Quantum treatment of two-stage sub-Doppler laser cooling of magnesium atoms
D.V. Brazhnikov, O.N. Prudnikov, A.V. Taichenachev, V.I. Yudin, A.E. Bonert, R.Ya. Il'enkov, A.N. Goncharov
aa r X i v : . [ phy s i c s . a t o m - ph ] A ug Quantum treatment of two-stage sub-Doppler laser cooling of magnesium atoms
D.V. Brazhnikov,
1, 2, ∗ O.N. Prudnikov, A.V. Taichenachev,
1, 2
V.I. Yudin,
1, 2, 3
A.E. Bonert, R.Ya. Il’enkov, and A.N. Goncharov Institute of Laser Physics SB RAS, Novosibirsk 630090, Russia Novosibirsk State University, Novosibirsk 630090, Russia Novosibirsk State Technical University, Novosibirsk 630073, Russia (Dated: September 17, 2018)The problem of deep laser cooling of Mg atoms is theoretically studied. We propose two-stagesub-Doppler cooling strategy using electro-dipole transition 3 P → D ( λ = 383.9 nm). The firststage implies exploiting magneto-optical trap with σ + and σ − light beams, while the second one usesa lin ⊥ lin molasses. We focus on achieving large number of ultracold atoms (T eff < µ K) in a coldatomic cloud. The calculations have been done out of many widely used approximations and basedon quantum treatment with taking full account of recoil effect. Steady-state average kinetic energiesand linear momentum distributions of cold atoms are analyzed for various light field intensities andfrequency detunings. The results of conducted quantum analysis have revealed noticeable differencesfrom results of semiclassical approach based on the Fokker-Planck equation. At certain conditionsthe second cooling stage can provide sufficiently lower kinetic energies of atomic cloud as well asincreased fraction of ultracold atoms than the first one. We hope that the obtained results can assistovercoming current experimental problems in deep cooling of Mg atoms by means of laser fields.Cold magnesium atoms, being cooled in large number down to several µ K, have certain interest, forexample, in quantum metrology.
PACS numbers: 37.10.De, 05.10.Gg, 06.30.Ft
I. INTRODUCTION
Laser cooling and trapping of neutral atoms playsimportant role for many directions of modern quantumphysics. One of the directions is quantum metrologythat experiences galloping progress nowadays. It is aimedon producing of standards of various physical quantitiesand on carrying out precise measurements with the helpof them (e.g., see [1]). At present the most precisemeasurements are possible for the physical quantitiesas frequency and time. It is due to success achievedin producing etalons (standards) for these quantities.Contemporary time standard is based on a frequencystandard, which defines its stability and accuracy to aconsiderable degree. At that, frequency etalons can beused not only as a basis for time standards, but alsofor conducting precise measurements of other physicalquantities and constants as, for instance, electricalcurrent and voltage, magnetic field, length, Rydberg andfine-structure constants.High-accuracy experiments for versatile examinationof relativistic and quantum theories have becomefeasible owing to modern frequency standards. Amongpractical applications of time and frequency standardsthe broadband communication networks, navigationaland global positioning systems should be mentionedespecially. Many laboratories in word-known scientificcenters do research in the field of frequency standards.One of the latest trends in this field is connected with theconcept of intercity or even international quantum clock ∗ [email protected] network that could combine time etalons from variouslaboratories and countries into one system [2–6].There are two main directions of primary frequencystandards development: the technology of a single ionconfined in a electro-quadrupole trap and the secondone based on many neutral atoms trapped in an opticallattice (e.g., see [7, 8]). The latter direction is muchnewer than the former and it undergoes intense progress.The idea of neutral atoms trapping in a periodic lightpotential is not new and it was actively studied in1970s (see monograph [9] and citations in it). Asfor metrological purposes this idea has experienced thesecond birth in the beginning of XXI century afternoticeable progress in technique and methods of lasercooling of atoms, development of the “magic”-wavelengthconcept [10, 11], and also experimental and theoreticalsuccess in the field of spectroscopy of forbidden atomictransitions [12–16]. At present, stability of optical-lattice-based frequency standards is on the same levelwith single-ion standards and in some cases even better.The state-of-the-art prototypes reached instability anduncertainty on the relative levels of 10 − − − [17–20].Ones of the main candidates for producing the new-generation frequency standards are alkaline earth andalkaline-earth-like atoms: Yb (for instance, see [21–23]), Ca [24], Sr [18, 19, 25], Hg [26] and Mg [27,28]. These atoms are the most appropriate becauseof narrow spectroscopic lines due to forbidden opticaltransitions from the ground state S to the lowestexcited triplet state P , , (see Fig. 1). Moreover, onemore key circumstance consists in the existence of so-called “magic” wavelength for these transitions. Underthe magic-wavelength optical field the linear (in theintensity) light shift is