Coupled two-component atomic gas in an optical lattice

0.57 MB·English

by  Jonas Larson

Coupled two-component atomic gas in an optical lattice
Jonas Larson and Jani-Petri Martikainen
NORDITA, 106 91 Stockholm, Sweden
(Dated: January 8, 2009)
We present an ab initio study of the ground state of an ideal coupled two-component gas of
ultracold atoms in aone dimensional optical lattice, either bosons or fermions. Dueto theinternal
two-level structure of the atoms, the Brillouin zone is twice as large as imposed by the periodicity
of the lattice potential. This is reflected in the Bloch dispersion curves, where the energy bands
regularly possess several local minima. As a consequence, when the system parameters are tuned
across a resonance condition, a non-zero temperature phase transition occurs which arises from an
9
interplay between internal and kinetic atomic energies. For fermions, this phase transition is of
0
topological character since the structure of the Fermi surface is changed across the critical value.
0
It is shown that these phenomenaare also expected to occur for two and three dimensional optical
2
lattices.

n
a PACSnumbers: 37.10.Jk,05.30.Jp,05.30.Fk,03.75.Mn
J

8
I. INTRODUCTION with kinetic and scattering energies.

]
r Consequently, we do not restrict the character of the
e The coupled dynamics between an ultracold atomic atom-field interaction to be dispersive. In fact, we al-
h
gasandanopticallattice has gainedenormousattention lowforavanishing atom-fielddetuning andinparticular
t
o during the last decade. Originally, D. Jaksch et al. pro- study the behaviorforbothpositive andnegativedetun-
t. posedthattheMott-superfluidquantumphasetransition ings. Couplingbetweeninternallevelsofatomsinoptical
a
(PT) can be realized using cold atoms dipole coupled to lattices is rather unexplored. The first analysis seem to
m
a standing wave laser field [1]. A mere four years later, date back to the works by Krutitsky et. al., where they
- this PT was observed in the seminal experiment by I. consideredeffectivecoupledtwoorthreelevelmodelsde-
d
n Bloch and coworkers [2]. One among the many reasons rivedfromRamaninteractions[12, 13]. Theyfoundthat
o for the great interest in these systems is the possibility the hopping term in the corresponding Bose-Hubbard
c to experimentally realize, and therefore verify, theoret- model may be tuned to be either positive or negative,
[ ical models developed in the field of condensed matter leading to new phases. An internal coupled model was
2 physics, see the review [3]. Several achievements in this as well studied by Garc`ıa-Ripoll et. al. in Ref. [14].
v field have been accomplished both theoretically and ex- Contrary to [12, 13], we consider here a direct coupling
1 perimentally, and to only mention a few; disorderedsys- between two atomic levels, which is actually what one
9 tems and matter localization [4], frustrated systems [5], gets in the standard optical lattice model by decreasing
8
the strong atom-atom interaction regime and the Tonks the amplitude of the detuning. Another crucial differ-
3
gas [6], cold atom quantum Hall counterp arts [7] and ence between the current contribution and the ones of
.
8 quantum information processing [8]. Ref. [12, 13] is that our model goes beyond any tight
0 binding approximation, in which hoppinig beyond near-
Almost exclusively, effective dispersive models have
8
est neighbours is allowed. The detuning in our model
0 been considered, where the internal level structure of
may be seen as having similar role as the parameter θ
: the atoms can be discarded and one is left with a single
v (relative phase between the two Raman lasers) in [12].
atomic level. Exceptions are works on spin-dependent
Xi atoms, which have shown to acquire new phases com- An important point to note is that in Ref. [14], on the
other hand, the coupling between internal levels is not
r pared to the spinless case [9]. Another, related system
a originating from a non-zero detuning (as in the current
is mixtures of different atomic species, again containing
paper) or from a non-zero angle θ (as in [12, 13]), but
novelphysics[10]. However,thesemodelsdonotanalyze
from atom-atom scattering. We explicitly show that the
directcouplingbetweentheinternalatomicstates,aswill
two-levelstructureofourmodelrendersdispersioncurves
be the topic of the present paper. Furthermore, atom-
having local minima. It is known that such phenomena
atomscatteringplaysacrucialroleinthedynamicsofall
can give rise to topological PTs [15, 16, 17], which in-
above references. For example, the Mott-superfluid PT
deed are found in our model as well. A topological PT
inthe Bose-Hubbard model arisesfromacompetitionbe-
(for fermions; a topological change in the Fermi surface)
tweentheatomickineticenergy(hoppingbetweenneigh-
does take place when the detuning changes sign. The
boring sites) and the onsite scattering interaction [11].
