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Atom cooling with an atom-optical diode on a ring A. Ruschhaupt1,∗ and J. G. Muga2,† 1Institut fu¨r Mathematische Physik, TU Braunschweig, Mendelssohnstrasse 3, 38106 Braunschweig, Germany 2 bDepartamento de Qu´ımica-F´ısica, Universidad del Pais Vasco 48080 Bilbao, Spain We propose a method to cool atoms on a ring by combining an atom diode –a laser valve for one-wayatomicmotionwhichinducesrobustinternalstateexcitation–andatrap. Wedemonstrate numericallythattheatomisefficientlysloweddownateachdiodecrossing, anditisfinallytrapped when its velocity is below thetrap threshold. PACSnumbers: 03.75.Be,42.50.Lc 8 0 0 Thereiscurrentlymuchinterestincontrollingthemo- Atom diode 2 a) tion of cold atoms for further (deeper) cooling,quantum n information processing, atom laser generation, metrol- Trap a ogy, interferometry, and the investigation of fundamen- J tal physical phenomena. Cold atoms are relatively easy 6 to produce and offer with respect to other particles 1 many possibilities for coherent external manipulation Atom ] with lasers, magnetic fields, or mechanical interactions. h They may be trapped in artificial lattices or even in- p b) 3 - dividually, can be guided in effectively one-dimensional W2 ΩQ nt wires, or adopt interesting collective behavior in Bose- 2 2 a Einstein condensates;also,their mutual interactions can Ω γ3 u be changed, or suppressed. All this flexibility facilitates P W1 q the translation of some of the concepts and applications 1 1 [ of electronic circuits into the atom-optical realm to im- W T 1 plement atom chips, atom circuits, or quite generally v “atomtronics” [1]. In this context, efficient elementary c) 8 circuit elements playing the role of diodes or transis- 2 2 4 v > 0 PUMPING 4 tors need to be developed. In particular, we have pro- 1 Ω (x) 1 1.2 prioesrefdoranadtosmtuidciemdoatiolanse[r2,de3v,i4ce, a5c,t6in],gaansdasoimnei-lwarayidbeaars-, W2 (x) P W1 (x) xQΩQ(x) 80 nReaairzeenxpaenrdimceonwtoarlkveerrsififcoartiaotno,mhacvoeolbinegen[7c,on8s].ide(rTedhebsye x2 xPx1 xT WT (x) 0 one-waymodelsrelyonatom-laserinteractionsinthein- 2 2 : v < 0 v dependent atom regime, but there are also complemen- 1 1 i tary proposals making use of interatomic interaction for X “diodic”one-waytransport[9].) Inthe followingwe pro- r a pose to combine, within a ring, the diode and a trap, to achieve cooling and trapping with phase-space com- FIG.1: (a) Settingofan atom diodeina ring,(b)schematic action of the different lasers on the atom levels for the two- pression. Ring-shaped traps for cold atoms have been level atom diode plus quenching, and (c) schematic spatial proposedorimplementedformatter-waveinterferometry location of the different laser potentials and their effect on and highly precise sensors [10], for studying the stability the moving atom. They are all taken as Gaussian functions: of persistent currents [11, 12], sound waves, solitons and ΩP(x) = ΩˆPΠ(x,xP,σ), Wα(x) = Wˆα Π(x,xα,σα), where vfoorrtcicoehserinenBtoascec-Eelienrsatteiionnc[o1n5d,e1n6s]a,tepsro[1d3u],ctcioolnlisoiofnhsig[1h4l]y, α=1,2,T,Q;σ1,2 =σ;and Π(x,x0,σ˜)=exp“−(x−2σ˜x20)2”. directional output beams [17], or quantum computation [18]. (For further applications see [19].) The ring traps are implemented by magnetic waveguides [10, 17, 20], In this paper, we assume