canceled. Also, one of the lasttendency in this field is connected with spectroscopy oftransition S → P in even isotopes (with zero nucleusspin), which is highly forbidden. Frequency of thistransition is immune to many frequency-shift effects,therefore this transition can be exploited as a good“clock” transition. In spite of the transition is highlyforbidden, it has already been observed by means ofmagnetic-field-induced spectroscopy [29] in Yb [21], Sr [25, 30–33] and Mg [34].To date, atoms of the first four elements (Yb, Ca,Sr and Hg) can be effectively cooled by the lasermethods down to ultralow temperatures, approachingto the recoil energy limit, what is required for effectiveloading an optical lattice ( ∼ µ K). Besides, sub-recoil temperatures can be obtained by using evaporativecooling technique, getting Bose-Einstein condensation[35–37]. Unfortunately, researchers have not been ableto reach the same great success with Mg atoms. Inparticular, neither two-photon laser cooling [38] nor laserquenching [39] methods have appeared to be ineffectivein the case with magnesium. The minimum temperatureof a magnesium cloud that has been obtained by lasercooling method is about 500 µ K, what is rather farfrom desirable range of values, in particular, from therecoil temperature (3–10 µ K, depending on an atomictransition).At the same time, magnesium atom has someadvantages with respect to the other candidates for thefrequency standard. Thus, black-body radiation (BBR)shift is one of the main limiting factors for accuracyand stability of quantum frequency standard (e.g., see[2, 8, 19]). BBR shift of the clock transition 3 S → P for magnesium is much smaller than for Yb, Ca, Srand just a little bit higher than for a mercury atom(see Tab.1). However, from an experimental viewpointMg has some advantages against Hg. In particular,mercury atom requires noticeably smaller wavelengthsfor laser spectroscopy, cooling and trapping than Mg.For instance, effective trapping of cold atoms in anondissipative optical lattice and producing Lamb-Dickeregime need for an optical potential depth at the levelof 50–300 in the recoil energy units (e.g., see [8, 19]).It means that a highly intensive laser field at the magicwavelength λ m should be applied. But, as it can be seenfrom Table 1, λ m (Mg) ≈
468 nm and λ m (Hg) ≈
363 nm.Therefore, from an experimental viewpoint, producingdeep optical potential for mercury atom is more difficultproblem than for magnesium one due to the muchsmaller wavelength. Besides of relatively small BBRshift, magnesium atom has one more advantage withrespect to Ca and Sr. It consists in absence of opticalpumping of atoms on the non-resonant level 3 D duringthe realization of laser precooling with the help of strongdipole transition 3 S → P (see Fig. 1) to reach thetemperature down to a few millikelvins [27, 28, 44, 45].Recent experiments [34, 46] showed some progress FIG. 1. Partial energy diagram of Mg atom. Solidlines denote the cooling transitions with correspondingtemperature limits, while dashed lines denote possible “clock”transitions, which can be used for laser stabilizing.TABLE I. Data for several atomic elements relevant for new-generation frequency standards: λ cl is a wavelength of theclock transition 3 S → P and λ m is its magic wavelength,BBR frequency shifts are indicated with respect to absolutefrequencies of clock transition ( λ cl is taken from NIST AtomicSpectra Database a ).Atom λ cl λ m BBR shiftSr 698.5 813.5 [12] − . × − [41]Yb 578.4 759.4 [21, 23] − . × − [41]Ca 659.7 735.5 [24] − . × − [41] Mg 457.7 ≈ [40] − . × − [41]Hg 265.6 362.6 [26] − . × − [42, 43] a http://physics.nist.gov/asd in cooling of Mg atoms. The atoms were cooleddown to the record temperature equaled to 1.3 µ K andconfined in an optical lattice. However, the final numberof atoms was of the order of 10 , what was about0.01 % from the initial number of atoms in a magneto-optical trap (MOT), involved the cyclic triplet dipoletransition 3 P → D . That great loss in atomic numberwas due to the fact that velocity selection technique,having similarities with evaporative cooling, was usedfor reaching such ultralow temperature, but the lasercooling method in MOT, unfortunately, showed the cloudtemperature equaled to only about 1 mK. That resultnoticeably yielded to successful results with the otheratomic elements (Ca, Sr, Yb, Hg). At the same time, webelieve that laser cooling strategy for magnesium atomscan be proper tuned for getting much better result oflaser-cooling temperature as well as number of atomstrapped.Therefore, we can state that problem of deep coolingof magnesium atoms by means of laser radiation isstill unsolved. Moreover, increasing of ultracold atomicnumber has principal importance for many applicationsof cold atoms. For instance, authors of the paper [47]managed to obtain Bose-Einstein condensation composedof ∼ strontium atoms. Besides, frequency-standardstability depends on an atomic number in an opticallattice and it increases with the number increases [1, 48].All these things considered, we can conclude that it isimportant to solve the problem of deep laser coolingof magnesium atoms (down to T ∼ µ K) as well asto provide much larger number of ultracold atoms in alattice.