same kind of PT indeed occurs also for bosons. The
Here we will show that by taking into accountfor dipole
non-zerotemperaturesituationisalsoconsideredandthe
induced transitions between coupled atomic states, PTs
topologicalPTisfoundtobe stableagainsttemperature
can occur even f or ideal gases lacking atom-atom inter-
fluctuations.
action. Thisoriginatesfromthecompetitionbetweenthe
atomic kinetic and the internal energies, in comparison Theoutlineofthis paperisasfollows. We firstdiscuss
2
the single particle Hamiltonian in Sec. II, and point out Asaperiodicoperator,theeigenstatesandeigenvalues
some of the symmetries associated with it. One of these of Hamiltonian (3) are characterized by two quantum
symmetryoperatorsindicatesthattheBrillouinzoneex- numbers, one discrete band index ν and a continuous
tendsovertwicethesizeexpectedfromtheperiodicityof quasi momentum q
the optical lattice, a fact that is clarified even further in
Hˆ ψ (q) =Eν(q)ψ (q) , ν =1, 2, 3,.... (5)
the proceeding Sec. III. In Sec. III we begin by analyz- ν ν
| i | i
ing the energy spectrum and the two lowest band Wan-
The range of q assigns the first Brillouin zone and is de-
nierfunctions. Usingthisknowledgewedemonstratethe
termined by the symmetries of the problem. For any
presence of a PT, both for fermions and bosons at zero parameters, the operator Tˆ = eiλpˆ commutes with the
and non-zero temperatures. We also discuss the effect
Hamiltonian, where λ = 2π is the field wavelength in
of weak atom-atom interaction. A discussion of possible
scaled units. Not as evident is that also the operator
extensions is left for the conclusions given in IV. Last,
in the appendix IV we also present an effective Raman Iˆ=σˆzeiλ2pˆ (6)
coupledmodelthatwouldprovidequalitativelythesame
results, but with the benefit of using two meta-stable is a symmetry of the Hamiltonian [20, 21]. Thus, there
atomic states. is a natural π periodic structure of the Hamiltonian.
This signals that the first Brillouin zone extends beyond
1/2 < q 1/2 as implied by the 2π periodicity of the
− ≤
II. SINGLE PARTICLE HAMILTONIAN Tˆ operator,and implicitly we have
q ( 1,1]. (7)
The Hamiltonian for a single two-level atom, whose ∈ −
internal states dipole couple via a standing wave laser However, for ∆ = 0 the spectrum is doubly degenerate
field, reads and in this limit it might be more convenient to define
the Brillouin zone within 1/2 < q 1/2. One may
Hˆ = pˆ˜2 + ¯h∆˜σˆz +2h¯g˜cos(kxˆ˜)σˆx. (1) note that Tˆ =Iˆ2. The phy−sicalbackgr≤oundof this addi-
2m 2 tional symmetry operator Iˆ emerges from the fact that
Here, pˆ˜and xˆ˜ are atomic center-of-massmomentum and absorptionoremissionofasinglephotonflipstheinternal
position, m its mass, ∆˜ the atom-field detuning, g˜ the states, , while only for a two-photonprocessis
|±i→|∓i
effective atom-field coupling and k˜ the field wave num- the internal state unchanged.
ber. The internal states of the atom are labeled
|±i
and the Pauli matrices operate as σˆ = and
z
σˆ = . Beforeproceedingweintr|o±diuced±im|±enision- III. STRUCTURE OF THE GROUND STATE
x
les|s±viaria|b∓leis through the characteristic length k 1 and
−
energy E =h¯2k2/2m; A. Model characteristics
r
¯h∆˜ ¯hg˜
xˆ=kxˆ˜, ∆= , g = . (2) To find the ground state of N non-interacting two-
Er Er level atoms we need to diagonalize (3); both within the
internal and the motional degrees of freedom. We first
1 0
Inthe + = and = nomenclature,Eq.(1) notethatforagivenquasimomentumq,theHamiltonian
| i 0 |−i 1
becomes in sc(cid:20)aled(cid:21) variables (cid:20) (cid:21) canbe writtenonthe block formH =Hϕ⊗Hφ, andthe
states coupled by the sub-Hamiltonians read
∆
∂2 2gcos(xˆ) q+η η even
Hˆ =−∂x2 + 2 ∆ , (3) |ϕη(q)i= |q+ηi|+−i η odd
2gcos(xˆ) (cid:26) | i| i
−2 (8)
 
  q+η + η even
wthheicehxtseerrnvaelsfiaesldoucorumploedsetlhHeabmarieltostnaitaens. Weannodtesimthualt- |φη(q)i=(cid:26)||q+ηii||−ii η odd,
|±i
taneously shift the momentum by either 1. For ∆=0,
wherethefirstketisthemomentumeigenstatepˆq+η =
the unitary operatorUˆ = 1 (σˆ +σˆ ) de±couples the in- | i
√2 x z (q +η)q +η and η is an integer corresponding to the
| i
ternallevels,andoneobtainstwoMathieuequations[18] momentum shift governed by absorption and emittance
with diabatic potentials Vd(x) = 2gcos(x) [19]. The of photons. This decoupling of the dynamics is of great
internal atomic states of th±e decou±pled equations are importance as a given number of atoms residing in each
subset is constant when the parameters are varied. The
1
1 =Uˆ + = (+ + ), states (8) are eigenstates of the Hamiltonian in the ab-
| i | i √2 | i |−i
sence of matter-field coupling g =0, with bare energies
(4)
2 =Uˆ = 1 (+ ). εµ =(q+η)2 ( 1)µ∆, (9)
| i |−i √2 | i−|−i ± − 2
3
where the -sign is different for the two sets (8). In
±
this trivial situation, the detuning ∆ simply shifts the
parabolic dispersions. The general eigenstate, Bloch
state, were introduced in Eq. (5). The eigenvalue Eν(q)
is the ν’th energy band/dispersion curve, given for q
∈
( 1,1]. The eigenstate in a position representation can
−
be written using its constituent Bloch states as
ψ (x)= xψ (q) = ψ+ (x)+ +ψ (x)
ν,q h | ν i ν,q | i ν−,q |−i
(10)
= ψ1 (x)1 +ψ2 (x)2 .
