Level crossings in a cavity QED model J. Larson Physics Department Royal Institute of Technology (KTH) Albanova, Roslagstullsbacken 21 SE-10691 Stockholm, Sweden (Dated: February 1, 2008) In this paper I study the dynamics of a two-level atom interacting with a standing wave field. Whentheatomissubjectedtoaweaklinearforce,theproblemcanbeturnedintoatimedependent one,andtheevolutionisunderstoodfromthebandstructureofthespectrum. Thepresenceoflevel crossings in the spectrum gives rise to Bloch oscillations of the atomic motion. Here I investigate the effects of the atom-field detuning parameter. A variety of different level crossings are obtained by changing the magnitude of the detuning, and the behaviour of the atomic motion is strongly 6 affectedduetothis. Ialsoconsiderthesituationinwhichthedetuningisoscillatingintimeandits 0 impact on theatomic motion. Wavepacket simulations of thefull problem are treated numerically 0 andtheresultsarecomparedwithanalyticalsolutionsgivenbythestandardLandau-Zenerandthe 2 three-level Landau-Zenermodels. n a PACSnumbers: 45.50.Ct,42.50.Pq,42.50.Vk J 9 I. INTRODUCTION quet or Bloch theory point of view. The dynamics 1 can also be understood using a Wannier-Stark lad- v der spectrum, consisting of equally spaced discrete 3 Thegeneraltheoryofsystemsevolvingwithape- complex eigenvalues [11]. 5 riodicHamiltonianshaslongbeenknownandunder- With no atom-field coupling, the atom moves 0 stood. However,it is only recently that some of the freely and the spectrum, consisting of an infinite 1 theoretical predictions have been tested experimen- set of bare energy dispersion curves, is continuous 0 6 tally. It has turned out that the physical systems E > 0. When the periodic potential is turned on, 0 one first had in mind, namely these of solid state the degeneracy points where the curves cross are / materials,haveseveraldrawbacksmakingithardto split and gaps are formed. The dispersion curves h realize such experiments. Instead of using the pe- become periodic in k, with a period equal the Bril- p - riodic structure in crystal lattices, one may use a louinzone. Theavoidedcrossingsarecentralforthe nt standing wave light field as an ’effective’ potential, evolution when k grows linearly in t. By lineare- a acting upon the internal electron states of an atom. lizing the dispersion curves around a crossing, the u Herethewavelengthofthefieldsetsthe period,and system serves as a testing ground for the Landau- q duetolackoflatticedefectsinthestandingwaveand Zener model [12, 13]. This model treats a linear : v long atomic decay times, the decoherence times ex- time-dependent two-level crossing system analyti- i ceedthoseinsolidstatesystemsbyseveralordersof cally, and the asymptotic solutions for transitions X magnitude. One purely quantum mechanical effect betweenthetwostatesaregiven. TheLandau-Zener r of the atomic motion is seen when the atom is sub- tunneling hasbeen testedexperimentallyin connec- a jected to a constant force F. The optical potential tion with Bloch oscillations [4, 7, 14] and also been then has a washboard shape. As long as the force the subject of theoretical investigations in the same is weak compared to the amplitude of the stand- system [10]. ing wave, the quantum numbers that describe the In all the works mentioned above, the particle system are still good numbers, apart from the fact lacks internal structure. For example, in the situa- that the quasi momentum will follow the classical tionofatomsinteractingwithastandingwavefield, evolution k = k Ft. Thus the system remains it is assumed that the detuning between the atomic 0 − in its initial energy band while it sweeps the quasi transitionfrequencyinvolvedandthefieldfrequency momentum. Since the energy bands are periodic in is large, implying that one of the two atomic levels k, the motion will also be periodic contrary to the can be omitted after an adiabatic elimination. This constantly