II. LASER COOLING IN MOT:SEMICLASSICAL APPROXIMATION
Laser cooling of neutral atoms in a magneto-opticaltrap is one of the main cooling methods. At first time thelaser field composed of six beams with orthogonal circularpolarizations ( σ + σ − configuration) was suggested in[49] as effective way for simultaneous cooling andtrapping of atoms. Narrow spectral lines allow lasercooling of various atoms down to a few tens andunits of microkelvin, and even lower. In particular,narrow intercombination transition 4 S → P in Ca( γ ≈ π ×
400 Hz) provides temperatures around 4–6 µ K[50, 51] just by the help of Doppler cooling process. Therewere good results with intercombination transitions alsofor the other elements: Sr [52], Yb [21, 53] and Hg [42], foreven as well as for odd isotopes. In certain aspects evenisotopes, having a zero nuclear spin, are more attractivefor frequency standards of new generation. However, as ithas been already noted in the Introduction, still there arenot satisfactory results of cooling Mg atoms by meansof laser radiation in contrast to the other elements.The first our attempt to solve the problem withdeep laser cooling of magnesium was undertook in therecent work [54], where the detailed theoretical studyof magnesium kinetics in 1D MOT using the dipoletransition 3 P → D was conducted. The theory wasbased on the semiclassical approach [9, 55], based on thewell-known assumptions: ω rec ≪ min { γ, γS } (1)and ∆ p ≫ ~ k . (2)Here ω rec = ~ k / M is the recoil frequency, M is massof an atom, k =2 π/λ is wave number. The saturationparameter S is defined as S = R ( γ/ + δ , (3)where γ is the spontaneous relaxation rate of excitedstate, δ = ω − ω is the detuning of laser radiationfrequency ω from the transition frequency ω , and R isthe Rabi frequency.Condition (1) implies that recoil frequency mustbe rather small in comparison with a typical rate ofestablishment of steady state among atomic internaldegrees of freedom. In particular, in the case of an atomwithout any degeneracy of the ground state this rate isdefined by γ . If there is a degenerate ground state andoptical pumping can occurs, this rate is defined by γ orthe pumping rate γS , depending on what is smaller. Thesecond semiclassical requirement (2) implies that typicalwidth of stationary linear momentum distribution f ( p )must be much larger than the recoil momentum fromemmision/absorption of a photon.Doppler limit for temperature of laser cooling T D ,which can be achieved at the frequency detuning δ = − γ/
2, can be figured out from equation for minimumkinetic energy in one-dimensional case: E minkin = 12 k B T D = 740 ~ γ. (4)Strictly speaking, this equation is valid for transition J g =0 → J e =1. It was found in [56] under σ + σ − configuration (also see [57]). If we use this formulafor getting estimate of T D in the case of transition3 P → D ( γ ≈ π . T D ≈ µ K. For effective trapping of atoms with suchrelatively high temperature the large intensity of cwoptical lattice field at the level of tens of MW/cm is required, what is hardly feasible in an experiment.Therefore, much lower temperature of atomic cloudis needed. At the same time, since the transitionconsidered has degenerate energy levels, one cananticipate activation of so-called sub-Doppler mechanismduring the laser cooling in MOT under the polarization-gradient field. In principal, this process would overcomethe Doppler limit (4) and show much lower temperaturethan in the case of J g =0 → J e =1.Semiclassical approach is based on kinetic equation ofFokker-Planck type on the Wigner distribution functionin phase space f ( z, p ). That equation can be acquiredby reducing of exact quantum kinetic equation onthe density matrix in the series on small parameter ~ k/ ∆ p ≪ pM ∂∂z f ( z, p ) = h − ∂∂pF ( z, p ) + ∂ ∂p D ( z, p ) i f ( z, p ) . (5) -40 -30 -20 -10 0 10 20 30 400,000,020,040,060,080,100,12 kp / P r obab ili t y den s i t y FIG. 2. Momentum distributions of magnesium atomsat δ = − γ ≈− π ×