for simplicity tight lateral purelyopticaldipoleforces[21],magnetoelectrostaticpo- confinement so that motion along the ring is effectively tentials [22], overlapping of magnetic and optical dipole onedimensionalor,moreexactly,circularwithalengthl. traps [23], or misalignment of counterpropagating laser Anatomdiodefollowedbyatrapareputonthering(see beam pairs in a magneto-optical trap [14]. Fig. 1a). Theinitialstateofground-stateatomswillhave some velocity and position width, but the anti-clockwise 2 moving atoms will be reflected by the diode -which can Nowweswitchtothequantummechanicaldescription. becrossedonlyinthedirectionofthearrow-soallatoms The scheme of the diode used here can be seen in Fig. will eventually approachthe diode clockwise. After each 1b and c and it has been explained before [2, 3, 4, 6]. crossing of the diode, the atom, which is now excited, In brief, there are three, partially overlapping laser re- is forced to emit a spontaneous photon (quenched), and gions: two of them are state-selective mirrors blocking ends up atthe bottom ofthe trap;the atomhasto loose the excited (level 2) and ground (level 1) state on the kinetic energy to escape from the trap, so it is slowed left and right, respectively of a central pumping region down at every crossing until being finally trapped when on resonance with the atomic transition. If the atom its velocity is below the trap threshold. The process is is traveling from the right and the velocity is not too reminiscent of Sysiphus cooling [24], a difference being high, it is reflected by the state-selective mirror poten- that both the transfer from ground to excited state in tial W ¯h/2. If the atom is traveling from the left in the 1 the diode and the quenched decay are highly controlled, ground state then it will be pumped to the second level robust and efficient processes. adiabatically (so that the process is robust and indepen- Beforelookingatthequantum-mechanicaldescription, dent of velocity in a broad velocity interval) and then we will examine a simple classical model to estimate the pass the potential W ¯h/2. Note that this setting, see 1 time scales and the cooling efficiency for different recoil Fig. 1b, can be realized by a detuned STIRAP trans- velocities. In this classical toy model, the diode and the fer [3] with just two overlapping lasers. To avoid back- traparereducedtoapointatpositionx ,andtheinitial wards motion after the atom has crossed the diode we D particlepositionsandmomentaaredistributedaccording assume a third level which decays to the ground state to Gaussian probability distributions corresponding to with a decay rate γ and we add a quenching laser cou- 3 the initial quantum distributions |Ψ (x)|2 resp. |Φ (k)|2 pling levels 2 and 3 with a Rabi frequency Ω , see Fig. 0 0 Q (see below). In each classical trajectory a random recoil 1b. A novelty with respect to previous diode models is kickisimpartedatthediodeclockwisepassage,asinthe the addition of a ground state well overlapping partially quantumjumpcalculationdonebelow. Thetrajectoryis withthe quenchinglaserregion. Theeffect ofthis wellis “trapped” (and eliminated from the ensemble) when the twofold: itsubtractskineticenergyfromthegroundstate velocity becomes smaller than the threshold imposed by atoms trying to escape from it, and it also traps eventu- the trap