOT, fordifferent values of the pump detuning. Blue upside downtriangles: δ/ Γ = 0, green diamonds: δ/ Γ = 0 .
2, puple cir-cles: δ/ Γ = 0 .
25, red triangles: δ/ Γ = 0 .
3, black squares: δ/ Γ = 0 .
34. The steady state temperature is extracted as afitting parameter. For δ = 0 .
34 Γ the temperature does notreach a steady state but all others do. Note that all detuningsare positive. σ + − σ − as in the MOT. For each value of δ the molassestime t mol is varied. If a steady state between the forceand the diffusion coefficient exists the temperature mustconverge towards it. In this way we determine the steadystate temperature T st . A few typical results are shownin Fig. 3. The temperature T ( t mol ) is fitted to T ( t mol ) = T st − ( T st − T ) e − t mol /τ , (1)where T st and the time constant τ are fitting parameters.The initial temperature T = 2 T D is set by us. We notethat the only laser parameter being varied is thus thedetuning δ of the pump, which is measured relative tothe main transition of the D -line, namely | F = 2 (cid:105) →| F (cid:48) = 3 (cid:105) . We have also observed, that the temperature at δ = 0 is largely insensitive to the repump laser detuningover a wide range of values.In Fig. 4 the fitting parameter T st is plotted as a func-tion of the laser pump detuning (black points with er-ror bars) and reflects the central result of this paper: at δ = 0 a steady state temperature of ∼ . T D is ob-tained. This is clearly seen in Fig. 3 where the evolutionof the temperature at δ = 0 is shown in blue color (upsidedown triangles). Also for other blue detuned molasses,namely δ = 0 . δ = 0 .
25 Γ (purplecircles) and δ = 0 . t mol and δ for which steady stateis reached. This is reflected by the small error bars inFig. 3. As δ → .
34 Γ (black squares in Fig. 3) no steadystate is obtained and the temperature diverges (see also - - - - ( Γ ) t e m pe r a t u r e ( T D ) FIG. 4. (Color online) Experimental data points are shownas black points. The error in the detuning is 0 .
05 Γ (see text).The error depicted for the temperature is one standard devi-ation of the fitting (dashed lines in Fig. 3). The standardtwo-level theory is shown as a blue dashed line. The esti-mated interval obtained by numerically solving the 24 level+ polarization model is shown as an orange region. The pa-rameters for the calculation are derived from the experimentand given by Ω p = 0 .
26 Γ, Ω r = 0 . δ r = 0. Fig. 4). In all of the experiments with δ ≥ t mol . This loss can be attributed to anintensity imbalance of the molasses laser beams to whichthe system naturally becomes more sensitive at these ex-treme conditions.This deviation from a simple two-level theory ofDoppler cooling is especially striking. We show the two-level theory prediction as a blue dashed line in Fig. 4 andits complete failure to describe any of the significant fea-tures of the experimental results is educational. We notealso that the Doppler limit is unreachable experimen-tally. The minimal temperature is larger by a factor of ∼ T D . It is very well known that the min-imal achievable temperature increases for a large numberof atoms and large densities due to photon re-scattering.But here we work with a very small number of atoms andthis effect can safely be neglected. This factor of ∼ Li under similarexperimental conditions [23].Next, the dependence of T st at resonance ( δ = 0) onthe intensities of pump I p and repump I r is investigated.For this, both detunings were set to resonance and thetemperature was measured as a function of pump inten-sity I p (while I r / . I sat as in the main experiment)and repump intensity I r (while I p / . I sat as in themain experiment). The results are shown in Fig. 5. Herewe set the molasses time to t mol = 1 . ms and find thetemperature through TOF measurements. We find thatthe temperature is largely insensitive to the repump in-tensity. These results indicate that the main role of therepumping laser is to prevent optical pumping to | F = 1 (cid:105) . ● ● ● ● ● ● ● ● ● ■ ■ ■■■■■ ■ ■ ● pump ■ repump / ( I sat ) t e m pe r a t u r e ( T D ) FIG. 5. (Color online) Measurement of the steady statetemperature at δ = 0 as a function of the pump or repumpintensity. As can be seen there is an optimum in the pumpintensity while the experiment is insensitive to the repumpintensity. The green dashed line is the average temperaturefor the repump measurements (green squares). For the pump laser though, the behavior qualitativelyagrees with the simple Doppler cooling model: the tem-perature decreases with decreasing intensity. For too lowintensities, laser cooling fails and the temperature risesagain.
