Level crossings in a cavity QED model

0.61 MB·English

by  J. Larson

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<k <1(scaledunits). Insomepapers,theBril-
−
tive parameters. The zero detuning case with the louin zone has been defined as 1/2 < k < 1/2,
−
force applied is consideredin III, and I show the re- with the consequence that two sets of energy bands
sults from numerical wave packet simulations. The areobtained,oneforeachinternalstate [15,21].
|±i
Landau-Zener transitions and the Bloch oscillations ThespectrumE (k)formsaninfinitesetofcontinu-
ν
arediscussed. The three-levelcrossingmodelis pre- ousbandsseparatedbygaps. Forsmoothpotentials,
sented in section IV, and the wave packet simula- the gap size decreases for increasing values of ν; in
tionsarecomparedwiththeanalyticresultsofpaper general the band-to-gap ratio falls off faster than
[16]. In the next section V, I look at the situation 1/ν. Sinceupper andlower electronicstates
|↑i |↓i
when there is no force acting on the system, but of the atom couple through absorption or emission
the detuning is oscillating in time. It is found that of one photon, the number of excitations of the sys-
a similar evolution may be obtained as in the case tem is preserved by the Jaynes-Cummings type of
3
interaction(1). We maythus, fora givennumber of
excitation, define the excitation eigenstates
+ = n 1
| i | − i|↑i (3)
= n ,
|−i | i|↓i
where n isthenphotonnumberstateofthemode.
| i
These will also be referred to as internal states. In
each exchange of energy between the field and the
atom, the atomic momentum is shifted by 1 (in
±
scaled units) from the mechanical ’kick’ due to ab-
sorption or emission of a photon. Consequently, an
FIG. 1: The lowest energy bands of the Hamiltonian
internalstate, + ( ),willonlyreturntothesame
| i |−i (1) as function of quasi momentum k for two different
internalstateafterhavingshiftedthemomentumby
detunings δ. The quasi momentum k runs between -
an even number of momentum kicks. A state with
1 and 1 in the plots. The Bragg scattering crossings
momentum p = k, k , and internal state , is are marked with circles and the Doppleron with square
| i |−i
coupled to the following set of states boxes. The coupling is V0 =0.2 in both plots.
k+µ , µ even
|ψµ(k)i= |k+µi|+−i, µ odd, (4)
(cid:26) | i| i kickshaveoppositemomenta,whilethesecondterm
where µ runs over all integers. Note that the mo- gives the processes when the absorbed and emitted
mentum of ψ (k) is k+µ, and the quasi momen- photons kick the atom in the same direction, either
µ
| i
tum k will therefore be imagined within the first leftorright. InFig.1twoexamplesofthelowestly-
Brillouin zone. There is an orthogonal set of states ing bands of the spectrum Eν(k) for 1 < k < 1
−
with even/odd reversed, which is obtained by using are shown. In the first plot (a) the detuning is
the initial state k + . These two sets do not cou- zero, δ = 0, while the second plot (b) has a pos-
| i| i
ple, and in the special case of zero detuning δ = 0 itive detuning δ = 0.7. The diagonal matrix ele-
theirspectrabecomeidentical. Thisdegeneracywill, ments of the Hamiltonian, in the bare state basis,
however,neverbeinterestingforus,sincewewillal- are εµ(k) = (k + µ)2 δ/2, which are, of course,
±
waysassumeoursystemtobeinoneofthetwosets. eigenenergies for zero coupling, and therefore called
The states of Eq. (4) are egenstates of the Hamilto- bare eigenenergies. For a non-zero coupling, the de-
nian in the absence of coupling g = 0, and I will generacies are split, forming the gaps in the spec-
0
refer to them as bare states. For a non-zero cou- trum. Theseavoidedcrossingswillbeofimportance
plingthe barestatesarenolongereigenstatesofthe in the remaining of the paper. In the two plots we
Hamiltonian, and the new eigenstates, seetwodifferentkindsofcrossings(thereexistalsoa
thirdtypediscussedinsectionIV),thosewhenboth
H φν(k) =Eν(k)φν(k) , ν =1,2,3,..., (5) of the bare crossing energies have the same internal
| i | i
state and those when they have different inter-
willbecalleddressedstates. Inthetheoryofperiodic |±i
nal states. In the first case we say that the crossing
operators,thedressedstatesarealsoknownasBloch
pointisaBraggresonanceandthesecondiscalleda
or Floquet states and their properties are familiar.