ν,q | i ν,q | i
Note that the above specifies just two possible con-
stituent Bloch states and any other internal basis would
define two new ones. In particular, ψ1,2(x) and ψ (x)
ν,q ν±,q
arerelatedviatheoperatorUˆ = 1 (σˆ +σˆ )ofEq.(4).
√2 x z
In terms of the the dispersions, the Hamiltonian is given
by FIG. 1: The three lowest energy bands of Hamiltonian (3),
where the scaled dimensionless parameters are g = 0.1 and
Hˆ = ∞ Eν(q) µ nˆν, (11) ∆=0, 0.25, 0.5, 1inplots(a)-(d)respectively. Thetwolow-
− q estbandsaredistinguishedbetweenthetwosub-Hamiltonians
Xν=1q∈X(−1,1](cid:2) (cid:3) Hϕ (solid line) and Hφ (dashed line).
wherenˆν istheoperatorgivingthenumberofatomswith
q
energyEν(q)andherewehaveintroducedachemicalpo-
tential µ. Note that for a given atom number N one has
∞ν=1 q ( 1,1]hnˆνqi=N, where hnˆνqi is the expectation shheorewtnh.eTenheergryessopneacntcruemcaissed,o∆ub=ly0d,egisengeivraetneians(pao)inatnedd
valuefort∈he−particularstateofinterest. TheBlochstates
P P out in the previous section. However, for non-zero ∆,
can be expressed in the bare basis (8)
the degeneracy is lifted (indicated in (b)-(d)) and as a
consequence, the first Brillouin zone extends over quasi
∞
|ψν(q)i = η=X∞−∞dcνην((qq))|ϕφη((qq))i . (12) wmtthhhoeaimtlleotetwhnheteeasdtoiqtsdhpi∈eseprrs(eli−orowsn1ioe,cs1nut].rclivuneTresvhdeheeaorvdfiveateshmsheefordrsoeumlbitnh-HHeanϕacmoianrirlsstetionsenpagidolaen.nmdNsHionttφioe-,
η | η i muminallplots. Thenon-monotonicityofthedispersion
Note that expanding ψνη(=Xq−)∞does not mix the states cduiffrvereesnctanenbeerguinesd;erksitnoeotdicfreonmergthyecianutesrinpglaythbeetpwaereanbothliec
ϕ (q) and φ (q) du|e to tihe block-diagonal form of structures,the coupling energywhichtypicallysplits the
η η
|the Haimilton|ian Hi= H H in this bare basis. We degeneracy and the internal atomic energy that (loosely
ϕ φ
emphasizedthata quasimo⊗mentumeigenstate,orBloch speaking) shifts the dispersions.