accelerating one expected from classical results in a two photon process and no single pho- mechanics. This phenomena is called Bloch oscil- ton exchanges are present. The main reason for us- lations after the investigations [1, 2]. These oscil- ing a large detuning is that only the ground state lations have been verified experimentally in various andnotthe upperatomic levelispopulatedandthe systems; atoms in optical potentials [3, 4, 5], BEC’s coherencetimesincreaseconsiderably. Intheexper- in optical lattices [6, 7] and solid state superlatti- iments,astrongdrivenlightfieldisusedmakingthe cies [8]. Some theoretical studies are presented in decay of the field negligible. refs. [9, 10], which analyze the problem from a Flo- In this paper I investigate the situation when the 2 detuning may be small, which is fairly unanalyzed, with a force. A discussion about how to minimize onereferenceis[15]. Itisunderstoodthatthepaper losses, of both the atom and the field subsystem is ismostlyoftheoreticalinterest,neglectingdecaysof considered in section VI, where I also show the ef- both the atom and the field. I do, however, men- fect of field losses on the atomic inversion. Finally I tion how to overcome the problem of losses. The conclude the paper with conclusions in VII. field may be driven by an external classical source, and one may use a Λ-type of atom where one of the lowerstatesis coupledto the upper state by the II. THE MODEL standingwaveandtheotherlowerstateiscoupledto theupperonebyanexternalclassicaltravelingwave Theevolutionofthe combinedatomplus a cavity field. If the upper state can be adiabatically elim- modeisdescribedbyaJaynes-Cummingstypeofin- inated, the effective two-level system has the same teraction[17,18],wherethecouplingisx-dependent form as the one used in this paper, and thus the followingfromthevariationoftheelectricfieldE¯(x) losses due to atomic excitations can be minimized. alongthe cavitymode, g(x)=d¯ E¯(x)/¯h. In partic- I argue how an effective model for describing losses ular, the mode is assumed to ha·ve a standing wave of the field may be derived in certain case, and nu- shape, g(x) =2g cos(qx). For cold atoms, their ki- 0 merically show the effect of the losses. I keep the netic energy term must be taken into account fully field quantized, assuming the field to be in an n quantum mechanically [19, 20], and we arrive at photon Fock state; this does not make the analy- sis more complicated than for a classical field. In ¯h2 d2 ∆˜ contrast to the paper [15] I consider not only the H =−2m˜ dx˜2 + 2σz +g˜02cos(qx˜) a†σ−+aσ+ , zero detuning situation. By controlling the detun- (1) (cid:0) (cid:1) ing, a new type of level crossing is achieved where where m is the atomic mass, ∆˜ is the atom-field threebareenergycurvescross. Thiskindofcrossing mode detuning, a† (a) are raising (lowering) boson have similarities with the three-level Landau-Zener operators for the mode state, q = 2π/λ is the pho- model [16], and comparisons between wave packet ton wave number and σ , σ± are the Pauli z, rais- z simulationsofthefullatom-fieldsystemandthean- ing/loweringoperatorsactingonthetwo-levelatom. alytically solvable three-level Landau-Zener model The dynamics governed by (1) has been studied in is carried out. Thus, this atom-field model provides numerous papers, see i.e [21, 22, 23, 24, 25, 26, 27]. one physically interesting system where this three- Inthefollowingwewillusescaledunits. Thepho- level Landau-Zener generalization is applicable. ton recoil energy E = h¯2q2/2m˜ defines a charac- R I also investigate the situation when the detun- teristicenergyscaleandq−1 setsalengthscale. The ing becomes time-dependent and particularly when new scaled parameters are it oscillates in time, which is the situation when the atom is strongly Stark shifted by an alternat- x=qx˜ p= p˜ t=t˜ER q h¯ ing external electric field. In these considerations, (2) the force is not