Doppleronresonance[23]. Notethatin(b)themost
Itisoftenconvenienttotransformbetweenbareand
pronounced avoided crossing is between the second
dressed states
and the third band and not between the first and
φ (k) = cµ(k)ψ (k) the second band. This is because the lowest bare
| ν i µ ν | µ i eigenenergies that build up the two lowest energy
(6)
ψ (k) = P cν(k)φ (k) , bands both belong to states with the same internal
| µ i ν µ | ν i state,andthesestatesareonlyindirectlycoupledby
where the coefficients cPµ(k) depend on k. the Hamiltonian (1). By controlling the detuning a
ν
varity of different spectra can be obtained. For ex-
The Hamiltonian (1) has close similarities to the
ample, the location of the gaps and also their size
Mathiue equation [28], and in the δ = 0 limit our
can be controlled to some degree.
’two-level’ Hamiltonian is, actually, separable by a
unitary transformation into two uncoupled Math- ThespectrumorthedispersioncurvesE (k),con-
ν
iue equations. In the opposite limit, g /δ 0, necting quasi momentum k with energy for a given
0
→
the Hamiltonianisalsodiagonalinits internal’two- band, contain information about the dynamics of
level’ structure. For the two disconnected equations the system. More precisely, the dispersion curves
obtained, the ’potentials’ go as cos2(x), see [4]. identify the behaviour of an atomic Gaussian wave
The’potentials’equal 1[1+cos(2∼x)],wherethefirst packetinthepresenceoftheperiodiclightfield. For
2
term describe the situation when the two photon us, there are two types of wave packets that are of
4
interest; Gaussian dressed or bare state wave pack- could, for example, be the force felt by a trapped
ets. The first one is a Gaussian wave packet built atom due to gravity. Clearly, by including the ex-
out of dressed states according to tra term, the periodicity is violated and the new
eigenstates will have different properties than the
1
former dressed states. The spectrum no longer con-
Φ = dkϕ (k)φ (k) , (7)
| νi ν | ν i sists of bands, but becomes purely continuous. It
Z−1
is also commonly known that the spectrum can be
whereϕ (k)isGaussiancenteredaroundsomequasi described by a discrete set of complex eigenvalues,
ν
momentum k within the first Brillouin zone. Note calledWannier-Starkladders[11]. Contrarytowhat
0
that we have assumed only a single band to be oc- could be expected, the atomic wave packet will not
cupied, thus we do not sum over ν. The second constantlyaccelerate,but ratherhaveanoscillatory
Gaussian state is behaviour. This characteristic has become known
as Bloch oscillation originating from [1, 2], and it is
1
usually understoodeither by Wannier-Stark ladders
Ψ = dpχ(p)p = dkχ(k+µ)ψ (k) ,
µ
| i | i|−i | i [11] or from the typical band structure of fig. 1 [9].
Z µ Z−1
X Here we will use the latter of the two approaches to
(8)
analyze the dynamics.
andχ(p)isanormalizedGaussiandistributionwith,
for simplicity, a spread within the size of one Bril- When the linear force F is weak enough, an ini-
louin zone. Using Eq. (6) for going between the two tialGaussiandressedstate willnotpopulate nearby
basis, it follows that χ(k) = ϕ (k µ)cµ(k µ). bands. It is the possible to show [1] that the quasi
If cµ(k) is smoothly varying oveνr th−e spreνad o−f the momentum distribution obeys
ν
GaussianthismeansthataGaussiandistributionin
quasi momentum space gives a set of nearly Gaus-
∂ ∂
siandistributions,eachshiftedbyunity ineitherdi- ϕ(k,t)2 = F ϕ(k,t)2, (11)
∂t| | − ∂k| |
rection, in real momentum space. The time evolu-
tion of a dressed Gaussian wave packet is
with solution
1
Φ (t) = dkϕ (k)e−iEν(k)t φ (k) . (9)
| ν i ν | ν i ϕ(k,t)2 = ϕ(k Ft)2. (12)