III. THEORY FOR MULTI-LEVEL ATOM
We describe here a 1-D semi-classical theory for lasercooling of a multi-level atom. As stated above, the tem-perature is given by the steady state of the cooling forceand the diffusion. The exact form of the force and the dif-fusion coefficient are found by solving the optical Blochequations (OBE) for the relevant internal degrees of free-dom at steady state [4]. Then, for Doppler cooling onbroad transitions (Γ > ν r , where ν r is the recoil fre-quency) one can show that the velocity distribution func-tion W ( v, t ) satisfies the following Fokker-Planck (FP)equation [4]: ∂W ( v, t ) ∂t = − ∂∂ ( mv ) [ F ( v ) W ( v, t )]+ ∂ ∂ ( mv ) [ D ( v ) W ( v, t )] . (2)Its general solution in steady state ( ∂ t W ( v, t ) = 0) is: W ( v ) = AD ( v ) exp (cid:20)(cid:90) F ( v ) D ( v ) d ( mv ) (cid:21) , (3)where the integration constant A is used for normaliza-tion. The limit of vanishing velocities allows a partic-ularly simple treatment. To the first order in velocity,the force is linear ( F = − αv ) and the diffusion coeffi-cient is constant ( D ( v ) = D ) and one immediately ob-tains a Gaussian distribution with standard deviation (cid:10) v (cid:11) = D/mα . The 1-D equipartition theorem (EPT)then implies the FDT: k B T = m (cid:10) v (cid:11) = Dα . (4)If the full velocity range is considered deviations froma Gaussian distribution become apparent for vanishingdetuning even for a two-level atom [28].In the following we first describe the OBE for lithiumatoms (sec. III A). Then we find the force F ( v ) (sec. III B)and the diffusion coefficient D ( v ) (sec. III C) by numer-ically solving the OBE. Then we are able to solve theFP equation resulting in the velocity distribution W ( v )(sec. III D). By fitting to a Gaussian we compute thewidth (cid:10) v (cid:11) of the distribution and by means of Eq. (4)find the temperature.A few complications come up when dealing with amulti-level atom. For bosonic Li atoms there are a totalof 24 levels (see Fig.1). This means that we will needa 24 ×
24 density matrix ρ to describe the problem. Inaddition, there are two lasers in the game (pump and re-pump). The Hamiltonian will thus have two parts, onedescribing the pump and its interaction with the atom( H p ) and one for the repump ( H r ). Also the laser polar-ization configuration has to be dealt with which leads toan extra term in the Hamiltonian. A. Optical Bloch Equations
The internal degrees of freedom of our system are de-scribed by a 24 ×
24 density matrix ρ ( t ) which satisfiesthe equation of motion: dρdt = − i (cid:126) [ H, ρ ( t )] + γ dec . (5)Here γ dec is the decay term due to spontaneous emissionand the Hamiltonian is given by H = H p + H r as indi-cated above. Each part of the Hamiltonian contains anatomic term H p,rA and an interaction term V p,rAL . In therotating frame the atomic terms are given by H p,rA = (cid:88) F (cid:48) ,m F (cid:48) (cid:126) δ p,rF (cid:48) | F (cid:48) , m F (cid:48) (cid:105)(cid:104) F (cid:48) , m F (cid:48) | (6)for the pump ( H pA ) and the repump ( H rA ). Here the sumextends over all relevant excited states ( | F (cid:48) = 1 , , (cid:105) for pump and | F (cid:48) = 0 , , (cid:105) for repump, see Fig. 1) and δ pF (cid:48) and δ rF (cid:48) are the detunings of the levels | F (cid:48) (cid:105) for thepump and repump laser respectively. The interactionterms V p,rAL are written in the long wavelength, electricdipole and rotating wave approximations. With help ofthe Rabi frequency of the pump and repump (Ω p and Ω r )we have V p,rAL = (cid:126) Ω p,r (cid:88) F,m F (cid:88) F (cid:48) ,m F (cid:48) C ( F (cid:48) ,m F (cid:48) ; F,m F ) × | F (cid:48) , m F (cid:48) (cid:105)(cid:104) F, m F | e ikz + h.c., (7) where C ( F (cid:48) ,m F (cid:48) ; F,m F ) are the Clebsch-Gordan coeffi-cients. Here the sum extends over all dipole