included, and hence the dependence E = E˜ V = 2g˜0 δ = ∆˜ , oftimearisefromtheoscillatingdetuningonly. The ER 0 ER ER atom again traverses a level crossing and is, in the where the tilde indicates unscaled quantities. adiabatic limit, transferred between different bare ∼ The scaled Hamiltonian has a spatial period d = states, and, consequently, its motion may have an 2π, and according to Bloch or Floquet theory, it is oscillating behaviour. known that the system eigenstates are determined The paper is organized as follows. In section II by two quantum numbers; a discrete number ν and theextendedJayness-CumingsHamiltoniandescrib- a continuous number k [23]. The first ν is the band ing the systemwithout anexternalconstantforceis index ranging from 1,2,... and k is the quasi mo- introduced. I discuss general features like the two mentumtakingvalueswithinthefirstBrillouinzone natural basis states, the band structure and effec- 1 In the previous two sections the atom, while in- 0) we have the effective Hamiltonian teracting with the standing wave field, was exposed to a constant force. The effect of this force could, (k 1)2+ δ(t) V0 insomelinearoradiabaticregime,be effectivelyde- He =" − V0 2 k2 2δ(t) #, (23) scribed by quasi momenta growing linearly in time. 2 − 2 FromtheeffectivesystemHamiltonianwecouldun- and for simplicity we pick k = 0.5, giving the 0 derstandhowthedynamicsoftheatombehavesand Schrödinger equation how the transitions between states take place. In- stead of adding a force F to the system and con- ∂ ϕ δ(t) V0 ϕ sequently introducing a time dependence, we could i 2 = 2 2 2 . (24) externally vary either the coupling V0 or the detun- ∂t(cid:20)ϕ1 (cid:21) " V20 −δ(2t) #(cid:20)ϕ1 (cid:21) ingδduringtheinteractioninordertoinsertexplicit The eigenvalues of the above Hamiltonian are time-dependence. Inthis sectionwenolongerapply 1 δ2(t)+V2 and thus, when δ(t) changes sign an external force, but instead assume the detuning ±2 0 we have an avoided level crossing. In the adiabatic tobetimedependent,andthechangeintimewillbe p limit, the population is transfered between the bare assumed slow, such that we remain in the adiabatic statesacrossthecrossing. TheSchrödingerequation regime throughout the evolution. (24) with δ(t)=δ cos(ωt) has been studied in [32]. 0 The adiabatic solution, assuming an infinitely slow change, for the inversion is [33] δ2(t) σ = 1 h zi − [V2+δ2]1/2[V2+δ2(t)]1/2 0 0 0 (25) V2cos(φ(t)) 0 , −[V2+δ2]1/2[V2+δ2(t)]1/2 0 0 0 where t φ(t)= dt′ V2+δ2(t′) (26) 0 Z0 q andwithδ(t)=δ cos(ωt),wecanexpressthisangle 0 as an elliptical integral FIG.7: ThetwolowestadiabaticenergybandsEν(k,t), obtained by diagonalization of the Hamiltonian with a φ(t)=V E(ωt|m), m = δ02 , (27) time dependent detuning δ(t) = δ0cos(ωt). It is seen 0√1 m m 1 V2 that all crossings are avoided. Theparameters are V0= − − 0 0.5, δ0 =2 and ω=0.1. where E(ωtm) is the elliptic integral of the second | kind [28]. In the adiabatic limit and first assuming Oneinterestingchoiceofatimedependentdetun- cos(φ(t)) = 1, the inversion (25) will oscillate with ing would be an oscillating one, δ(t) = δ cos(ωt), period t = π/ω. The oscillations coming from 0 osc where ω sets the time-scale. One could achieve cos(φ(t)) will be seenas ripples ontop ofthe other, this kind of detuning by, for example, Stark-shift slower oscillations, and the amplitudes of the two the atom with an external alternating electric field. kinds of oscillations are determined by V and δ . 