Z−1 | | | − |
By expanding the dispersion curve around k and
0 This is nothing but having an ’adiabatic’ evolution.
onlyincludingthefirstthreeterms(assumingE (k)
ν Withinthislimit,aGaussiandressedstatewillmove
to vary little within the spread of ϕ (k)), one may
ν itsmeanaccordingtok =k Ft,andasitexitsone
interprettheindividualtermsoftheexpansion: The 0−
Brillouin zone it enters the same zone on the oppo-
first gives an overall phase-shift, the second and
site side. All population remains in the same band
third ones are connected with group velocity and
as the initial state during its evolution, and we see
effective mass m as
2 from fig. 1 that, since the group velocity v defined
g
∂E (k) 1 ∂2E (k) in eq. (10) will flip sign while traversing the quasi
ν ν
vg = ∂k and m = ∂k2 . momentum, the atomic wave packetwill oscillate in
(cid:12)k=k0 2 (cid:12)k=k0 x-space.
(cid:12) (cid:12) (10)
(cid:12) (cid:12) Weknowthatthevalidityofadiabaticapproxima-
In the absent (cid:12)of external forces, v determin(cid:12)es the
g
tions is related to the ’distance’ between adiabatic
velocity of the wave packet and m determines how
2
energy eigen-curves [29]. Thus, the approximation
’fast’the wavepacketspreads,see[27]foradetailed
that no other bands are populated will, most likely,
discussion. The wave packet thus behaves as freely
break down near the avoided crossings. In this sec-
evolvingbutwithsomeeffectiveparameters. Thisis,
tion we chose the zero detuning situation, δ = 0.
ofcourse,allwellknownfromthetheoryofelectrons
By neglecting coupling to other bands except close
in crystal lattices.
to a crossing, we may treat the problem as a two-
levelsystemwithalowestordereffectiveSchr¨odinge
equation
III. LANDAU-ZENER TRANSITIONS AND
BLOCH OSCILLATIONS
∂ d (k Ft+1)2 V0 d
i µ+1 = 0− 2 µ+1 ,
Averyinterestingphenomenonofthe atomicmo- ∂t(cid:20) dµ (cid:21) (cid:20) V20 (k0−Ft)2(cid:21)(cid:20) dµ (cid:21)
tion occurs when a linear force is allowed to act (13)
on the atom, i.e. the Hamiltonian (1) will include where we have assumed a Doppleron resonance
an extra term F˜x˜, where the coefficient in scaled crossing. Hered istheamplitudeforthebarestate
µ
dimensionless units becomes F = F˜/qE . This ψ (k Ft) . Thus, if we consider bands ν and
R µ 0
| − i
5
ν+1, the various amplitudes are
cµ+1, before crossing
d = ν
µ+1 cµ+1, after crossing
(cid:26) ν+1
(14)
cµ , before crossing
d = ν+1
µ cµ, after crossing
(cid:26) ν
Bylinearalizingtheenergydispersioncurvesaround
the first crossing and omitting the constant energy-
shifts, we obtain
∂ d Ft V0 d
i µ+1 = 2 µ+1 . (15)
∂t(cid:20) dµ (cid:21) (cid:20) V20 −Ft (cid:21)(cid:20) dµ (cid:21)
ThisistheanalyticallysolvableLandau-Zenermodel
[12, 13], which has been widely used in different ar-
eas of physics. The asymptotic solution at t= is
∞
dµ 2 =1 e−Λ, (16) FIG. 2: Bloch oscillations of an initial Gaussian bare
| | −
state in the lowest band for zero detuning δ = 0. Con-
where the initial condition at t= is dµ+1 = 1 tourswithfilledgrayscalesshowtheupperatomicwave
andΛ=πV2/4F istheadiabaticit−y∞param|eter.|The packetandthenon-filledcontoursthelowerwavepacket.