state, is a linear combination of the two internal atomic Let us briefly discuss the dispersion curves and their
states . The same holds for the Wannier functions limiting cases in more detail. In the regular dispersive
|±i situation one has ∆ g and the coupling between the
≫
w (x R) eiqRψ (x) internaladiabaticstatescanbeneglected. Toverifythis,
ν ν,q
− ≡
let us introduce the adiabatic states [22] as columns of
q∈X(−1,1] the unitary operator
(13)
= w+(x R)+ +w (x R)
ν − | i ν− − |−i
cos(θ/2) sin(θ/2)
Uˆ = , (14)
= w1(x R)1 +w2(x R)2 . ad sin(θ/2) cos(θ/2)
ν − | i ν − | i (cid:20) − (cid:21)
As defined above, the constituent Wannier func- where
tions w+(x R) and w (x R) are normalized as
νwi(−x R)2dx =ν−1, an−d likewise for w1(x R) 4gcos(x)
i= | ν − | ν − tan(θ)= . (15)
andw±2(x R). LikefortheBlochstates,theconstituent ∆
P νR −
parts depend on the internal basis. Equipped with this
machinery, we now turn to analyze the complex band The operator Uˆ then diagonalizes the 2 2 matrix of
ad
×
structure of the Hamiltonian (3). the Hamiltonian (3). However, due to its x-dependence
Figure1displayssomeexamplesofthe bandstructure willitnotcommutewiththemomentumoperatorpˆ. This
- the dispersion curves, where the first three bands are causes non-diagonal terms in the transformed Hamilto-
4
nian [22], most properly defined within q ( 1/2,1/2]. Notewor-
∈ −
thy is the fact that the diabatic potentials have minima
H˜ = UˆHˆUˆ−1 =−∂∂x22 +Vcent(xˆ) weihthileertfhoer mxdmi−nim=a2fnoπr tohrexadmd+ia=ba(t2icnp+ot1e)nπtifaolrs ianrteegeeitrhner,
+ ΩV∗+a(dxˆ(,xˆpˆ)) VΩ(axˆd(,xpˆˆ)) . (16) xthame−c=ornreπspoornxdami+ng=d(cid:0)ianb+ati12c(cid:1)oπr.aBdyiatbhaeti±c-psoigtnenwtieailn.dicate
(cid:20) − (cid:21)
Here, V (x) is a centrifugal term turning up on the
cent R=0 R=π/2
diagonal [23], Ω(x,p) is the non-adiabatic coupling [22]
and specifically (a) (b)
2 0.1
)|
2 R
Vad(x)= ∆ +4g2cos2(x) (17) x-
± ±s(cid:18)2(cid:19) i(w1
aretheadiabaticpotentials. Itfollowsthatthecentrifugal |
0
and adiabatic correction terms are small in the ∆ g
≫
regime [22], giving the adiabatic potentials 2 (c) (d)
∆ 4g2cos2(x) R)| 0.1
Vad(x) . (18) -
x
± ≈±2 ± ∆ (
iw2
Thus, we derive the regular situation most commonly
|
0
considered in the literature and we especially note that
-20 0 20 -20 0 20
the first Brillouin zone extends over q ( 1,1]. Choos- x x
∈ −
ing ∆ > 0 and considering the weak coupling limit
4g2/∆ 0,thestates q withq ( 1,1]haveeigen-
FIG. 2: The constituent squared amplitudes of the Wannier
→ | i|−i ∈ −
values functions of the first, (a) and (b), and second, (c) and (d),
∆ Bloch band. The left plots display the Wannierfunctions for
E0(q)=ε0 =q2 . (19) R=0,whiletheintherightplotsR=π/2,correspondingto
− 2
the minima of Vad,d(x) and Vad(x) respectively. Solid lines
− +
For the states q 1 on the other hand, we have the depict|w+(x−R)|2 anddashedline|w−(x−R)|2. Theother
| ± i|−i 1 1
dispersions dimensionless parameters are g=0.1 and ∆=0.25.
∆
E0(q)=(q 1)2 . (20)
± − 2
However, as is noticeable from Eq. (8), the states q
| i|−i
and q 1 arenotcoupledandwhenonerestrictsthe Thecoupledtwo-levelcharacterofthesystemgivesrise
| ± i|−i
analysis to just one of these sets and one consequently toratherpeculiarconstituentWannierfunctionsw (x
regains a regular spectrum, by which we mean disper- R) or w1,2(x R). In general, R is chosen such thaν±t th−e
ν −
sion curves that posses only a single minimum within potential is minimal at x = R. However, in the present
one Brillouin length, dE1(q)/dq 0 for 0 q 1. More model R is not a priori given since the potential has a
≥ ≤ ≤
interesting is the intermediate regime, neither adiabatic complex two-level structure. From the symmetry of the
nor diabatic (∆=0), where the abovestatesmay indeed Hamiltonian it follows that w+(x R) w (x R)
coexist if we assume that both of the states q can for ∆ ∆, while w1,2(x νR)−are in↔variaν−nt u−nder
| i|±i ↔ − | ν − |
be present. Thus, when the coupling between states such sign change. In Fig. 2 (a)-(d) we show examples of
|±i
cannot be neglected, one must take into account all the |w1±(x−R)|2 (a) and (b) and |w2±(x−R)|2 (c) and (d),
sqt=ates1aansdwtehlel aloswfoerstqe=ne0r.gy band contains minima at fRo)r2Ra=nd0tohreRso=lidπ/w2.+(Txhe dRa)sh2e.dNlinoeteshthowatsf|owr1−R(x−=
± | | 1 − |
As pointed out, here we allow for atomic states + 0, the two constituent Wannier functions are identical,
| i
and to have the same momentum. It is understood which, however, does not hold for R = π/2. For R = 0,
|−i
that the same arguments hold for ∆ < 0, making the the constituent Wannier functions resemble typical ones
replacement + . We should mention that even if obtained in one component systems [25]. This is not tru
weonlyconsi|deir↔ato|m−siwithinonesubset(8),thedisper- e for R=π/2 (corresponding to the minima of Vad(x)),
+
sioncurveswillcontainseverallocalminima[21,24],and where only w (x R)2 shows the regular shape and
the results presented are valid also in such cases. In the w+(x R)|2 lν−ooks−as if|R coincide with a maximum of
| 1 − |
diabatic limit, ∆ = 0, we saw that we can separate the the potential. Note that the result of the figure presents
dynamics into two uncoupled problems with diabatic po- an intermediate regime where neither the adiabatic nor
tentials Vd(x) = 2gcos(x). The spectrum is then dou- the diabatic approximations can be considered; g = 0.1
bly degen±erate an±d the first Brillouin zone is therefore and ∆=0.3.
