Electron-like and photon-like excitations in an ultracold Bose-Fermi atom mixture

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by  Yue Yu

Electron-like and photon-like excitations in an ultracold Bose-Fermi atom mixture
Yue Yu1,2 and S. T. Chui2
1. Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China and
2. Bartol Research Institute, University of Delaware, Newark, DE 19716, USA
(Dated: February 6, 2008)
Weshowthattheelectron-likeandphoton-likeexcitationsmayexistinathree-dimensionalBose-
6 Fermi Hubbard model describing ultracold Bose-Fermi atom mixtures in optical lattices. In a
0 Mott insulating phase of the Bose atoms, these excitations are stabilized by an induced repulsive
0 interaction between ’electrons’ if the Fermi atoms are nearly half filling. We suggest to create
2 ’external electric field’ so that the electron-like excitation can be observed by measuring the linear
density-densityresponseof the’electron’ gas tothe’externalfield’inatime-of-flight experiment of
n
the mixture. The Fermi surface of the ’electron’ gas may also be expected to be observed in the
a
J time-of-flight.

3
PACSnumbers: 03.75.Lm,67.40.-w,39.25.+k,71.30.+h

]
r
e Ultracoldatomsinopticallatticeshaveofferedahighly where the lattice spacing λ/2 is set to unit. ( We also
h
tunable platform to study various physical phenomena set ~ = c = 1.) na = a†a and nf = f†f . t and t
t i i i i i i B F
o which may not be definitely clarified in condensed mat- arethehoppingamplitudesofthebosonandfermionbe-
t. ter systems [1]. On the other hand, new systems which tween a pair of nearest neighbor sites ij . µB and µF
a maynotberealizedincondensedmattercontentarepre- are chemical potentials. And U andhUi are the on-
m BB BF
sented, for example, mixtures of Bose-Fermi atoms as siteinteractionsbetweenbosons,andbetweenbosonand
- constitution particles. Experimentally, the Bose-Fermi fermion. In this work, we use U >U >0 although
d BB BF
n atom mixtures in optical lattices have been realized for this is not necessary in general. The microscopic calcu-
o 87Rb-40K [2], and 23Na-6Li [3]. lations of these model parameters in terms of the cold
c Microscopically, these mixtures may be described by atom mixture have been established, e.g, in Ref. [4].
[ a Bose-Fermi Hubbard model [4]. The constitution par- To deduce the low energy theory in strong couplings,

1 ticles in this model are spinless boson(ai) and fermion we will use the slave particle technique, which has been
v (fi) with i the lattice site index. In this Letter, we con- applied to the coldboson system[9, 10, 11]. In the slave
6 siderthree-dimensionalcubiclattices. Wewillshowthat, particle language, the Hamiltonian reads H = H +H
2 4
3 in high temperatures, there are only excitations of these whereH andH arethetwo-operatorandfour-operator
2 4
0 constitution particles. We call this a confinement phase. terms, respectively. Namely,
1
The systemundergoesaphasetransition,incertaincrit-
0
ical temperature, to a uniform mean field (UMF) state H = [µ α(nα +nα )+µ nα ] (2)
06 inwhichelectron-likeandphoton-likeexcitationsemerge 2 −Xi Xα B c,i h,i F c,i
/ if the fermion occupation is nearly half filling [5]. At the U
at exact half filling, this UMF state turns to a long range + B2B α(α−1)(nαc,i+nαh,i)+UBF αnαc,i,
m ordered state, the checkerboard crystal of ’electrons’ [6]. Xi Xα Xi Xα
We show that this UMF state can only be stable if the
- and
d bosons are in a Mott insulator (MI) ground state.
n
We suggest an experiment to create an ’external field’ H = t c† h h† c
o by changing the depth of the fermion’s optical potential 4 − F α,i α,i β,j β,j
c Xhiji Xα,β
: [7]. The responsefunction ofthe mixtureto the external
v field may be measured by the density distribution image tB √α+1 β+1 (3)
i −
X in a time-of-flight of the mixture cloud. The behavior Xhiji Xα,β p
r of the response functions may be used to identify the ( h† h +c† c )(h† h +c† c ),
a electron-like excitation. We expect the ’electron’ Fermi α+1,i α,i α+1,i α,i β,j β+1,j β,j β+1,j
surfacecanbeobservedbythetime-of-flightexperiment,
where nα = c† c and nα = h† h . We explain
which has been used to observe the Fermi surface of the c,i α,i α,i h,i α,i α,i
briefly the derivation of this slave particle Hamiltonian.
pure cold Fermi atoms [8].