0
breakdownofnothavingexactLandau-Zenertransi- The scaled coupling is rather weak in this example,
tionsbetweenneighboringstatesintheBlochoscilla- V0=0.2,implyingthatbareanddressedstatesmayhave
tionschemehasbeenthoroughlystudiedin[10],and alargeoverlap. Thus,astheGaussianquasi-momentum
wave packet traverses the lowest band, the atom will
will not be the subject of this paper. The solution
Doppleron scatter at each Brillouin zone and therefore
(16) is only an approximation, but, nevertheless, it
theinternalstate|±iisflipped,whichisseeninthelower
gives a rough idea of the parameter dependence; a
plot showing the atomic inversion hσzi. Here the force
small force F, the velocity of the momentum wave
F =0.005.
packetislow,andastrongcouplingV ,thebandgap
0
is large and the energy curves are far apart, which
gives an adiabatic transition between the two bare
weakenoughthat only a few percentageof the pop-
states ψ (k + Ft) and ψ (k + Ft) . Thus,
µ 0 µ+1 0 ulationisinhigherdressedstates. Thisisseeninthe
| i | i
according to eq. (14), no transition takes place be-
lower plot in fig. 2 where the atomic inversion σ
z
tweenthebandsinsuchasituation. Populationthat h i
is shown as function of the scaled time. From the
escapes into higher bands due to non adiabatic evo-
atomic inversion we see that the atom ’jumps’ be-
lution,willmovemorefreely,since the higherbands
tween upper and lower states and we also note that
are less coupled to one another. Note that in the
the evolution is very adiabatic and not much pop-
Landau-Zener solution, the system is integrated be-
ulation leaves the lowest band. As argued above, a
tween to + and it is therefore not fully accu-
larger force F should make the population of the
rate to−i∞dentify∞the amplitudes d 2 with the ones
µ oscillating part damp out. This is seen in fig. 3
| |
in the Landau-Zener model.
where we plot the same as in fig. 2 except that F
In fig. 2 we show the evolutionof aninitial Gaus-
is three times as large. Note that the Bloch period
sian bare state in the presence of a force F. The
T = 1/F is a third as long, and how the atomic
B
atomis initially in its lower state with a spatialdis-
inversion damps out much faster than in fig. 2 due
tribution
to the non-adiabatic transitions.
χ˜(x)= 1 e−4x∆22x, (17)
4 2π∆2
x
IV. THREE-LEVEL LANDAU-ZENER
with the width ∆2 =p50. Thus, it has an average TRANSITIONS AND BLOCH
x
OSCILLATIONS
initial quasi momentum k = 0 and in x-space it
0
extends over several periods of the standing wave
meaningthat ∆2 is muchsmallerthanthe Brillouin The previous section dealt with the dynamics of
k
zone. Note that, since the initialstateis a bareone, a wave packet, evolving under the Hamiltonian (1)
some of the higher bands will be populated already with the additional ’force’ potential Fx, in the sit-
before the first crossing. However, the coupling is uation of zero detuning δ =0. Loosely speaking, as
6
FIG.4: Thesameenergybandsasinfig.1,butfortwo
differentdetunings. Forthesedetunings,δ=1,-1,there
is a new kind of level crossing where three curves inter-
sect. Hence,thesecrossingsareneitherpurelyBraggnor
Doppleron resonances.
crossing the dressed state is a linear combination of
bare states containing both upper and lower atomic
states. The parameters are the same as in the ear-
FIG. 3: This shows the evolution of the same initial lier figs. 2 and 3, except that the force is weaker
state as in fig. 2 except that the force is three times F =0.0025.
as large, F = 0.015. The non-adiabatic transitions due
to the increased force are seen both in the atomic wave
packetsandintheatomicinversion,whichdampsoutat
each transition.
thewavepacketapproachedalevelcrossing,itadia-
batically absorbedor emitted a photon, resulting in
a flip of internal state and a momentum ’kick’. We
said that the atom was Doppleron scattered. The
maindifferencebetweenthoseBlochoscillationsand
the ones mostoftenstudied is the internalstructure
of the system; with no internal structure, the adia-
batictransitiononlyresultsintwoornomomentum
’kicks’andnoflip,whichisthatofBraggscattering.