130 MHz, I =20 mW/cm (solid) and I =470 mW/cm (dashed). Here F ( z, p ) is laser-field force on an atom, D ( z, p ) isdiffusion of an atom in the light field. This equationmust be completed with normalizing condition that inone-dimensional periodic laser field has the form:1 λ + λ/ Z − λ/ dz + ∞ Z −∞ f ( p, z ) dp = 1 . One-dimensional σ + σ − laser-field configuration allowssignificant simplifying of (5), at that the dependence f on z vanishes (see section III B).Our semiclassical calculations [54] have been donebeyond many widely used approximations (for instance,slow atoms and weak field approximations). As ithas been shown the minimum kinetic energy achievablein MOT is close to 30 × E rec , where E rec = ~ ω rec isthe recoil energy. The effective temperature, whichcan be associated to this value, T eff ≈ µ K. It isapproximately three times lower than the estimate ofDoppler limit T D ≈ µ K, but, unfortunately, it is stillvery far from desirable range of values and, in particular,the recoil temperature T rec =5 µ K.Let us consider a question on validity of semiclassicalapproach to the magnesium problem. Indeed, as it willbe shown further on the basis of semiclassical treatment,the optimal parameters of cooling field for transition3 P → D can be chosen as δ = − π ×
130 MHzand I =500 mW/cm . The corresponding saturationparameter S ≈ × − . In spite of such low saturation,the first semiclassical requirement (1) is still satisfied,because ω rec =2 × − γ . At the same time, typicalmomentum distribution width ∆ p may not satisfied thesecond semiclassical condition (2). Indeed, in generalcase the distribution can have complex shape. Inparticular, Fig. 2 shows two examples of momentumdistributions for different values of light field intensity.The distribution acquires two-peaked profile at low intensity I =20 mW/cm : there is the high-contrast spikeon top of the wide background. This backgroundconditionally describes “hot” fraction of atoms in a cloudwith effective temperature T eff ∼ −
10 mK, while thespike corresponds to ultracold fraction with T eff ∼ µ K.Similar distributions were observed earlier (e.g., see [62]with semiclassical low-saturation-limited calculations ofSisyphus cooling of Cs atoms with the help of transition F g =4 → F e =5 or the quantum-treatment calculations foratomic W-type scheme in [63]). In our case the narrow-spike width is about ~ k and the requirement ∆ p ≫ ~ k is not satisfied at all. With increasing the intensity( I =470 mW/cm ) the two-peaked shape disappears.However, the distribution as a whole is still sufficientlynarrow and the second requirement (2) is satisfied with agood margin. Also it should be noted that the condition(2), as a matter of fact, depends on the value of totalangular momentum F g . In other words, at the samesaturation parameter S the requirement (2) can be validfor small F g and getting not valid with its increasing.Basing on the aforesaid, we can conclude that moreprecise theoretical treatment is needed in the casewith magnesium for adequate description of kinetics ofultracold atoms. This treatment can be based on thedensity matrix formalism with full account for the recoileffect (e.g., see [9, 58, 64]). Moreover, as it will beseen in the next section, the quantum-treatment resultsnoticeably differs from the semiclassical ones, based onthe equation (5). That difference, in particular, gave usan idea for exploiting the second stage of sub-Dopplerlaser cooling for getting the desirable results. III. FULL ACCOUNT FOR THE RECOILEFFECT
Let us consider the problem of laser cooling ofmagnesium atoms out of semiclassical approximationlimit as well as some other widely used approximations(weak-saturation limit, secular approximation, etc.).