5
for bosons, the ground state is degenerate and n and
0
n may pertain any positive values such that the total
+1
number gives N (we have not included q = 1 as it lies
−
outside the Brillouin zone), and in fact any linear com-
bination of these degenerate states is adequate. Further,
the state (22) is valid for any filling factors ξ. In fact,
the same holds in general for any non-integer filling fac-
tor ξ. As states (21) and (22) are prodcut states, there
existno correlationbetweenthe atoms. Nonetheless,en-
tanglement between atomic motion and internal atomic
states occurs for each atom (provided ∆, g =0). This is
6
clear since
xψ (q) =χ(x)(a+ +b ), (23)
ν
h | i6 | i |−i
for some normalized function χ(x) and a2 + b2 = 1.
| | | |
The expansion of the Bloch eigenstates in terms of bare
statesisgiveninEq.(12). Thecorrespondingcoefficients
are displayed in Fig. 3, for the lowest Bloch band with
parameters g = 0.5 and ∆ = 0.25. For q < 1/2, the
c1(q) are all zero, while d1(q) = 0 for 1|/2| < q < 1.
µ µ | |
A result deriving from the block structure of the Hamil-
tonian (8). The c1(q) coefficients are dominated by the
µ
ones with either µ = 1 or µ = 1, and µ = 0 is the
most prominent coefficient of the−d1(q). Note that the
µ
probability for the atom to be found in the state for
|−i
a given eigenstate ψ (q) is
ν
P( ;ψ (q))= cν(q)2+ dν(q)2 (24)
− ν | η | | η |
ηeven ηodd
X X
and P(+;ψ (q))=1 P( ;ψ (q)).
ν ν
− −
FIG. 3: Coefficients c1(q) (a) and d1(q) (b) defined in We define the internal density per particle as
η η
Eq. (12). The parameters are as in Fig. 1 (b), but with a
larger coupling g=0.5. ρ±(x)= N1 |ψν±,q(x)|2, (25)
ν q
XX∈Q
where the second sum runs over occupied momentum
B. Zero temperature many-body ground state
states,andN isthenumberofatoms. Thedensityρ (x)
−
for fermions is shown in Fig. 4 (a) as function of the
Wewillfocusonthelowestband,ν =1,thusassuming
detuning ∆ where the lowest Bloch band is half filled,
afillingfactorξ = N <1,whereN isthetotalnumberof
K ξ =1/2. Forotherfilling factorsoneregainsverysimilar
atomsandK numberofsites. Inthenumerics,K willbe plots. Bysubstituting∆ ∆,theρ+(x)isidenticalto
taken large enough (typically K > 100) to assure small ↔−
ρ (x). Noticeable is the discontinuity at ∆ = 0. Thus,
−
boundary effects. The N many-body ground states for
around resonance, ∆=0, the population of + and
fermions and bosons, at zero temperature and for filling | i |−i
atomsmayfluctuateandinparticularafirstorderPTin
factors ξ <1, are
the collective atomic inversion (per particle)
|ΨiF = fˆq†|0i= |nqi, (21) j 1 [P(+;ψ (q)) P( ;ψ (q))] (26)
qY∈Q qY∈Q z ≡ N ν − − ν
Xν {Xnq}
Ψ = n ,n , n +n =N (22)
| iB | 0 +1i 0 +1 is expected. Note that the above inversion (26) may be
derived from the expectation value of the collective in-
respectively and contains those values of q which are
inside the FermiQsea, fˆ is the Fermi creation operator version Jˆz
q†
of mode q of the lowest Bloch band, 0 is the vacuum
| i Jˆ 1 N
ianndqunaqsi(=mhonˆmqei)ntaugmainmcohdaeraqc;terizestnhen=umNb.erNoofteattohmats jz = hNzi ≡ N hσˆz(i)i, (27)
q q i=1
∈Q X
P
6
H ). Thus, the single particle ground state energy is ei-
ϕ
therE ( 1)orE (0)dependingonthesignof∆,andas
1 1
±
one tunes ∆ across resonance the atom will absorb/emit
one photon and W W.