The state configurations at an arbitrary given site con-
The Bose-Fermi Hubbard Hamiltonian we are inter-
sists of α,s α = 0,1,2,... ; s = 0,1 where α and s
ested in reads {| i | }
arethebosonandfermionoccupations,respectively. The
H = (t a†a +t f†f ) (µ na+µ nf) Bose and Fermi creation operators can be expressed as
− B i j F i j − B i F i a† = √α+1[α+1,0 α,0 + α+1,1 α,1],f† =
Xhiji Xi α | ih | | ih |
αP,1 α,0. The mapping to the slave particle reads
+ UB2B Xi nai(nai −1)+UBF Xi nainfi, (1) |Pfαer,αm0|iion→i(hCh†αF,)||α[6,1]ian→d ch†αα .thWe eslacvaellbcoαsotnh.e cTomheponsoitre-
2
malized condition (α,0 α,0 + α,1 α,1) = 1 im- frequencies. Near the critical point where χh,c 0,
plies a constraint Pαα(n|αc +ihnαh)|= 1| atiehach|site. The one can expand Dαβ by χh,c . For α =αβ,iβj,≈the
sbeclαieaϕ,rviice→cαo,pnie,asiherαtrαϕiv,ccialαet,is→io,nha.rαei,PsSiieϕi→nihacαee,iUiαt.hϕ(e1hT)αsh,leigaavreguefllgopeebcatarsstlyitcmhUleem(1ptee)atcrrhystynicmiclqeαmu,nieeutem→rsy-- mχhαeβa,nij fi=eltdFχeqcβuα,ajtiiµnoααhn−−is6=µnjaββhrewχitcαhβα,niβjαh,ij== [teF−χβhβµααB,j−iµn16αFαc]−−−µn1βFβc,nαcan=d
B B
sentiallyworksinthestrongcouplingregion,wefocuson [e−βµαF +1]−1. The solutions of these mean field equa-
UBB/(6tB) 1. tions can only exist in the weak coupling limit, i.e.,
≫
There are two types of four slave particle terms in µα t : Tαβ = t . Therefore, in the parameters
F,B ≪ F c F
H4, the tB-terms and tF-terms. We first neglect the we are considering, χh,c = 0 for α = β. For α = β,
t -terms in the mean field level of the CF. To decou- αβ,ij 6
B the mean field equations are Tαχc = Tαχh and
ple the t -terms , we introduce Hubbard-Stratonovich c αα,ij F αα,ji
F
fields ηˆαc,βh,ij and χˆcα,βh,ij. The partition function is given Tcαχhαα,ij =TBαχcαα,ji with TFα,B =tFnαc,h(1∓nαc,h). The
by Z = DχˆDηˆDc¯DcDh¯DhDΛ e−Seff where the effec- critical temperatures are then given by Tcα = TFαTBα.
tive actiRon reads Below Tcα, the minimized free energy includingpvariables
χh,c is F t (τα)2(S2α)2 where τα = Tcα−T and
1/T αα ∝ − F α S4α T
Seff[χˆ,ηˆ,c,h,Λ]= dτ H2+ iΛi S2α,S4α arethefraPctionsofnon-zerotermscorresponding
Z0 (cid:26) Xi otopttihmealseχcocnd2ordτeαrSaαn/dSαfo.ur order of |χαα| [13]. The
+ c¯α,i(∂τ iΛi)cα,i+h¯α,i(∂τ iΛi)hα,i We con|sαidαe|r a∝n inte2ger4bosonfilling na =1 for U¯
(cid:20) − − (cid:21) BB
Xi Xα 1. Since nα6=1 and nα6=1 in this region are very sma≫ll,
h c
+ t [ηˆh (χˆc c¯ c ) χˆc χˆh there are no solutions of the mean field equations with
F βα,ji αβ,ij − α,i β,j − αβ,ij βα,ji
hijXi;α,β Tcα6=1 ≥ 0 for the critical temperature equations. Thus,
all slave bosons and CFs with α = 1 are confined. Only
+ηˆαcβ,ij(χˆhβα,ji−h¯β,jhα,i)]+tB terms(cid:27), (4) Tc1 >0 canbe found. BelowTc1, t6he CF c1 andthe slave
boson h are deconfined in the mean field sense. In the
1
where Λi is a Lagrange multiplier for the constraint inset of Fig. 1, we plot Tc1 for a set of given parameters.
α(nαc,i + nαh,i) = 1. Rewriting ηˆαc,βh,ij = ηαc,βh,ijeiAij, The curve Tp(UBB) is corresponding to µ1F(Tp)=0, i.e.,
the effective chemical potential of c vanishes.