transitions(see dotted lines in Fig.1).This is the full Hamiltonian in the lab frame. Nowwe must pass into a spatially rotating frame. This isnot to be confused with the usual (temporal) rotatingframe. This spatial rotating frame is introduced by thelaser polarization configuration. It is well known thata σ + − σ − configuration (along the z -axis) results in alocally linear but spatially rotating polarization. As theatom moves in the laser beam the direction of the linearpolarization changes as a function of its location z = vt .This is best dealt with by transforming to a referenceframe that rotates together with the polarization. Sincethis frame is not inertial the Hamiltonian picks up anextra term: V rot = kv ˆ J z , (8)where k = ω/c is the wave number of the laser and ˆ J z isthe angular momentum operator in the z direction andthe generator of rotation around the z -axis [7]. Notethat this term has an explicit velocity dependence. Thetransform of the other terms of the Hamiltonian to thespatially rotating frame is trivial and will not be furtherdiscussed here. We just mention that it gets rid of thephase exp( ikz ) in Eq. (7).Now the OBE at steady state can be readily ob-tained by plugging all Hamiltonian parts (Eqs. (6-8)) intoEq. (5) and setting ∂ t ρ = 0. B. Force
The force operator is given by the spatial derivative ofthe Hamiltonian ˆ F = −∇ H . Since we are dealing withplane waves the spatial dependence is found only in thephase exp( ikz ) of Eq. (7) and the force operator is:ˆ F = i (cid:126) k Ω p (cid:88) pump C ( F (cid:48) ,m F (cid:48) ; F,m F ) | F (cid:48) , m F (cid:48) (cid:105)(cid:104) F, m F | + i (cid:126) k Ω r (cid:88) repump C ( F (cid:48) ,m F (cid:48) ; F,m F ) | F (cid:48) , m F (cid:48) (cid:105)(cid:104) F, m F | + h.c., (9)where we have already made the transition into the spa-tially rotating frame. The semi-classical force is thenobtained by taking the trace of ˆ F with the steady statesolution of the OBE ρ st : (cid:104) F (cid:105) = T r (cid:16) ˆ F ρ st (cid:17) . (10)This operation will single out the optical coherences ofthe density matrix. Due to the minus sign in the h.c. ofEq. (9) and the overall i , just the imaginary part will beselected.The numerical results are shown in Fig. 6 for δ = − . - totalpumprepump - -
50 0 50 100 - - ( v r ) f o r c e ( ℏ k Γ ) FIG. 6. (Color online) The numerically calculated force ona lithium atom (red, solid) for δ = − . v r = (cid:126) k/m . The Rabifrequencies Ω p = 0 .
26 Γ and Ω r = 0 . acting on an atom the purple dotted and green dashedlines show the contributions of the pump and repumplaser respectively. The sub-Doppler feature typical fora multi-level atom in a σ + − σ − laser configuration canclearly be seen for vanishing velocities and will be dis-cussed shortly. First though, notice that the force of therepump laser is negligible for all velocities. This was to beexpected since it is tuned to resonance and explains whythe experiment is insensitive to its intensity (see Fig. 5).The total force is largely dominated by the force of thepump laser.Regarding the sub-Doppler features, we first focus onred detuned laser light (see upper inset of Fig. 6 andblue dotted line in Fig. 7(a) and its inset). As can beseen the force strengthens as v →
0. Hence, if the forcewere naively expanded to first order around v = 0 atemperature far bellow the Doppler limit would be ex-pected. It is well known, however, that no sub-Dopplertemperatures can be obtained for lithium atoms. Andhere the reason is visible: the region of the sub-Dopplerforce is extremely narrow. It is so narrow that the cap-ture velocity for sub-Doppler cooling is well below theDoppler limit itself and approaches v ∼ v r ( v r = (cid:126) k/m is the recoil velocity) where the semi-classical approachbecomes invalid. An atom at the Doppler limit has anaverage velocity of v D ≈ v r and, taken in 3-D, has anegligible probability to be at the capture velocity forsub-Doppler cooling (well within the range indicated bythe grey shaded region in Fig. 7). Thus, the sub-Dopplermechanism is extremely inefficient in lithium and to re-flect this known fact in the 1-D theory we shall cast awaythe shaded region from further consideration. This lineof reasoning was first proposed in Ref [21]. a ) δ / Γ = - δ / Γ = δ / Γ = - -