0 0 9 As already mentioned, in the limit of large de- VI. THE PROBLEM OF LOSSES tuning, bare and dressed states are identical, and since the change is assumed to be adiabatic, the So far losses have been neglected, this is clearly a states will remain dressed also for small detunings. severeapproximation,especially since the dynamics In fig. 8 we show the evolution of an initial bare isassumedtobeadiabatic. InthissectionIintendto Gaussian wave packet, when the detuning oscillates discuss ways to minimize the losses, and also argue asδ(t)=δ cos(ωt). Butsincetheinitialdetuningis 0 about the effects of losses of the field. ratherlarge,thestatehasdressedfeaturesalsowhen Throughoutthe paper I have kept the field quan- the detuning is small, which is seen in the oscilla- tized, and assumed it to be in a Fock state. The tions of the wavepacket,see [9, 10]. Fromthe lower Hamiltonian then has exactly the same form as it plot one notes that the oscillations of the inversion wouldhavewithadrivenfieldwithalargeamplitude die out (compared to the stable adiabatic solution andthesamemodeshape. Sotheanalysisisapplica- (25)), whichisdueto thenon-adiabaticevolutionof ble to such a situation as well. One advantage with thewavepacket. However,intermsofthetransition such fields is that losses are negligible. Thus, losses rate between the two states, the adiabatic solution ofthe field may be decreasedby drive the field with and the numerical one agree very well. The ampli- an external classical source. However, in order to tude of the detuning is here δ = 80, and by using have a stable periodic pattern of the field, it should typical experimental parameters (see next section) befixedusingsomemirrors/cavity. Itisunderstood the unscaled amplitude is of the order of MHz. that since the upper atomic level is populated dur- ∼ Achieving such strong Stark-shifts should be possi- ingtheinteractionitismostlikelythatlosseswillbe ble [34]. ofgreatimportance. Howeverwemayestimate typical time-scales in physical units for the system. In ref.[5]thephotonrecoilenergyisE =1.03 10−10 R eV,whichgives a characteristictime t˜=h¯t/×E 1 R ≈ ms, where we took t = 200 from fig. 2. This rather longtimemaybedecreasedbyusingastrongerforce F, which, however, increase the non-adiabatic con- tributions. As mentioned, losses of the field is mini- mizedbyusingastrongdrivenfieldwhileonewayto overcomethe losses of the excited atomic state is to useaΛ-typeofatom[35]asinfig.9,wherethefields are assumed driven. When the detuning ∆ in the figure is large, the upper level can be adiabatically eliminated [36] to give the interaction Hamiltonian (scaled units) δ V cos(x) H = 2 0 , (28) int V cos(x) δ (cid:20) 0 −2 (cid:21) where the effective coupling is V = Ω13Ω23. Thus, 0 ∆ when the elimination of level 3 has been carried | i out, the dynamics of this model is exactly the same as for the two-levelatom used in this paper. Finally one may use some arguments to get a rough idea about how the losses of the field affects the dynamicalquantities. To solvethe problemcor- FIG. 8: The wave packet evolution for the time de- rectlythemasterequationmostbeconsidered. How- pendent detuning δ(t) = δ0cos(ωt). It is seen that the ever, we may use some approximations to derive an wavepacketsmovesbackandforthasthedetuningflips effective model. By assuming that the field is ini- sign. Thisisduetotheadiabatictransitionbetweenbare tially in a coherent state with a large amplitude, it stateswithdifferentgroupvelocitiesvg. Theoscillations is likely to assume that, in spite of the interaction inthewavepacketaretypicalforGaussiandressedstates with the atom, the field remains in some coherent [9, 10]. In the lower plot, the atomic inversion is shown state or in a linear combination of coherent states from the numerical simulation (solid line) and for the with the same amplitude. The effect of a zero tem- adiabatic result (25) (dotted line). The parameters are V0 =10, δ0 =80, ∆x =300 and ω=0.1. perature environment coupled to a coherentstate is thatthe amplitude decreaseasα=α e−κt,whereκ 0 isthecouplingconstanttotheenvironment,see[37]. Astheamplitudeoftheinitialcoherentfieldislarge, 10 FIG. 9: Λ-configuration. The 2-3 transition couples to a standing wave mode, while the 1-3 transition to a traveling wave field. FIG. 10: This shows the results of having a decaying coupling V0e = V0e−κt for κ−1 = 