A. Problem statement
We assume the laser field to be one-dimensional, composed of two plane monochromaticcounterpropagating light waves with equal frequenciesand amplitudes (the quantization axis z is collinear tothe wave vectors): E ( z, t )= E e e − i ( ωt − kz ) + E e e − i ( ωt + kz ) + c.c. == E e ( z ) e − iωt + c.c., (6)where e , are the unit complex vectors of waves’polarizations, while e ( z ) is the following complex vector e ( z ) = e e ikz + e e − ikz . (7)Nonzero components of the vectors e , in the sphericalbasis are e − = − sin( ε − π/ , e +11 = − cos( ε − π/ ,e − = − sin( ε − π/ e iϕ ,e +12 = − cos( ε − π/ e − iϕ . (8)Here ε , are the ellipticity parameters (in particular, ε = ± π/ ε =0 is for linear polarization), ϕ is the anglebetween main axes of polarization ellipses. For instance,the case with ε , =0 and ϕ = π/ lin ⊥ lin field configuration.Here quantum treatment of atomic kinetics under thelaser field (6) is based on the equation on single-atomdensity matrix in coordinate two-point representationthat has the form (e.g., see [9, 55, 64]): ∂ b ρ ( z , z , t ) ∂t = − i ~ h b H ( z , t ) b ρ − b ρ b H ( z , t ) i + b Γ (cid:8)b ρ (cid:9) , (9)with the Hamiltonian b H ( z i , t ) = ( b p i / M ) + b H + b V ( z i , t ) . (10)The first term in the Hamiltonian is the operatorof kinetic energy of an atom ( b p i is the linearmomentum operator), b H describes intratomic degrees offreedom, operator b V corresponds to the atom-field dipoleinteraction, and the linear operator functional b Γ { . . . } isrespective for relaxation processes in an atom. Let usintroduce the projection operator onto the excited atomstate: b P e = X m e | F e , m e ih F e , m e | , (11)and the Wigner vector operator b T , whose sphericalcomponents are: b T σ = X m e ,m g C F e ,m e F g ,m g ;1 σ | F e , m e ih F g , m g | , (12)with σ =0 , ± C F e ,m e F g ,m g ;1 σ the Clebsch-Gordancoefficients (e.g., see [65]). Then the terms b H and b V from (10) in the resonant approximation can be writtenas b H = − ~ δ b P e (13)and b V ( z i ) = − ~ R b T · e ( z i ) + h.c. = − ~ R b V eg ( z i ) + h.c. (14) Here R = E d/ ~ is the Rabi frequency (with d thereduced matrix element of dipole operator of an atom), b V eg ( z i )= b T · e ( z i ) is the dimensionless operator of atom-field interaction, depending on the coordinate in generalcase, h.c. means Hermitian-conjugate term.Introduce the new coordinates: z = k ( z + z ) / , q = k ( z − z ) , (15)in which the spontaneous relaxation operator from (9)acquires the form: b Γ = − γ (cid:16) b P e b ρ + b ρ b P e (cid:17) + γ X σ =0 , ± ζ σ ( q ) b T † σ ρ b T σ , (16)with ζ ± = 32 (cid:16) sin( q ) q − sin( q ) q + cos( q ) q (cid:17) ,ζ = 3 (cid:16) sin( q ) q − cos( q ) q (cid:17) . (17)Note that in the absence of recoil effect, i.e. in the limit q →
0, we have ζ σ = 1.The density matrix can be divided into four matrixblocks: b ρ = b ρ gg b ρ ge b ρ eg b ρ ee ! . (18)Matrix blocks b ρ gg and b ρ ee describes populations of theground and the excited states as well as low-frequency(Zeeman) coherences. Blocks b ρ ge and b ρ eg are responsiblefor optical coherences. For the new coordinates (15) andusing all introduced notations the new equations on thedensity matrix blocks can be easily acquired from (9).So, in the steady state we have − iω r ∂ ∂q∂z b ρ gg ( z, q ) = γ X σ =0 , ± ζ σ ( q ) b T † σ ρ b T σ ++ iR h b V eg † ( z + q b ρ eg − b ρ ge b V eg ( z − q i , (19) (cid:16) γ − iω r ∂ ∂q∂z (cid:17)b ρ ee ( z, q ) == iR h b V eg ( z + q b ρ ge − b ρ eg b V eg † ( z − q i , (20) (cid:16) γ iδ − iω r ∂ ∂q∂z (cid:17)b ρ ge ( z, q ) == iR h b V eg † ( z + q b ρ ee − b ρ gg b V eg † ( z − q i , (21) (cid:16) γ − iδ − iω r ∂ ∂q∂z (cid:17)b ρ eg ( z, q ) == iR h b V eg ( z + q b ρ gg − b ρ ee b V eg ( z − q i . (22)These equations compose a basis for further theoreticalanalysis. For instance, probability density of atoms inthe momentum space can be found from the formula: f ( p ) = 1(2 π ) ∞ Z −∞ dq π Z − π dz Tr (cid:8)b ρ ( z, q ) (cid:9) e − ipq . (23)Here the linear momentum of an atom evaluated inthe recoil momentum units ~ k and Tr[ . . . ] denotestrace operation. Momentum distribution f ( p ) must benormalized: + ∞ Z −∞ f ( p ) dp = 1 . (24)This means that the set of equations (19)-(22) must besupplemented with the condition:12 π π Z − π Tr (cid:8)b ρ ( z, q = 0) (cid:9) dz = 1 . (25)Average kinetic energy of an atom in the recoil energyunits can be evaluated, for instance, with the help of thefollowing formula: E k = + ∞ Z −∞ p f ( p ) dp . (26) B. The first stage: Cooling in MOT