→−
The same argument holds also for many atoms where
eachone of them swapsinternalstates, , while
|±i↔|∓i
passing through ∆ = 0. For a fermionic system, the
PT is of topological character, since the structure of the
Fermisurfacechangesacrossthecriticalpointduetothe
fact that each atom emits/absorbs a momentum kick.
The mechanism behind the PT is the interplay between
internal and external atomic energies. The collective in-
version,relatedtointernalstructure,revealsthePT,but
(b) its topological character is, however, only manifested in
1
the external degrees of freedom. Even for bosons can
g=1/4
the PT be termed topological becuase the nature of the
g=1/2 atomic phase space distribtion is altered while crossing
0.5
thecriticalpoint. TopologicalPTs,orLifshitztransition,
g=1
havebeen studiedcomprehensivelyin quantumHalland
superconducting systems [15, 16]. However,a recent pa-
z 0
j
per considered a topological PT of ultracold fermionic
atoms in an anisotropic three dimensional optical lattice
-0.5 [17].
Thenon-linearity,imposedbyincludingatom-atomin-
-1 teraction,is knownto causechangessuchasloopsin the
-2 -1 0 1 2
∆ band structure around the avoided crossings of the dis-
persions [27]. This effect occurs for rather strong cou-
plings, or more precisely when the effective scattering
FIG.4: Theupperplot(a)displaystheFermiparticledensity
ρ−(x)(25)fortheinternalstate|−iathalffillingofthelowest amplitude becomescomparableorlargerthanthe lattice
Bloch band. The lower plot (b) shows three examples of the depth [27]. For reasonable temperatures is the Fermi
collectiveatomicinversionj (26)asfunctionof∆forbosons surface well below the first band gap where the greatest
z
(dashed lines) and for fermions (solid lines). In (a) g=1. effects of the non-linearity is expected to manifest itself.
Consequently, the topological PT is supposed to be in-
sensitive for such modifications, since the general struc-
ture of the spectrum remains and thus the topological
where σˆz(i) is the i’th atoms z Pauli matrix measuring change in the Fermi sea would still take place as the de-
the i’th particle inversion. In the Dicke model of N two- tuning is tuned across resonance. Nonetheless, one may
level atoms interacting with a quantized field mode, jz expect that even for weak interactions, the interaction
is often considered as an order parameter having a dis- induced splitting of the degeneracy of the ground state
continuousfirstorderderivativeatthecriticalatom-field might cause a re-formation of the ground state into a
couplingwhichdefines thenormal-superradient quantum correlatedstatewhichisfundamentallydifferentfromthe
PT [26]. The inversion jz as function of ∆, presented simpleproductstate. Thus,thegroundstateisnolonger
in Fig. 4 (b), illustrates the discontinuity incorporated asimpleproductstateofFockstatesasgiveninEqs.(21)
in our model. Note that this PT is different from the and (22). On the other hand, the inversion is invari-
one of the Dicke model, where the PT originates from antforsuchre-configurationofthe groundstate because
competition between field and interaction energies and ψ (q)σˆ ψ (q) = ψ ( q)σˆ ψ ( q) , q ( 1,1].
ν z ν ν z ν
not between kinetic and internal atomic energies as in h | | i h − | | − i ∈ −
this paper. In fact, the Dicke model lacks any sort of One limitation of our model is spontaneous emission
periodic lattice potential. of the excited atomic state. This can be circumvented
We have verified that the PT obtained in our model by considering instead a Raman coupled Λ-atom setup,
alsooccurswhenonerestrictsthedynamicstooneofthe inwhichtheexcitedstateisadiabaticallyeliminateddue
subsets (8). Assuming a single particle in either of the to dispersive interactions. The resulting effective Hamil-
subsets (8), one has E ( 1) = E (0) for ∆ = 0, while tonian, derived in the appendix, has a similar form to
1 1
±
E ( 1) = E (0) for ∆ = 0 (clear for example from the the one of Eq. (3) apart from two additional Stark shift
1 1
± 6 6
dashed curve if Fig. 1). Note that this is not true if we terms. By proper choices of parameters can these two
include both subsets where E ( 1) = E (0) is always terms be made small and the corresponding spectrum
1 1
±
true(hereE (q)isthe lowestBlochbandofeitherH or then possesses the same properties as Hamiltonian (3).