20 0 20 40 - - f o r c e ( ℏ k Γ ) -
10 0 10 b ) - -
20 0 20 400.00.20.40.60.81.01.2 d i ff u s i on ( ℏ k Γ ) -
10 0 10 c ) - -
20 0 20 400.00.51.01.5 atomic velocity ( v r ) v e l o c i t y d i s t r i bu t i on ( a r b . ) -
10 0 10
FIG. 7. (Color online) Results of the numerical calculationsfor different values of the detuning δ . The Rabi frequencies arechosen as in Fig. 6. The grey shaded region is ± v r which isequivalent to | v | < . v D , where v D = (cid:112) (cid:126) Γ /m is the Dopplertemperature. Plot of (a) the semi-classical force as a functionof the atomic velocity, (b) the diffusion coefficient and (c) thevelocity distribution for three different detunings: δ = − . δ = 0 (solid red) and δ = 0 . When the detuning of the laser vanishes (see red linein Fig. 7(a)) or is blue detuned (see green dashed linein Fig. 7(a)) the slope of the force changes sign in theregion v → δ <
0. The negativeslope found for higher velocities (Doppler region) is whatthe atom is subject to and therefore the force remainscooling. Furthermore, this slope is affected by the sub-Doppler region. It is steeper because it crosses zero at afinite velocity instead of zero velocity. In a sense, the factthat sub-Doppler cooling fails to work in lithium makesthe Doppler force stronger at vanishing detuning.The non-vanishing cooling force at resonance can beunderstood intuitively by looking at the level diagramof Li (Fig.1). For δ < δ = 0 the transition | F = 2 (cid:105) → | F (cid:48) = 3 (cid:105) is at resonanceand contributes exclusively to diffusion (heating). How-ever, the transitions | F = 2 (cid:105) → | F (cid:48) = 2 , (cid:105) are still reddetuned. Hence the total force manages to maintain adamping character (negative slope). Because the transi-tion | F = 2 (cid:105) → | F (cid:48) = 3 (cid:105) is far more dominant than theothers the total force ultimately becomes a heating forcefor some positive detuning which we find experimentallyto be δ ≈ Γ / C. Diffusion Coefficient
The diffusion coefficient has two contributions. Onecomes from the coupling to the vacuum of the quan-tized electromagnetic field and reflects the fluctuationsin spontaneous emission. Up to a constant it is the sumof all excited state populations D vac = 12 (cid:126) k Γ × T r (cid:88) F (cid:48) ,m F (cid:48) | F (cid:48) , m F (cid:48) (cid:105)(cid:104) F (cid:48) , m F (cid:48) | ρ st (11)The second contribution is due to the fluctuations in theforce. It is given by the integral of the two-time aver-age [3, 4] D las = (cid:60) (cid:20) (cid:90) ∞ dτ (cid:68) ˆ F ( t ) ˆ F ( t − τ ) (cid:69) − (cid:68) ˆ F ( t ) (cid:69) (cid:68) ˆ F ( t − τ ) (cid:69) (cid:21) (12)The key to doing this computation is the quantum re-gression theorem (QRT) [29, 30] which states that thetwo-time average of two density matrix elements evolves δ / Γ = - - -
20 0 20 400.00.51.01.5 atomic velocity ( v r ) v e l . d i s t. ( a r b . ) δ / Γ = - -
20 0 20 400.00.51.01.5 atomic velocity ( v r ) FIG. 8. (Color online) Fitting to a Gaussian of the velocitydistribution. The solid line is the calculated distribution for δ = − δ = 0 (right). The dashed line is the fitto Gaussian excluding | v | < v r and the dotted line for | v | < v r . This gives an estimated interval for the temperatureshown in Fig. 4. according to the OBE, i.e. the same way as a single ele-ment.Like the force, the diffusion coefficient D = D vac + D las also has sub-Doppler features (Fig. 7(b)) but, again likethe force, the relevant velocities are inaccessible for theatoms. Also, it seems that as the velocity increases thediffusion coefficient approaches a constant value. This isa by product of the transformation to a spatially rotatingframe which is applicable only in the low velocities limit( kv/ Γ (cid:28)