The existence of Doppleron resonances in this
modelisjustoneexampleofhowthismodelbehaves
differently from the standard systems lacking inter-
nal structure of the particle. By varying the detun-
ing,athirdkindofresonancecanbeachieved,which
is seen in fig. 4. In the figure δ = 1 or δ = 1, and
−
in both situations there is a level crossing between
three bare energy curves. This scattering contains
zero, one or two photon exchanges, giving a com- FIG. 5: Bloch oscillations of an initial Gaussian bare
bination of both Bragg and Doppleron resonances. state wave packet for δ = 1 and k0 = 0. Filled con-
tour curves represent the atom in its lower state, while
More generally we have a three level crossing for
thenon-filled curves(hardly seen in theplot) shows the
δ = (2j +1), j = ..., 1, 0, 1, ..., where negative
− upper atomic wave packet. From the lower plot, show-
detunings are located at k = 0 and positive ones at
ingtheatomic inversion,wenotethat theupperatomic
k = 1.
state is populated almost only close to the curve cross-
±
The evolution of an initial Gaussian bare state ings, see fig. 4. This does, however, not mean that the
with the atom in its lower state when δ = 1 is second band is populated when the quasi momentum
showninfig.5. Fromtheatomicinversion,weagain wave packet passes a crossing, rather that the ground
note that most of the population is within the low- dressed state is a linear combination of both upper and
est band. However, as the wave packet traverses loweratomicstatesclosetoacrossing. Otherparameters
through a crossing, the upper atomic state is be- are, F =0.0025, V0 =0.2 and ∆2x =50.
ing populated. This does not mean that the sec-
ond energy band is populated, but that close to a In order to get a deeper insight into the tran-
7
sition we may, similarly to the previous section,
treat the Hamiltonian as an effective time depen-
dentthreelevelsystembyneglectingallotherlevels
and putting k = k Ft. Let us shift the crossing
0
−
to t=0 and the energy E =0, and then we get the
linearalized Hamiltonian
2Ft V0 0
2
H = V0 0 V0 . (18)
l  2 2 
0 V0 2Ft
2 −
 
This is a three-level version of the Landau-Zener
model and it has been solved analytically in a more
generalized form in [16]. I may introduce a tran-
sition matrix, , that takes the initial probability
T
amplitudes squared at t = to their final values
−∞
at t= , provided that only one level is populated
∞
originally. In the above model, bare and dressed FIG. 6: This figure shows the probability (21) for the
states become identical at t= , except for a flip atom tobein its excited state|+i and with momentum
in subscripts 1 3, and the m±at∞rix has identical −1 < p < 0 as function of force F (dotted line), after
↔ T an initial Gaussian bare wave packet in the |+i state
elements irrespective if it is givenin bare or dressed
has been propagated across the crossing at k = 0 seen
basis. Thus we have for bare states
in fig. 4. Dashed line displays the probability (22) of
d∞ 2 P2 2P(1 P) (1 P)2 endingupinthelowestbandν =1. Thesolidcurvegives
| dµ∞−12| = 2P(1 P) (1 2−P)2 2P(−1 P) the result of eq. (19) with initial condition |d−µ+∞1| = 1.
 d|∞µ | 2  (1 −P)2 2P−(1 P) P−2 × Finallythedotted-dashedlinerepresentstheresultfrom
| µ+1| − − numerical integration of eq. (18) between t = −τ and
    t = τ corresponding to one half Brillouin zone (from
|d−µ−∞1|2 k0 = −1/2 to k0 = 1/2). All four curves approaches
d−∞ 2 , zerowhentheforceisincreased,leadingtoalargernon-
 |d−µ∞|2  adiabatic transitions. The other parameters are V0 =
 | µ+1|  (19) 0.2, k0 =−0.5 and ∆2x =50.