χPˆcα,βh,ij =χcα,βh,ijeiAij and Λi =Λ+A0,i, A0i and Aij are We now go to concrete mean fie1ld solutions and fo-
U(1) gauge field corresponding to the gauge symmetry
cus on the near half filling of the fermions. Because we
[12]. Before going to a mean field state, we first require
workinathree-dimensionallattice,thefluxquantapass-
the mixture is stable against the Bose-Fermi phase sep-
ing a cubic cell are zero. Thus, there is no a flux mean
aration. For example, it was known that the mixture
field state. (The flux mean field state may exist in a
is stable in the MI phase if 4πt sin(πnf)U > U2
F BB BF two-dimensionalBose-Fermi Hubbard model.) Two pos-
[4]. Near the half filling of Fermi atoms, this condition
sible solutions are the dimer and uniform phases. In a
is easy to be satisfied. Fixing a gauge, ηˆh ηh ,
αβ,ij ≈ αβ,ij cubic lattice, each lattice site has six nearest neighbor
χˆh χh andΛ Λ(whichisasaddlepointvalue
αβ,ij ≈ αβ,ij i ≈ sites. A pair of slave boson and CF located at the near-
ofΛi). This is correspondingto a meanfield approxima- est neighbor sites may form a bond. The dimer phase
tion. The effective mean field action is given by means for any given site, only one bond ended at the
site is endowedwith a non-zeroχh,c and other five bond
αα
S = iN βΛ+ dτ (t χc χh ) (5) carry χh,c = 0. In the uniform phase, each bond is en-
MF s Z (cid:26)Xhiji F Xαβ αβ,ij βα,ji dowed wααith the same real values of χcα,h if a CF (i.e.,
a†f† 0 = c† vac ) is surrounded by six slave bosons
+ Xi6=j Xαβ(c¯α,i(Dcαβ)−ij1cβ,j +¯hα,i(Dhαβ)−ij1hβ,j)(cid:27) (aei.CgiF.,|aci†jh|e0cik=er1bih|o†1aj|rvdiaccir)y.stIanltshteatfeer[m6]i.onSlhigahlftlfiyllainwga,ythfrisomis
the half filling, this is a UMF state with h -c bond. In
where (Dαβ)−1 = (∂ µα)δ δ t χh δ 1 1
c ij τ − F ij αβ − F ~τ αβ,ji j,i+~τ the uniform phase, the dispersions of the CFs and slave
w(Ditαhβµ)−αF1===µB(∂α+µµFα−)δUδB2Bα(αt−1)−χcUPBFδα−iΛ. wainthd bosons are ξαc,h(k)=tF|χhα,c|| icoski|.
h ij τ − B ij αβ− F ~τ αβ,ji j,i+~τ At the fermion half filling,Pthe mean field free energy
µαB = µBα− UB2Bα(α−1)−iΛ. (~τPis the unit vector of favorsforthedimerstate. However,asweshallseesoon,
the lattice.) intheMottinsulatorphaseofthebosons,thebosonhop-
The mean field equations are χcαβ,ij = hc†α,icβ,ji = ping term (tB-term) we have neglected at the mean field
T Dαβ(p ),χh = h† h = T Dαβ (ω ) levelwillcontributeanearestneighborrepulsionbetween
n c,ij n αβ,ij h α,i β,ji n h,ij n CFsorslavebosonsifU >U . Thisrepulsionpoten-
whPere Dcα,βij(pn) = dτeipnτDcα,βij(τ) andPDhα,βij(ωn) = tialwillraisetheenergyBoBfdimeBrFphasewhiletheuniform
dτeiωnτDhα,βij(τ) wiRth pn and ωn the Fermi and Bose phaseis notaffected. Thus,forourpurpose,we focuson
R
3
the uniform phase below. lengthlimit(k 0),integratingouth firstandkeeping
1
→
In the mean field approximation, we neglect the bo- the Gaussian fluctuations of the gauge field, an effective
son hopping term and the gauge fluctuations , action in continuum limit is given by
0i ij
A A
which must be considered if the mean field state is
Hstuabblbea.rdW-SterafitrosntovdiecahlfiwelidthΦtih=e tBα-t√erαm+. 1I[cn†αtr+o1d,iucαce,i+a S[c1,Aµ]=T Xn Z d3k(cid:20)c¯1(ipn−µ1c +ie0A0)c1(7)
h† h ] to decouple t -term.PΦ may be thought as 1 1
α+1,i α,i B i + c¯ (k +ie A )2c + A A ΠB (k,ω ) ,
the order parameter field of the Bose condensation. The 2m 1 a 0 a 1 2 µ ν µν n (cid:21)
c
phase diagramof the bosonmay be determined by mini-