As a rule, in magneto-optical trap atoms are localizedin weak-magnetic-field region (in the vicinity of trap’scenter). Therefore, magnetic field does not affectsignificantly on the temperature of a cloud and weomit it here from our analysis. In other words, weconsider kinetics of atoms in 1D laser field, composed oftwo counterpropagating beams with orthogonal circularpolarizations ( σ + σ − configuration). Then in equation(8) we can take ε = π/ ε = − π/ ϕ =0 and polarizationvector of total light field in spherical basis takes the form: e ( z i ) = e − e − ikz i − e +1 e ikz i . (27)This form corresponds to the laser field with linearpolarization, which rotates by an angle α = − kz i duringpropagation along z -axis. At that the dimensionlessoperator b V eg from (14) is b V eg ( z i ) = b T − e − ikz i − b T +1 e ikz i . (28) FIG. 3. Transformation the old coordinate frame K to thenew one K ′ . The considered field configuration has some uniquefeatures. First of all, the field has homogeneous intensity(it does not depend on z -coordinate). And the second,the field polarization also can be made homogeneous(e.g., see [63, 66]). Indeed, let us pass to the newcoordinate system K ′ , in which z ′ -axis coincides with z -axis in the old K -system, while the axes x ′ and y ′ rotate around z -axis by the angle α = − z = − k ( z + z ) / K ′ -system the linearly polarizedtotal-field vector does not rotate anymore (without lossof generality it can be considered to be directed along x ′ -axis). Then in the new system the interaction operator b V eg from (14) does not depend on the coordinate z i : b V eg ( z ) = ⇒ b V ( q ) = b T − e − iq/ − b T +1 e iq/ , (29) b V eg ( z ) = ⇒ b V ( q ) = b V ∗ ( q ) . (30)Since the relaxation operator b Γ from (16) also does notdepend on z , the density matrix in the new coordinatesystem is not the function of z and it depends only on q -coordinate. This circumstance significantly simplifiesnumerical evaluations of the density matrix equations.The operator of rotation b D ( n , α ) can be exploited forgetting the equations on density matrix in the new basis(e.g., see [65]). Here the unit vector n defines a rotationaxis, while α is a rotation angle. In our case it is naturalto coincide n with quantization axis z and take α = − z .Then action of the rotation operator on a wave function | F a , m a , z i i reduces to a simple multiplication by e imα ,i.e. b D ( n , α ) | F a , m a , z i i = e im a α | F a , m a , z i i , (31)with ( a = e, g ). In the system K ′ the set of equations(19)-(22) take the form:2 ω r ∂∂q h b F z , b ρ ( q ) i = b Γ { b ρ ( q ) } + iδ h b P e , b ρ i ++ iR h b V ( q ) b ρ − b ρ b V ( q ) i . (32) (b) K i ne t i c ene r g y ( E un i t s ) Intensity (W/cm ) r e c (a) U l t r a c o l d f r a c t i on ( % ) Intensity (W/cm ) FIG. 4. Comparison the results of semiclassical (dashed) and quantum (solid) treatments at δ = − γ ≈− π ×
130 MHz. (a)Average kinetic energy of an atom as the function of light field intensity, (b) Ultracold fraction of atoms in a cloud. -40 -30 -20 -10 0 10 20 30 400,000,020,040,060,080,100,12 kp / P r obab ili t y den s i t y FIG. 5. Momentum distributions of magnesium atoms:comparison of semiclassical (dashed) and quantum (solid)treatments, δ = − γ , R ≈ . γ ( I ≈
20 mW/cm ). At that the normalizing condition (25) becomes rathersimple: Tr (cid:8)b ρ ( q = 0) (cid:9) = 1 . (33)Figure 4a shows average kinetic energy of an atomas the function of light field intensity, calculatedon the basis of numerical solving the equation (32).Analogical dependence is also presented, gained byapplying semiclassical approach on the basis of Fokker-Planck equation (5). As it is seen from the figure thesemiclassical approach (dashed line) gives the minimumkinetic energy of an atom at the level of E min ≈ E rec ,what is several times smaller than the Doppler limit E D ≈ . E rec . At the same time the quantum approach(solid line) shows the result for energy just a little bit smaller than the Doppler limit ( E min ≈ E rec ).Hence the quantum treatment of the problem showsthat it is hardly possible to cool magnesium atomsin MOT down to desirable range of temperatures onthe basis of transition 3 P → D . All this agreeswith the experiments of research group from theUniversity of Hannover [34, 46]. Effective temperature,corresponding to the minimum at the plot E ( I ) forquantum-treatment result, is about 310 µ K at frequencydetuning δ = − γ ≈− π ×