1, for Li this corresponds to v (cid:28) v r ). Theactual diffusion coefficient vanishes for growing velocity. D. Extracting the Temperature
In the previous two subsections we have obtained theforce F ( v ) and the diffusion coefficient D ( v ). Now weplug them into the solution of the FP equation (Eq. (3))and solve the integral numerically. After normalizationwe find the velocity distribution W ( v ) shown in Fig. 7(c).Since only the ratio F ( v ) /D ( v ) enters the exponentialfunction in the FP equation the fact that D ( v ) doesnot vanish for large velocities does not prevent conver-gence of W ( v ). Due to the vanishing of the force, i.e. F ( v → ∞ ) →
0, the integral will do so as well and the ob-tained velocity distribution behaves normally. Althoughfor | v | > v r the calculated distribution might deviatefrom the actual distribution due to the limit of validityof the spatially rotating frame.As can be seen the distributions are non-Gaussian.This is due to the sub-Doppler features in the force andthe diffusion coefficient. Following our discussion above(Sec. III B) the sub-Doppler mechanism is inefficient inthe case of lithium and the appearance of these featurescan be attributed to a simplified 1-D approach. To ex-tract meaningful information from the theory, we thusignore the region | v | < v cut and fit the remaining dis-tribution to a Gaussian. This is depicted in Fig. 8 for δ = − δ = 0. Using the width (cid:10) v (cid:11) as a fittingparameter we obtain the temperature from Eq. (4). Werepeat this for different values of v cut ranging from 5 v r (dashed line in Fig. 8) to 1 5 v r (dotted line in Fig. 8).For large negative detunings ( δ (cid:46) − . v → v → ±∞ which are insignificant here. The sub-Dopplerfeatures ( v →
0) become very narrow and small when δ is larger than the hyperfine splitting of the excited statewhere the multi-level atom starts resembling a three-level atom (two ground states and one excited state).In contrast, for δ (cid:38) v (cid:28) v r . Due toboth reasons the distribution deviates from a Gaussianand we do not expect a good correspondence betweentheory and experiment in this region. However, it is in-teresting to note that the distribution remains finite at δ = 0 (no divergence is seen) and thus the theory doespredict a finite kinetic energy at resonance. IV. COMPARISON OF DATA AND THEORY
We calculate the force and the diffusion coefficient forpump and repump laser beam parameters which matchthose used in the experiment (Ω p = 0 .
26 Γ, Ω r = 0 . δ r = 0 while δ p = δ is varied). In Fig. 4 the steadystate temperature is plotted as a function of the pumplaser detuning as two orange solid lines with a shaded re-gion between them. These two lines are derived from lim-iting values obtained for the range of v cut (see sec. III D).We observe a good quantitative agreement between theexperiment and the numerical calculations in the regionof δ < − Γ /
2. This region can be trusted especially wellbecause the sub-Doppler feature excluded from the Gaus-sian fit is extremely narrow and the fit quality is verygood (see Fig. 8). This is confirmed by the observationthat the shaded region in the theory curve is very nar-row here. In addition, there is a steady state temperatureall the way up to δ ≈ +Γ /
3. The theory agrees quali-tatively with this experimental fact however we do notexpect quantitative agreement because of the reasons dis-cussed in sec. III D. Finally, we note that the minimumattainable temperature agrees well with the experimentalvalue.