1 φ
7
C. Non-zero temperature ground state coupling; g = 1/2) in the large temperature limit. Even
at fairly large couplings, this asymptotic value is non-
zero.
0.6
0.5 T=0.02
0.5
0.48 T=0.1
0.4 0.46 T=0.2
0.44
ex0.3 -0.1 0 0.1
P
0.2
0.1
0
-1.5 -1 -0.5 0 0.5 1 1.5
∆
FIG. 6: Population of excited bands P , in the case of
ex
fermions, as function of the detuning and of temperature;
T = 0.02 (solid line), T = 0.1 (dashed line) and T = 0.2
(dot-dashed line). At resonance and moderate temperatures,
theexcited population approximates 1/2 duetothedegener-
acy at that point. For large detuning, the gap between the
two lowest bands increases causing a decrease in the excited
band population. The inset shows a close-up of the popu-
lation around ∆ = 0, indicating that for large temperatures
P exceeds 1/2 at resonance due to occupation of third and
ex
higher bands. Hereg=1/2 and ξ=1/2.
FIG. 5: The collective atomic inversion j (a) for fermions
z Anotherrelevantquestionistheamountofpopulations
(black lines) and bosons (gray lines) as function of the de-
of excited quasi momentum states ψ (q) for non-zero
tuning ∆ and different temperatures, fermions: T = 0 (solid | ν i
temperatures, especially for fermions. The two lowest
line),T =0.02(dotted)andT =0.1(dot-dashed)andbosons:
Blochbandsareidenticalatresonance,∆=0,andconse-
T = 0 (solid line) and T = 0.1 (dashed line). The inset is a
close-up of the curves to the right of ∆=0. The second fig- quently equallypopulated. For smalltemperatures,only
ure (b) depicts the temperature dependence of the inversion statescorrespondingtothesetwobandsaresubstantially
gap δj of Eq. 28, for Fermions (solid black line) and bosons occupied,whileforlargetemperaturesalsothethirdand
z
(dashed gray line). In both plots, g = 1/2 and the chemical forthbandwillbepopulated. Fornon-zerodetuning,the
potential is chosen such that the number of particles is half degeneracy is lifted and the populations of the two low-
thelattice number. est bands are no longer balanced. We may define the
individual band population
We turn now to the situation of non-zero temperature
P = nˆν , (29)
and thermal excitations. As argued above, each eigen- ν h qi
state ψ1(q) in the zero temperature ground state un- q∈X(−1,1]
| i
dergoes a collective Rabi flip by sweeping the detuning
where nˆν = Tr ρ nˆν is the expectation value of the
acrossresonance,whichsuggeststhatalsoforexcitations numberhoqpieratornˆνTforqthestateρ givenattemperature
around the Fermi surface or of the Boson ground state (cid:2)q (cid:3) T
T. Thus, P measures the population in band ν, and
willtheatomicinversionbediscontinuousaround∆=0. ν
in particular for ∆ = 0 do we have P = P for i =
Let us introduce the gap function i i+1
1, 3, 5, ... due to the degeneracy. The total population
δj = lim j lim j (28) of excited bands is given by
z z z
∆ 0− −∆ 0+
→ →
andstudythebehaviorofδj forvarioustemperaturesT. ∞
z P = P . (30)
ex ν
The temperature dependencies of the collective atomic
ν=2
inversion j and the inversion gap δj are presented in X
z z
Figs.5(a)and(b)respectively. Thegapisslightlylarger Note that P includes the second band ν =2 which for
ex
for Bosons than for Fermions. Surprisingly, the gap δj ∆ = 0 is degenerate with the lowest band ν = 1 and in
z
approaches a non-zero value ( 0.75 for this choice of this specialcaseis the subscriptex (excited) misleading.
≈
8
Naturally, 0 P < 1 and also 1/2 P (∆ = 0) < 1.
ex ex
≤ ≤
Figure 6 displays P as function of ∆ and for three dif-
ex
ferent temperatures. As expected, the excitations in-
crease for a large temperature and decrease for a large
detuning. The inset gives the population close to reso-
nance,anditisinparticularseenthatP (∆=0)>1/2
ex
(P (∆ = 0) = 0.5025 for T = 0.2) from the fact that
ex
the bands ν =3, 4, ... begin to be populated. The shape
of P seems fairly Lorentzian,but the inset revealsthat
ex
P is indeed not Lorentzian in the vicinity of ∆ = 0.
ex
Again we restrict the analysis to a filling factor ξ =1/2,
namelychosethechemicalpotentialµsuchthatitresults
in half filling.