130 MHz and light field intensity I ≈ .Beside the temperature of an atomic ensemble it is alsoimportant to know a profile of momentum distributionof atoms in a cloud. It may be found very useful, inparticular, for realization of evaporative cooling stagefor achieving ultralow temperatures ( ∼ µ K). Let usconsider a group of atom in the momentum space with p ≤ ~ k . Tentatively speaking we call this fraction as“ultracold” one. Figure 4b shows number of atoms inthe ultracold fraction N c as the function of light fieldintensity I . As it is seen from the figure there is amaximum in vicinity of 500 mW/cm . It should benoted that position of this optimum is not immediatelythe same as for the minimum of the dependence E kin ( I ).Figure 4b demonstrates that about 40 % of atoms can beconcentrated in the ultracold fraction. For comparisonanalogical dependence is presented, calculated on thebasis of semiclassical approach (dashed line), whichlies noticeably higher than the former one. Thedependencies E kin ( I ) and N c ( I ) lose to the semiclassicalones, because quantum treatment provides significantlydifferent result for the momentum distribution in thevicinity of p ≈
0. Indeed, Fig. 5 shows very sharp spikein the semiclassical case and a tiny peak as the result ofquantum calculations. (a) r e c Intensity (W/cm ) K i ne t i c ene r g y ( E un i t s ) (b) U l t r a c o l d f r a c t i on ( % ) Intensity (W/cm )
FIG. 6. (a) Average kinetic energy of an atom under lin ⊥ lin light field configuration calculated on the basis of quantumtreatment, (b) ultracold fraction of atoms in a cloud. Light field detunings are δ = − γ (dashed) and δ = − γ (solid). C. The second stage: Cooling in an opticalmolasses
Fortunately, solution of the problem of deep lasercooling of magnesium atoms can be found by involvingthe second stage of sub-Doppler cooling with the helpof one-dimensional optical molasses. The molasses iscomposed of two counterpropagating light waves withorthogonal linear polarizations ( lin ⊥ lin configuration).In contrast to σ + σ − configuration, in the case of lin ⊥ lin field the total-field polarization transforms fromlinear to circular (and back) along z-axis (e.g., see [63]).Consequently, there is no any rotating transformationof coordinate frame K that would make density matrixindependent of z -coordinate. Therefore, we must solvethe set of equations (19)-(22) on matrix b ρ ( z, q ). Wehave solved the equation numerically on the basis ofmatrix continued fractions method. The details of themethod can be found, for example, in [64] and we do notreproduce it here. Instead of that, we just present thenumerical results.Figure 6a demonstrates much lower minimum kineticenergy than in the case of σ + σ − field (see Fig. 4a,solid line). In particular, the minimum corresponds to E ≈ E rec at I ≈
300 mW/cm ( T eff ≈ µ K). Besides,as it is seen from Fig. 6b, ultracold fraction of atomsunder lin ⊥ lin light field can be higher than in the caseof σ + σ − (compare with Fig. 4b). Narrow structurein momentum profile in vicinity of p ≈ σ + σ − field (compare Fig. 7 andFig. 5). Therefore, the second cooling stage involvingoptical molasses can provide lower temperature as wellas larger number of atoms in ultracold fraction (upto 60%). After the second sub-Doppler cooling stageatoms may be loaded, for instance, to a dipole trap.At that the “hot” fraction of atoms (wide backgroundat Fig. 7) can be evaporated by proper choice of the FIG. 7. Quantum calculations of momentum distributionsat δ = − γ . Field strengths: R ≈ . γ , I ≈
20 mW/cm (solid)and R ≈ . γ , I ≈
600 mW/cm (dashed). light potential depth, saving only the ultracold fractionin a trap (with effective temperature ∼ µ K). It shouldbe noted that realization of the second sub-Dopplerstage should eventually provide much more number ofultracold atoms in a dipole trap (or an optical lattice)after evaporation than without this stage.