depth; otherwise a fixed amount of kinetic en- ally the cooled atoms. The corresponding Hamiltonian ergycorrespondingtothewelldepth 1mv2 issubtracted using|1i=(1,0,0)T,|2i=(0,1,0)T,and|3i=(0,0,1)T, 2 T and the motion continues. The results for the trapping where T means “transpose”,may now be written as probability are shown in Fig. 2a. Random recoil affects theresultintwoways: higherrecoilvelocitiesacceleratea W (x)+W (x) Ω (x) 0 p2 ¯h 1 T P rapid initial trapping, but slightly increase the time nec- H = x +  ΩP(x) W2(x) ΩQ(x) , 2m 2 essary for cooling and trapping the complete ensemble. 0 Ω (x) 0 Q   Thenumberofdiodecrossingsbeforetheatomistrapped for an initial velocity v >0 and no recoil is given by the where p is the momentum operator, and Ω (x) is the x P smallest integer n fulfilling v−nvT <vT. The time un- Rabifrequencyfortheresonanttransition. Allpotentials til this particle has been trapped is given by the time to arechosenasGaussianfunctionsaccordingtothecaption reachthediodethefirstcrossing,t0 =−(x0−xD)/vplus ofFig. 1. The “velocitydepth” ofthe trapused isvT := thetotaltimeforthenrounds,t =l n (v−jv )−1. n j=1 T h¯ Wˆ ≈ 1.8cm/s, and it corresponds to the trap For the parameters of Fig. 2a and v =Pv0 we get n = 2 rm(cid:12) T(cid:12) and t +t ≈41ms. depth(cid:12) use(cid:12)d in the classical simulation. 0 n (cid:12) (cid:12) We examine the time evolution by means of a one-dimensional master equation, ∂ i γ 1 3 mv mv ρ = − [H,ρ] − 3{|3ih3|,ρ} +γ du (1+u2) exp i recux |1ih3|ρ|3ih1| exp −i recux , (1) − + 3 ∂t ¯h 2 Z−1 8 (cid:16) ¯h (cid:17) (cid:16) ¯h (cid:17) where v is the recoil velocity and x is the position operator. The initial condition is taken as a pure state rec ρ(0)=|Ψ ihΨ |, namely a Gaussian wave packet (see the caption of Fig. 2). 0 0 The master equation (1) is solved by using the quantum tonian H =H −ih¯γ |3ih3|. For large γ (see [26]), eff 2 3 3 jump approach [27]. A basic step is to solve a time- dependentSchrödingerequationwithaneffectiveHamil- Ω (x) Q h3|Ψ(t)i≈−i h2|Ψ(t)i. (2) γ 3 3 aaa))) 1 aa)) 0.8 0.07 0.6 0.06 PT 0.05 0.4 0.1 0.04 [[sspp//((ccvvmm))]] 0.03 0.2 0.05 0.02 0.01 0 0 00 14 0 50 100 150 200 250 300 350 400 50 12 t [ms] 100 150 6 8 10 200 4 bbb))) 1 t [ms] 250 300 350 -4 -2 0 2 v [cm/s] 0.8 400 0.6 PT,x bb)) 0.4 600 0.2 500 400 0 0 50 100 150 200 250 300 350 400 [[11pp//µµ((xxmm))]] 500 230000 t [ms] 100 0 ccc))) 1 00 200 50 150 100 100 0.8 150 50 200 0 PT,v 00..46 t [ms] 250 300 350 400-200-150-100 -50 x [µm] 0.2 0 0 50 100 150 200 250 300 350 400 FIG.3: Evolutionoftheprobabilitydensitiesversustime(a) t [ms] in velocity space, (b) in coordinate space; vrec =0, theother parameters can be found in thecaption of Fig. 2. FIG. 2: (a) Trapping probability in the classical model; (b) trapping probability PT,x in the quantum model (xmin = is chosen with the probability density 3(1 + u2), 10µmandxmax =200µm);and(c)trappingprobabilityPT,v all other amplitudes are set to zero, and8 the wave in the quantum model. Thick, green, dotted line: vrec = 0 function is normalized. Because of Eq. (2), this (in the quantum model: averaged over N =200 trajectories, can also be done in the two-level approximation, ΩˆP =4×104/s,Wˆ1=Wˆ2 =4×106/s);solid,redline: vrec = −i W (x)exp imvrecux h2|Ψ(t)i −→ h1|Ψ(t)i, then 3.5cm/s(inthequantummodel: averagedoverN =190tra- Q h¯ jectories, ΩˆP =1 