Jaynes-Cummings Model

Transcription

1 Jayes-Cummigs Model The JC model plays a importat role i atom-field iteractio. It is a fully quatized model ad yet aalytically solvable. It describes the iteractio betwee a two-level atom ad a quatized field. I. TWO-LEVEL ATOM The level structures of a real atom look aythig but two-level. So how ca a two-level-atom TLA) be a good approximatio? The reaso lies i two factors: 1) Resoace excitatio ad ) Selectio rules. The absorptio cross sectio of a atom absorbig a off-resoat photo is geerally of the order of 1Å. But whe the frequecy of the photo matches with the trasitio frequecy from the iitial state to some fial state, the cross sectio ca be ehaced by may orders of magitude. This is why the itesities of the lasers used i labs are much less tha that required to produce a electric field with oe atomic uit W/cm ). Uder the resoace coditio, may levels lyig far away from the resoace ca be simply igored. I additio, the dipole sectio rules dictates oly certai magetic sublevels are excited. I most cases, the field therefore oly causes trasitios betwee a small umber of discrete states, i the simplest of which oly two states are ivolved. The states for a two-level atom: g ad e. Their eergies are separated by ω 0, the atomic trasitio eergy. The two states are assumed to have opposite parity hece dipole trasitio is allowed) ad orthogoal to each other. From these oe ca costruct four idepedet operators: g g, g e, e g, e e, which form a complete basis. Ay arbitrary operator, Ô, ca the be expaded oto this basis as Ô = O gg ˆσ gg + O geˆσ ge + O eg ˆσ eg + O eeˆσ ee where ˆσ ij = i j, ad O ij = i Ô j. I particular, the dipole operator ˆd = eˆr ca be expressed as ˆd = d geˆσ ge + d eg ˆσ eg where we have used the property that states g ad e have opposite parity such that g ˆr g = e ˆr e = 0. II. HAMILTONIAN AND THE ROTATING WAVE APPROXIMATION The total Hamiltoia, eglectig the ceter-of-mass atomic motio, describig the iteractio betwee the twolevel ad a sigle mode quatized field reads: where H A = ω 0ˆσ ee, H F = ωâ â H it = ˆd Ê = d geˆσ ge + d eg ˆσ eg ) H = H A + H it + H F ω ε 0 V [ur) â + h.c.] = gˆσ ge + ˆσ eg )â + â ) where ur) is the mode fuctio of the field at the ceter-of-mass coordiate R, ad we have assumed g = d ge ur) ω/ε 0 V ) to be real by a proper choice of phase. Let us ow cocetrate o the iteractio Hamiltoia H it = gˆσ ge + ˆσ eg )â + â ) = gˆσ eg â + ˆσ ge â + ˆσ ge â + ˆσ eg â ) which cotais 4 terms. To further simplify this, we ivoke the rotatig wave approximatio ad eglect the last two terms. There are two ways to uderstad the RWA.