mizing the Landaufree energy associatedwith the order where e = J , A = /e , m 1/(t χcδ) is the
parameter 0Φ 0 . The vanishing of the coefficient of 0 q4π µ Aµ 0 c ∼ F 1
the secondhor|deir|tierm in the free energy gives the phase effective mass of CF. The coupling constant 2πe20 = J/2
is a small quantity means that the Gaussian approxi-
boundary[9,10]. Ithastopointoutthatherethefluctu-
mation to the gauge field is reasonable. ΠB is the re-
ation has been neglectedand a cut-offto the type of µν
the slaAv0eibosons has to be introduced. However, the ex- sponse function by integrating over h1. The density-
density response function is given by [ΠB(ω,k)]−1 =
periencetoworkoutthepureBosephasediagramshowed 00
[ΠB(0)(ω,k)]−1 + V(k) where V(k) = J (1 k2/2)
thattheseapproximationscouldbeacceptable[9,10,11]. 00 a −
As expected, the phase diagram of the constitution is the Fourier component of the interacPtion (6) and
boson consists of the Bose superfluid (BSF), the normal ΠB(0)(ω,k) is the free boson response function. The
00
liquid and the MI, in which the MI phase only exists in current-current response function has a form ΠB =
ij
the zerotemperatureandanintegerbosonfillingfactor. (δ k k /k2)ΠB+(k k /k2)ΠB. Inthelongwavelength
IntheinsetofFig. 1,weshowthebosonphasediagramin limiji−t, tihej transvLerse ipajrt ΠBT= ω2ΠB. The longitu-
thesameparametersasthoseintheCFmeanfieldphase T −k2 00
dinal part ΠB(k,ω) 0h 0 2 as k,ω 0. However,
diagram. In the Bose condensate, the boson number in L ∝ |h | 1| i| →
out of the BSF phase, h does not condense. Thus, A
1 µ
eachsiteistotallyuncertainty. Thismeansthevanishing
is a transverse field and the action (7) is very similar to
bond number or S S 0. Thus, the mean field
2 ≈ 4 ≈ electron coupled to a photon field. We, therefore, call
state is not stable in the BSF. In the MI phase of the
c and A the electron-like and photon-like excitation,
1 µ
bosons,ontheotherhand,thebosonnumberineachsite
respectively. Since the hopping of the Bose atom in the
isexactlyoneforna =1. Thus,themeanfieldstatesmay
optical lattice is short range, the repulsive interaction
be stable. Furthermore, the t -term induces a nearest
B between ’electrons’ is also short range. If it was possible
neighbor interaction between CFs or slave bosons. This
to design the hopping of bosons t 1/ i j, the
B,ij
may be seen by taking the t term as a perturbation if ∝ | − |
B interaction between ’electrons’ would be puprely coulom-
UBB and UBB −UBF are much larger than tB. To the bic, i.e., V(k) 1/k2. The stability of the UMF state
secondorderoftheperturbation,thet -termcontributes ∝
B againstto the gauge fluctuations may be seen after inte-
aneffectiverepulsiveinteractionbetweentheCFsorslave
grating out c field. This leads to a CF response func-
1
bosons in the nearest neighbor sites
tion [ΠF (ω ,k)]−1 =[ΠF(0)(ω ,k)]−1+[ΠB (ω ,k)]−1.
µν n µν n µν n
J =XhijiJn1cin1cj +const=XhijiJn1hin1hj +const′, (6) lTenhgetUhMlimFiits,isttaisbpleowsihtievneΠifFµ6νJ(0>,0m)πc>k2F10.wIhnertehekF1lonisgFweramvei
with J = UBB1(6UtB2B2BU−B2FUB2F). For the dimer state, this re- mhoolmdsenontulymwohfetnhethCeFfe.rSminiocne 1fi/llmincg∝istsFliδg,httlhyisacwoanydfitrioomn
pulsive potential contributes an energy J/2 to a pair of the half filling.
adjacent bonds. For the UMF state, if the fermion is in WehaveshownthestabilityoftheUMFstatenearthe
halffilling, the checkerboarddistributionofthe fermions fermionhalffilling when the bosons arein the MI phase.