105/s, Wˆ1 =Wˆ2 =1 107/s). Common thepsecond level(cid:0)is set to (cid:1)zero and the wave function is parameters: l = 4×00µm ( 200µm x <×200µm), m mass normalized. ofNeon,initialwavepacke−tΨ0(x)=≤√12πR dkΦ0(k)eikx with We start by looking at the case with negligible re- Φ0(k) = √42π1√∆k(1,0,0)T exph− (k4−∆kk02)2 −i(k −k0)(x0 − cthoeil tvrealpopciintyg, pir.eo.babiwlitityhinvrcecoor=dina0t.e spWacee,caPlculat=e vmh¯0t0=k05)c−mi/msh¯,t0∆k2v2i=, k40c=m/msh¯,vt00, =∆k1=msm;h¯o∆thv,erx0pa=ram−2et0e0rµsmin, RxxvmTmianxdvdxhvh|xρ|ρ1|1v|ix.i,Thaendresiunltsvaelroecisthyowsnpaicne,FiPg.TT,v,2xb/=c the classical model: vT = 1.8cm/s, xD = 80µm; other pa- −vT 11 rametersinthequantummodel: WˆT = 105/s,WˆQ=105/s, (Rthick green dotted line) averaging over N = 200 tra- xW2 = 90µm, xP = 40µm, xW1 =−10µm, xT = 80µm, jectories; a numerical error defined by the difference of xQ=10−0µm, σ=15µm−, σT =30µm, σQ=10/√2µm. the result between averaging over N and N/2 trajecto- riesisalsoplottedinFig. 2b/c. Theparametersusedfor the atom diode with v = 0 result in a range for per- rec Therefore, the three-level Schrödinger equation can be fect “diodic” behavior −11cm/s< v < 11cm/s (defined approximatedbyatwo-levelonewiththeeffectiveHamil- as in [3]). Fig. 3 shows the evolution of the probabil- tonian ity density in velocity space, p(v) = hv|(ρ + ρ )|vi, 11 22 H = p2x + ¯h W1(x)+WT(x) ΩP(x) , and in coordinate space, p(x) = hx|(ρ11 +ρ22)|xi. The approx 2m 2 (cid:18) ΩP(x) W2(x)−iWQ(x) (cid:19) final probability densities can be seen in more detail in Fig. 4 (thick, dashed, green line). In this figure we may where W = ΩQ(x)2. The second element of the verifytheoccurrenceofcoolingandphasespacecompres- Q γ3 approach is the resetting operation at each jump, sion, namely, a narrower distribution both in coordinate exp imvrecux h3|Ψ(t)i −→ h1|Ψ(t)i, where u ∈ [−1,1] and velocity space. The final trapping probabilities are h¯ (cid:0) (cid:1) 4 aa)) 0.06 † Electronic address: [email protected] 0.05 [1] B. T. Seaman, M. Krämer, D. Z. Anderson, and M. J. m] 0.04 Holland Phys. Rev.A 75, 023615 (2007). p (v) [s/c 00..0023 [2] A06.16R0u4s(cRh)h(a2u0p0t4)a.nd J. G. Muga, Phys. Rev. A 70, [3] A.RuschhauptandJ.G.Muga,Phys.Rev.A73,013608 0.01 (2006). 0 -10 -5 0 5 10 15 20 25 30 [4] A.Ruschhaupt,J.G.Muga, andM.G.Raizen,J.Phys. v [cm/s] B: At.Mol. Opt.Phys. 39, L133 (2006). bb)) 1000 [5] A.Ruschhaupt,J.G.Muga, andM.G.Raizen,J.Phys. 890000 B 39, 3833 (2006). m] 700 [6] A.RuschhauptandJ.G.Muga,Phys.Rev.A76,013619 µp (x) [1/ 345600000000 [7] M(20.0G7.).Raizen,A.M.Dudarev,QianNiu,andN.J.Fisch, 200 Phys. Rev.Lett. 94, 053003 (2005). 100 0 [8] A. M. Dudarev, M. Marder, Qian Niu, N. J. Fisch, and -200 -150 -100 -50 0 50 100 150 200 M. G. Raizen, Europhysics Letters 70, 761 (2005). x [µm] [9] R.A.Pepino,J.Cooper,D.Z.Anderson,M.J.Holland, arXiv:0705.3268v1 [10] S. Gupta, K. W. Murch, K. 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B. supported by MEC (FIS2006-10268-C03-01) and UPV- PlenioandP.L.Knight,Rev.Mod.Phys.70,101(1998); EHU(00039.310-15968/2004). ARacknowledgessupport J. Dalibard, , Y. Castin and K. Mølmer, Phys. Rev. by the Joint Optical Metrology Center (JOMC), Braun- Lett., 68, 580 (1992); H. Carmichael, An Open Systems schweig. Approach to Quantum Optics, Lecture Notes in Physics m18, (Springer, Berlin, 1993). ∗ Email address: [email protected]