2 A. Violatio of Eergy Coservatio The terms ˆσ ge â ad ˆσ eg â violate eergy coservatio. The former de-excites the atom ad simultaeously absorbs a photo, ad the latter excites the atom while it emits a photo. By cotrast, the two terms we have kept coserve eergy. B. Method of Averagig The above argumet is rather heuristic. We ca make it more rigorous. For this purpose, it is easiest to work i the iteractio picture. Decompose the total Hamiltoia as H = H 0 + H it where H 0 = H A + H F. I the iteractio picture, we have It s easy to show that H I) it = eih0t/ H it e ih0t/ = ge ih At/ ˆσ ge + ˆσ eg )e ih At/ e ih F t/ â + â )e ih F t/ e ih At/ ˆσ ge e ih At/ = e iω 0tˆσ ge, e ih At/ ˆσ eg e ih At/ = e iω 0tˆσ eg Therefore e ih F t/ âe ih F t/ = e iωt â, e ih F t/ â e ih F t/ = e iωt â H I) it = ge iω0tˆσ ge + e iω0tˆσ eg )e iωt â + e iωt â ) We ote that the terms ˆσ ge â ad ˆσ eg â that do ot coserve eergy are multiplied by oscillatory terms which ivolve the sum of the frequecies of the field ad the atomic trasitio, while terms ˆσ eg â ad ˆσ ge â which do coserve eergy are multiplied by terms ivolvig the differece of the two frequecies. For the ear resoat case we are most iterested i, ω ω 0 ω + ω 0. Sice the Schrödiger equatio is a differetial equatio of first order i time we have to itegrate i time. This time itegratio brigs the frequecy sum ad differece ito the deomiator. Hece the domiat coributio must come from the slowly varyig part. The iteractio Hamiltoia i the iteractio picture is the give by H I) it = gˆσ geâ e i t + ˆσ eg â e i t ) with = ω ω 0. This correspodig i the origial Schrödiger picture to the iteractio Hamiltoia H it = gˆσ ge â + ˆσ eg â) The eglected couter-rotatig terms would cause a frequecy shift of the atomic levels o the order of g /ω 0, this is called the Bloch-Siegert Shift. III. DRESSED STATES OF THE JC MODEL Let us ow go back to the Schrödiger picture uder the RWA. The Hamiltoia reads This Hamiltoia possesses a costat of motio: H = ω 0ˆσ ee + ωâ â gˆσ eg â + ˆσ ge â ) 1) N ex = â â + ˆσ ee which ca be uderstood as the excitatio umber of the system. We use Fock states as our basis for the field, the the state with N ex = > 0 cotais two ad oly two states: 1; e ad ; g. Therefore the Hamiltoia matrix ca be decomposed ito blocks of matrices, each block correspodig to a particular N ex. States 1; e ad ; g are eigestates of H A +H F, i.e., the ucoupled system. For strog atom-field couplig, they are o loger accurate descriptio of the combied atom-field system. Istead, we should try to fid the eigevalues ad eigestates of the etire Hamiltoia. I most cases this is a impossible task. The beauty of the JC model is

3 3 ;g / ω / / -1;e + φ φ E+ E FIG. 1: Dressed states of the JC model. that o-trivial but exact solutios to the full Hamiltoia ca be obtaied i the RWA. This is precisely due to the fact that we ca decompose the Hamiltoia matrix ito sub-blocks. Let us focus o the th block spaed by 1; e ad ; g. The submatrix ca be easily obtaied as 1)ω + ω0 ) g = ω ) 1 + g ω ) R R where = ω ω 0 is the laser detuig ad R = g is the Rabi frequecy for the th block. The eigevalues ca be easily foud E ± = ω ) ± + R with correspodig eigestates where φ + = cos θ 1; e + si θ ; g ) φ = si θ 1; e + cos θ ; g 3) cos θ = Ω, si θ Ω = Ω + Ω with Ω = + R = + 4g beig the geeralized Rabi frequecy. The auxilliary agle θ ca also be defied as θ = ta 1 R / ). IV. ENTANGLEMENT AND VACUUM RABI OSCILLATION The eigestates of the JC Hamiltoia φ ± are called dressed states, they are also oseparable etagled states, i.e., states that caot be factored ito a product of a atom state ad a field state. Etaglemet ca be a useful property i cavity QED experimet. For example, cosider the iitial atom-field state cosists a excited atom outside the empty cavity, i.e., Ψ0) = 0; e. After the atom passig through the cavity, the state evolves ito ΨT ) = α 0; e + β 1; g. I this way the atom ca be used as a poiter. That is, the observatio of the atom s quatum state whether i e or g ) after it has left the cavity yields a umber that perfectly correlates with the quatum state of the field, without i ay way touchig the cavity field. This is a example of what is called a QND quatum o-demolitio) measuremet. The above example shows that the iitial state Ψ0) = 0; e is ot statioary for the coupled system. We ca make it more quatitatively to study the dyamics of this state. To this ed, let us fid the iverse relatios of ) ad 3): 1; e = cos θ φ+ + si θ φ 4) ; g = si θ φ+ + cos θ φ 5)