where only one of the three levels is populated at
time t= . The probability is given by
−∞ wouldbemorecorrecttocomparethemwiththecor-
P =e−Λ2, (20) responding probabilities obtained from integrating
the eq. (18) between τ and +τ. The corrections
−
of integrating over a finite time interval depends,
where the adiabaticity parameter is the same as in
the original Landau-Zener model; Λ=πV2/4F. of course, on how fast the solutions approache their
0 asymptoticlimits,forathoroughdiscussionsee[10].
A comment is in order about the result (19). In
In fig. 6, the dotted line gives the probability
thelimitoft ,theeigenvaluesoftheHamilto-
→±∞
nian (18) are ,0, and the three states do not 0
couple. Howev−e∞r, wit∞h the full Hamiltonian (1), in P ( 1<p<0)= dp p,+Ψ(t=τ) 2, (21)
+
− |h | i|
the approximation k = k0 Ft, the crossings are Z−1
−
periodic and the adiabatic eigenenergies are never
where Ψ(t=τ) isthefinalstatepropagatedacross
infinitely far apart,so the states will alwaysbe cou- | i
the crossing at k = 0 shown in the right plot of
pled. If we start with an initial bare state wave
fig. 4, as a function of the force F. The initial state
packet in the middle between two crossings, corre-
is a Gaussian bare state with internal state + and
spondingtoatimet= τ,the overlapwithdressed | i
k = 1/2. Dashed line shows the probability of
− 0
stateswillbenon-zeroforseveralstates. Att= τ, −
remaining in the lowest band
−
the state is propagated across the crossing till it
reaches the next middle point at t = +τ, and there 1
P(ν =1)= dk φ (k)Ψ(t=τ) 2. (22)
the populations of different bare or dressed states ν=1
|h | i|
arecalculated. Inthelimitofweakcouplingwemay Z−1
expectthatonlythe threelowestbandswillbepop- Finally the solid line and dotted-dashed curve dis-
ulated after the passage of the crossing, assuming play d∞ 2 obtainedwiththe transitionmatrix ac-
| µ−1|
that the initial bare state corresponded to the low- cording to eq. (19) and from numerical integration
est ground dressed state. These three probabilities of (18) between t = τ respectively. It is seen that
arecomparedwiththeasymptoticones d∞ 2 the probability from±numerical integration follows
| µ−1,µ,µ+1|
obtained from eq. (19). One may expect that it the asymptotic solution very well. The coupling is
8
ratherweakinthisexample,implyingthatbareand Usually the Stark-shift is quadratic in the field
dressed states have a large overlap. amplitude, so in order to have the above time-
Idonotgointodetailsinunderstandingthediffer- dependencethefieldvariationsmustbechosencare-
encesbetweenthethreecurves. Inprincipleasimilar fully. By making an instantaneous diagonalization
analysisasthe oneperformedin[10]forthe original ofthetime-dependentHamiltonian,wefindtimede-
Landau-Zener result could be done. However, the pendent adiabatic eigenvalues, E (k,t). In fig. 7 we
ν
Dykhne-Davis-Pechukas method [30, 31] for calcu- plot the time variation of the two lowest bands, for
lating transition probabilities for two-level systems, a fairly low detuning amplitude δ =2. For an adi-
0
which is used in [10] is, obviously, not justified on abatic process, a Gaussian dressed wave packet will
our three-level system. staywithinonebandandevolveaccordingtotheef-
fectiveparameters(10),whichwill,ofcourse,change
in time.
V. CHIRPED ADIABATIC TRANSITIONS Assuming thatwe canapproximatethe systemto
contain just the two lowest dressed states (and k >
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 Schr¨odinger 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. TheSchr¨odingerequation
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 typ-
ical 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 elimi-
nated,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-