250, 500 and 1000. andhencethe photondistributionissharplypeaked Here V0e = 0.5, F = 0.02, δ = 0 and δx2 = 50. The around its mean n¯, we introduce the effective cou- decayof theinversion is seen to increase for larger κ, as pling V (t) = Vee−κt with Ve = V √n¯. Thus, the expected. n 0 0 0 coupling is assumed to decrease in a similar way as the squarerootofthe meanphotonnumbern¯. This is, of course, a rough approximation; the evolution and the Hamiltonian can be instantaneously diag- is still coherent, contrary to the one obtained from onalized. The system, initially in an eigenstate of solving a master equation, and the spread of the the Hamiltonian in the absence of the force, will re- photondistribution is not takeninto account. How- main in the same energy band even when the force ever,itshouldgiveameasureoftypicaldecoherence is affecting the dynamics; φ (k = k Ft) . As ν 0 times. In fig. 10 I plot the atomic inversion σz the quasi momentum is ch|anged, the −systemi tra- h i as a function of time for three different couplings verses so called level crossings where the bare en- κ−1 = 250, 500, 1000, where we have assumed an ergies cross. It is at these crossings that the adi- initial Fock state 1 ofthe field so no n dependence abaticity constrains are most likely to be violated, | i on the coupling. It is seen that, the larger coupling, meaningthatotherstates/bandsmaybe populated. the faster decay of the inversion; σz 0. This If the population still remains in a single band even h i → corresponds to typical decay times 1.5-7 ms. These through the passage of a level crossings, the atomic are rather long for optical cavities ( µs), but not motion will be oscillating, contrary to standard be- ∼ formicrowavecavities( 10ms). Inordertoseethe haviour were the particle accelerates to higher and ∼ effect of the width of the photon distribution I have higher velocities due to the force. This is usually calculated σz for various n (consequently various calledBloch oscillationsand is a consequence of the h i couplings,chosenasV n/n¯e−κt)andweightthem energy gaps/crossings in the otherwice continuous 0 withthedistributionofapoisoning(coherentstate). spectrum. p This has been done for the examples of fig. 10 with TheBlochoscillationsoftheatomareinvestigated n¯ =50andvery similarresults as in thatfigure was for various values of the atom-field detuning. Bloch obtained, and are therefor not shown here. Hence, oscillationsof atoms interacting with standing wave withinitial coherentstateswith relativelylargeam- fields have long been known and also seen experi- plitudesthesamekindofevolutionasforFockstates mentally. But it has almost always been assumed is expected. that the detuning is large so that one internal elec- tronic state of the atom can be adiabatically eliminated,andtheinternalstructureisoverlooked. Here VII. CONCLUSION I have studied the situations when both atomic lev- elsareincluded, andthe differencesinthedynamics I haveanalyzedthe dynamicsofasingletwo-level duetothetwo-levelstructure. Inthesituationwhen atom inside a cavity, interacting with a standing just one level needs to be taken into account, the wave mode field. By exposing the atom to a con- atomundergoesatwo-photonprocessasittraverses stant force of moderate amplitude it is possible to a level crossing adiabatically, it is Bragg scattered. transform the time independent problem into one Withsmalldetunings,one-photonprocessesarealso wherethemomentumgrowslinearlywithtime. This possible,makingtheatomflipitselectronicstate. In transformation is only valid for weak forces, sug- the zero detuning situation, we have seen that for a gesting that we can apply the adiabatic theorem, in weak coupling and a small force the atom oscillates other words, time can be considered as a parameter both in its motion but also between the two elec-

Level crossings in a cavity QED model

Similar Level crossings in a cavity QED model