4 4 The state Ψ0) = 0; e will the evolve i time as Ψt) = e iω /)t cos θ 1 e iω 1t/ φ si θ ) 1 eiω 1t/ φ The probability that the system flips from 0; e to 1; g, i.e., the probability that a photo will be emitted ito a empty cavity, is give by p = 1; g Ψt) = si θ 1 si Ω 1t = R 1 Ω si Ω 1t 1 Spotaeous emissio i free space produces mootoic ad irreversible decay of upper-level amplitude, whereas here we fid the so-called vacuum Rabi oscillatio. V. COHERENT FIELD STATE: COLLAPSE AND REVIVAL Let us ow cosider the field is iitially i a coheret state α while the atom is iitially i its groud state g, i.e., Ψ0) = α g which ca be writte i terms of the dressed states: Ψ0) = α g = e α / α ; g = e α / α si θ!! φ+ + cos θ ) φ so that the state s time evolutio is give by Ψt) = e α / α si θ! φ+ t) + cos θ ) φ t) 6) where we have icorporated the time depedeces ito the dressed states: φ ± t) = e ie±t/ φ ± Now the probability P e t) of fidig the atom i state e at time t is give by e ρ a t) e, where ρ a t) = Tr f [ρt)] = Tr f [ Ψt) Ψt) ] is the reduced desity matrix for atom. Usig 6), we fid P e t) = e Tr f [ Ψt) Ψt) ] e = e Ψt) Ψt) e = ; e Ψt) 7) With 4) ad 6) we ca readily fid apart from a phase factor which yields ; e Ψt) = e α / α )! si θ +1 si Ω +1t P e t) = e α =1 α! si θ si Ω t = =1 p α ) si θ si Ω t 8) where p α ) = α = e α α /! is the probability that the coheret state has exactly photos. Similarly we ca fid that the probability for the atom to be i groud state is P g t) = =0 p α ) 1 si θ si Ω ) t = 1 P e t) Now let us examie the populatio iversio wt) = P e t) P g t). For the sake of simplicity, we cosider the o-resoace case = 0. From the results above we have wt) = e α p α ) cos R t 9) =1

5 Let us examie the sum whe the coheret state field is strog α 1), cotaiig may photos, ad thus is closest to a classical field. First we recall that for the Poisso distributio p α ), the average ad variatio of the photo umber is give by 5 = α, ) = = α The sum i 9) caot be evaluated aalytically, so the calculatio has to be doe umerically. A example is show i Fig.. Near to t = 0 the curves show Rabi oscillatios of with a average Rabi period give by RT = π, where R = g = g α is the average Rabi frequecy p) wt) gt FIG. : Collapse ad revival i JC Model. Here the coheret state has ˆ = 5. After some time there is a collapse of the evelope of Rabi oscillatios ad w 0. This ca be uderstood as the sum icludes cotributios from may differet Rabi frequecies. Whe eough time has passed, differet oscillatory terms get out-of-phase ad cacel with each other. The oset of this quatum collapse ca be estimated as δrτ col π where δr is the width of the Rabi frequecy distributio. The above coditio idicates that terms i the sum that are about half the distributio s width away from each other are π out of phase. So we ca use the followig to estimate δr: δr = R + ) / R ) / = R + / R / For 1, we ca expad R ± / aroud which gives Thus we have δr = g ad the collapse time is the R ± / = g ± / = g ) 1 ± 1 4 τ col = π/g which is idepedet of α as log as α 1. The truly remarkable feature of the curve is the later restoratio to early its iitial value. This pheomeo is called revival. The reaso for this revival is that the frequecies i the sum are discretized ad ear the ceter of the distributio, the frequecies are approximately equally spaced. For large α, the ceter will be very broad ad will cotai may terms. These terms will regai a commo phase if every term gets π ahead of the phase of the precedig term i the sum, i.e., whe δω τ rev = π, where δω = R +1 R = g + 1 ) evaluated ear the ceter of distributio with = = α. Hece we have δω = g + 1 ) = g/, ad the revival time is give by i.e., after Rabi periods. τ rev = π /g

6 I fact there is a ifiite sequece of equally spaced revivals. Revivals are a purely quatized-field pheomeo, arisig from the fact that the photo umber distributio is ot cotiuous, whereas the collapse is completely classical. These revivals were idetified ad amed ad their sigificace explaied i 1980 by Eberly ad co-workers after almost 15 years of repeated ear-discovery i more ad more accurate umerical evaluatios of the sum i 9). Quatum collapse ad revival was first observed experimetally i 1987 by Rempe ad Walter [PRL 58, )]. 6