IV. CONCLUSION
In conclusion we would like to summarize main resultsof the work. We have suggested using the second sub-Doppler cooling stage to solve the problem of deeplaser cooling of magnesium atoms. The first stageimplies using of magneto-optical trap involving dipoletransition between triplet states 3 P and 3 D . Inparticular, this stage was used in the experiment ofresearches from Hannover University [46]. In spiteof that the level 3 P is degenerate and one couldanticipate activation of effective sub-Doppler mechanismof cooling in polarization-gradient field [63], however,the conducted theoretical analysis has figured out theminimum temperature at the level of 310 µ K, what isjust a little bit lower than the estimate for Doppler limitof cooling ( T D ≈ µ K). To reduce this value by severaltimes we have proposed using the second laser-coolingstage with the help of optical molasses composed oftwo counterpropagating orthogonally linearly polarizedwaves ( lin ⊥ lin field configuration). In contrast to σ + σ − field, applied in MOT, the optical molasses demonstratesmuch lower temperature (80 µK ). At the same time,the minimum achievable temperature of laser coolingin the case of Mg still is noticeably higher thanfor some other atoms, where sub-Doppler mechanismprovides much better results. Most likely, it is due torelatively large recoil energy of magnesium atom. Forinstance, for considered transition in magnesium recoilfrequency ω rec ≈ π ×
53 kHz, while for
Cs ( S / , F=4 → P / , F=5) this frequency is significantly smaller( ω rec ≈ π × µ K [67].Besides temperature (average kinetic energy) ofensemble of atoms we have also paid attention tothe linear momentum distributions in the steady state.In particular, we have investigated the problem ofincreasing concentration of atoms in ultracold fraction(a region in momentum space in the vicinity of p =0).Conducted numerical calculations have revealed theoptimum parameters of laser field for maximization ofthe ultracold fraction ( T eff ∼ µ K). This fraction can beeasily localized in a optical trap, while the other fraction(“hot” atoms) can be evaporated from the trap by properchoice of the optical depth. At that, it is the second stageof sub-Doppler cooling that can provide great increaseof ultracold atomic number in comparison with the casewith only the first stage realized (as in the experiments[34, 46]).In our theoretical analysis quantum treatment withfull account for the recoil effect has been exploited, i.e.we have not been limited by semiclassical or secularapproximations as well as weak-field limit. It hasallowed us studying kinetics of cold magnesium in a widerange of intensity and frequency detuning to determinethe optimum parameters of laser field. Also we havecompared data provided by quantum and semiclassicalapproaches. As the result of that comparison we canconclude that semiclassical approach in the case of transition 3 P → D in Mg is not valid for adequateunderstanding the kinetics of ultracold magnesium atomsfor a wide range of light-field parameters. Moreover, wecan also conclude that for getting the adequate estimateof cooling parameters and understanding the problems indeep laser cooling of atoms it is quite necessary to treatthe problems with the help of quantum approach.At the end we should note that in spite of theoreticalanalysis has been done out of many widely usedapproximations, we have assumed the problem to beone-dimensional. However, light-field configuration usedin a magneto-optical trap is always three-dimensional(three pairs of circularly polarized beams). Therefore,obviously, the results of such 1D analysis may differfrom the real experiment with 3D field. For example,one can refer to the papers [68, 69] for getting theestimate of such kind of difference. In these paperscalculations were done for 1D and 3D configurationsby the example of simple transition F g =0 → F e =1 inlimits of semiclassical and slow-atoms approximations.At the same time, an optical molasses (the secondcooling stage suggested), which is of the most interestfrom the viewpoint of deep laser cooling, can beimplemented in 3D as well as in 1D configuration. Three-dimensional optical molasses for various transition ofthe type F g = F → F e = F +1 was investigated in ref. [70]under weak-saturation approximation and with the helpof adiabatic reduction of density matrix equations toground state. Unfortunately that approximation givesgood results not for wide range of parameters δ and I that can have an interest from the laser coolingview of point (e.g., see the work [64], where results ofadiabatic approximation were compared with results offull quantum treatment). Three-dimensional quantumtreatment with full account for the recoil effect andbeyond the aforesaid approximations is quite difficulttask that requires separate study. ACKNOWLEDGMENTS
The work was partially supported by the Ministry ofEducation and Science of the Russian Federation (gov.order no. 2014/139, project no. 825), Presidium of theSiberian Branch of the Russian Academy of Sciences,and by the grants of RFBR (nos. 15-02-06087, 15-32-20330, 14-02-00806, 14-02-00712, 14-02-00939) andRussian Presidential Grants (MK-4680.2014.2 and NSh-4096.2014.2). [1] F. Riehle,
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