V. MONTE-CARLO SIMULATIONS
As mentioned in the discussion of the multi-level the-ory, the non-Gaussian velocity distribution for all detun-ings shown in Fig. 7(c) can be attributed to the theory be-ing one dimensional. In the experimental part (sec. II) we ● ● ● ● ● ● ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ - - - - ( Γ ) t e m pe r a t u r e ( T D ) FIG. 9. (Color online) Comparison of the results of a semi-classical multi-level atom Monte-Carlo simulation to the an-alytic solution of a two-level atom (blue, dashed). The reddots are obtained when using random laser light polarization.In the σ + − σ − configuration the orange squares are obtained.The dotted lines are to guide the eye. These results are ob-tained for small laser intensities so they reflect the minimaltemperature. emphasized that this non-Gaussianity is not supportedby the experiments performed in 3-D. However, exten-sion of the semi-classical theory to a 3-D case is notwithin reach. The only good alternative is to simulatethe system by means of Monte-Carlo (MC) simulations.However, building full scale 3-D MC wave function simu-lations which would take coherent two-photon processesresponsible for sub-Doppler features in the velocity dis-tribution into account, is beyond the scope of the presentstudy. Instead, we build a semi-classical, simplified ver-sion of MC simulations (sMC) still taking into account all24 Zeeman sublevels, the two laser frequencies and theirpolarizations and the 3-D character of the cooling processbut including only incoherent one-photon transitions.The purpose of the this sMC approach is twofold. Inthe theoretical part of this paper (sec. III) we insist onartificially excluding the region of the sub-Doppler fea-tures in order to approximate the velocity distributionto a Gaussian. It thus might be tempting to claim thatcoherent two-photon processes, which are responsible forthese features, are not important for the treatment ofthe experimental results. Therefore, the first purpose ofthe sMC simulations is to answer this precise question.Then, if this question is answered positively (i.e. the co-herent processes can be neglected), the second purposeis to simulate the 3-D character of the cooling.The results of the sMC simulations are shown in Fig. 9as orange squares and are compared to the well-known1-D two-level theory (blue, dashed line). Apart from ageneral factor of ∼
2, both curves show identical behav-ior indicating that the sMC simulations miss nearly allexperimental features except for one: the minimal attain-able temperature is ∼ T D even for vanishing intensities.To verify the origin of this value we perform the simu-lations with random laser polarization. The results areshown in Fig. 9 as red circles and rediscover the two-levelsystem with high precision. The extra heating present inthe σ + − σ − polarization configuration has been iden-tified in earlier analyses of laser cooling in Refs. [7, 31]as being caused by an increased step size of the randomwalk in momentum space. Here we recover these resultswhich may be considered as a candidate to explain the in-creased minimal temperature observed in the experiment(see Fig. 4). However, the minimal temperature obtainedin the multi-level theory (sec. III) taken at vanishing laserintensities does not support this claim.Although the sMC simulations predict a finite temper-ature at resonance due to other hyper-fine states it ex-ceeds the experimentally observed value by a very largefactor and divergence occurs at vanishingly small positivedetunings. But even more striking is the failure of thesMC simulations to accurately predict the temperatureat large and negative detunings where, naively thinking,one could consider coherent processes to play a negligiblerole. Thus the sMC simulations are instructive and em-phasize the role of coherent processes in the whole rangeof laser detunings. VI. CONCLUSION
In conclusion, we experimentally identify an unex-pected regime for laser cooling of lithium atoms on the D -line: it persists up to a vanishing detuning, i.e. whenthe laser is tuned to resonance with the main coolingtransition. We show that a simple two-level theory isinconsistent with observations not only in this exoticregime but for all of the studied range of the laser detun-ing despite the known fact that sub-Doppler cooling failsto work on the D -line of lithium atoms. Therefore, todescribe the experiment we build a realistic theory whichtakes into account all 24 Zeeman sub-levels, pump andrepump lasers and the laser polarization. This theoryagrees especially well with the experimental results forlarge and negative detunings. Although the theory be- comes less convincing close to resonance it does predictthe steady-state velocity distribution at resonance andlike-wise even for small and positive detunings. Thus,the success of cooling at resonance can be explained bythe specific hyper-fine structure of the excited state oflithium’s D -line. On the one hand the hyper-fine struc-ture is inverted, such that the closed transition is thelowest in energy. On the other, the hyper-fine split-ting is very small keeping other excited states at relativeproximity to the closed cooling transition. Although thisproperty is responsible for the failure of the sub-Dopplermechanism, it permits efficient cooling even if the laseris tuned exactly to resonance with the closed transition.By means of sMC simulations we show that the coher-ent processes are crucial for explaining the experimentalresults. This is also confirmed by considering the coolingforce while neglecting density matrix coherences in themulti-level theory. In both cases we predict a finite butvery large temperature of the atoms at resonance. As asubject of future research, it is desirable to build a fullMC wave function simulation in order to fully describethe experimental results.Cooling at resonance realizes a perfect combinationof maximal photon scattering rate with effective coolingconditions. This can be directly applied in accurate atomcounting experiments with single atom resolution whichwould clearly benefit from this favourable combination.Finally we note that the specific hyper-fine structureof the excited state of lithium atoms might signify thatthese atoms are unique to exemplify on-resonance cool-ing. But since laser cooled atomic species are far frombeing exhausted, other such examples could be found infuture research. VII. ACKMOWLEDGEMENTS
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