D. Extensionto a higher dimensionaloptical lattice
FIG. 7: First energy band E1(qx,qy) of the two dimensional
model (31), with equal coupling strengths g =g =0.1 and
We conclude this section by discussing the situation x y
∆=0.25.
of a two dimensional optical lattice. The two fields are
assumed to share the same wavelength and both inter-
act with the same dipole transition of the atom. The
Hamiltonian takes the form
inthecollectiveatomicinversion. Moreover,itwasfound
∂2
Hˆ = + to be a non-zero temperature PT.
2D −∂x2
To conclude, we showed that quantum PTs can oc-
∆
curinmulti-componentultracoldatomicsystemsdespite
2g cos(xˆ)+2g cos(yˆ)
x y
+ 2 . the lack of atom-atom scattering. It is believed that
 ∆ 
2g cos(xˆ)+2g cos(yˆ) adding interaction between atoms can give rise
x y
2
  to novel phenomena. In such a scenario, two
 (31)
relevant questions directly come to mind: How
The lowest energy band in the symmetrical case with
does the coupled internal structure of atoms af-
g = g = 0.1 and ∆ = 0.25 is shown in Fig. 5. As
x y fect the phase diagrams of non-coupled models,
for the one dimensional case, for q , q = 1/2 does the
x y ± and how does strong interaction influence the
lowest energy band attain its maximum. Importantly,
topological PT demonstrated in this paper? The
multipleminimaofthedispersionisalsofoundinthetwo
secondquestionwasbrieflydiscussedattheendof
dimensional case. It is easy to convince oneself that this
Sec.IIIB,andbothquestionsarecurrentlyunder
holds also in three dimensional, and consequently that
investigations for future publications. We further
the topological PT is not limited to the one dimensional
plan to analyze the present system, and especially the
lattice.
topologicalPT,inacavityQEDmodelwherethe fieldis
treated quantum mechanically.
IV. CONCLUSIONS
In this paper have we studied an ideal gas of ultra-
coldtwo-levelatomscoupledtoanopticallattice. Itwas
shown that the coupled two-level character of the prob-
APPENDIX A: EFFECTIVE DISSIPATIONLESS
lemgiveriseto novelphenomena,notpresentinthe reg- TWO-LEVEL MODEL
ular dispersive case of large detuning where one atomic
level has been adiabatically eliminated rendering an in-
ternal structureless system. In particular, the Brillouin Decayoftheexcitedatomiclevelwilltypicallysetlim-
zoneis twice aslargeinthis systemcomparedto ainter- its on any experimental consideration. One way to min-
nal structureless one and the Bloch bands possess mul- imize such effects is to effectively couple two metastable
tiple minima. This comes about due to the competition atomic states using an additional laser. We consider
between the various involved terms of the Hamiltonian; therefore a three-level Λ-atom with metastable lower
kinetic and internal energy. An outcome of the peculiar states 1 and 2 andexcitedstate 3 ,withrespectiveen-
| i | i | i
energybandspectrumisthepresenceofatopological,or ergies E , i=1, 2, 3. State 1 and 3 are coupled through
i
Lifshitz, PT as the detuning is tuned across resonance. our optical lattice, while 2 and 3 couple via an external
This transition is of first order nature and is manifested laser whose field amplitude is assumedconstant overthe
9
atomic sample. The Hamiltonian becomes [28] where σˆ = 2 2 1 1 and σˆ = 1 2 + 2 1. We
z x
| ih |−| ih | | ih | | ih |
have numerically confirmed that the band structure of
3 (A2) looks very similar to the ones presented in Fig. 1.
HˆΛ = Eiσˆii+Ω σˆ23eiωLt+σˆ32e−iωLt Therefore,ourresultsandconclusionsofSec.III arealso
i=1 (A1) reproducible for a Hamiltonian such as (A2).
X (cid:0) (cid:1)
+2gcos(xˆ+ω t)(σˆ +σˆ ),
O 13 31
where Ω is the external laser coupling amplitude, ω
L
and ω the two field frequencies, and σˆ = i j . The
O ij
| ih |
excited state can be adiabatically eliminated if the in-
teraction is dispersive. Thus, we assume a highly de-
tuned configuration, ∆ = E E ω Ω and
1 33 22 L
− − ≫
∆ = E E ω 2g, and at the same time
2 33 11 O
− − ≫
∆ = ∆ ∆ is of the same order as Ω and g. One
3 1 2
| − |
thenderives,afterapplicationofarotatingwaveapprox-
imation, an effective two-level model for the metastable II. ACKNOWLEDGEMENTS
states, which is given by [28]
∆ Ω2 4g2
Hˆ = 3σˆ σˆ + cos2(xˆ)σˆ
eff 2 z − ∆ 22 ∆ 11 We wish to thank Prof. Maciej Lewenstein and Dr.
1 2
(A2) Giovanna Morigi for inspiring discussions.
1 1
+2gΩcos(xˆ) + σˆ ,
x
∆ ∆
(cid:18) 1 2(cid:19)
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