(then CFs) makes no contribution to the energy. For We may figure out the phase diagram of the CF in Fig.
a doping δ, i. e., sJlightly away from the half filling, the 1. The mean field phase transition temperature T1 is
c
energy raises a small amount of the order Jδ. Thus, the suppressedgreatly to T1∗ which is determined by the T
c c
UMF state minimizes this induced nearest neighbor re- and T . The dash line is the estimated crossover line
p
pulsion. from the ’electron’ gas (the MI of bosons) to the Fermi
We have shown the mean field states are not stable liquidoftheconstitutionFermiatoms(thenormalliquid
in the BSF. Therefore, we focus on the MI phase of the of bosons as the incompressibility of the MI is gradually
bosons below and examine the gauge fluctuations. The disappearing).
zeroth component restores the exact constraint of The experimental implications of the UMF phase are
i0
A
one type of slave particles per site while restoresthe discussed as follows. We consider the ’electron’ response
ij
A
gauge invariance of the effective action (4). These fluc- to an external ’electric’ field, ’made’ by a change of the
tuations may destabilize the mean fieldstate. To see the lattice potential of the Fermi atoms. This technique has
stability ofthe UMFstate againstthe gaugefluctuation, been used to study the excitation spectrum of atom su-
one should integrate out h and c . In the long wave perfluid in optical lattices [7]. This disturbs the density
1 1
4
experiment. A better experimentally measurable quan-

tity is the visibility = nf(rmax)−nf(rmin) where r
1.6 12..64 Tc1 and rmin are chosenVsuch tnhfa(rtmatxh)e+nWf(armninni)er envelopmaixs
0.8 Tc Tp cancelled [14]. The difference between the disturbed
1.2 and undisturbed visibility may directly correspond to
1*
T(6t)B0.8 0.0C2ON4FINE6ME8NT Tc nth0fe(rrmeasxp)o+nsne0ff(urnmcinti)oinnbdeecnaoumseinantfo(rr.mTaxo)d+irencftl(yrmseine)th≈e
imageofthe fermioncloud,onemayuseamagneticfield
to separate the fermion cloud from boson cloud before
0.4
recordingthefermionimageinthetime-of-flightassplit-
’ELECTRON’
GAS
ting components in a spinor Bose atom condensate [15].
0.0
4 6 8 10 12
The Fermi surface of pure cold fermions has been ob-
UBB(6tB)
served in a recent experiment [8] by the time-of-flight
experiment. When the mixture in the UMF state, as
we have discussed in the previous paragraph, n (r) =
f
FIG. 1: The phase diagram of the CFs in the UBB-T plane (mf)w˜ (k)2n1(k= mfr). Therefore,itisexpectedthat
for na = 1, UBF = UBB/2, tF/6tB = √610∗, nf = 0.55. The insttea|dfofth|e Fcermi sutrfaceofthe free Fermi atoms,one
solidcurvesarethecriticaltemperatureT afterconsidering
c mayobservethe ’electron’Fermisurfaceofthe ’electron’
thegaugefluctuationandtheshrinkoftheCFFermisurface.
gas. Namely, in the image of the time-of-flight, most of
Fermi atoms are inside of area with r < k1 t/m but
| | | F| f
flofigFhetremxipeartiommesntw, itthheHdi′ffe=ren−cePbientwfiϕeein. dIinstuarbtiemdea-onfd- nFoertm|ri|at<om|ks.F).|t/mf ( |kF| is Fermi surface of the free
undisturbed fermion densities by external field is given
In conclusions, we showedthat there may be electron-
by n (r) n0(r) = (mf)w˜ (k = mfr)2δn (k = mfr)
f − f t | f t | f t like and photon-like excitations in mixtures of Bose-
where nf(r) is the image after the time-of-flight and Fermi atoms in optical lattices in which the bosons are
n (k) is the Fourier component of nf; t is the flying in the MI phase (U and U U t ) and the
f i BB BB − BF ≫ B
time, w˜ is the Fourier component of the fermion Wan- fermions are nearly half filling. (To avoid the demix-
f
nier function and m the Fermi atom mass. If T >T1∗, ing, 4πt sin(πnf)U > U2 .) It was suggested that
f c F BB BF
the density response of the system is simply given by through the time-of-flight experiment, the electron-like
the free fermion one, and then δn (k) ΠF(0). When response function and the ’electron’ Fermi surface may
the bosons are in the MI phase (nfa = 1∝), n0f0= n1 im- bemeasured. Anelectron-likeresponsefunctionalsoim-
plies δnf(k) = δn1(k). Thus, foriT < T1∗i, especciially pliesagaugefieldeffect. However,anexperimenthowto
c c
below the dash line in Fig. 1, δn (k) Π (k,0) with directly observe a ’photon’ was not designed yet.
f 00
∝
Π−1 =(ΠF(0))−1+(ΠB(0))−1+V(k). Then, this differ- This work was supported in part by Chinese National
00 00 00
encebetweentheresponsefunctionsmaybemeasuredin Natural Science Foundation and the NSF of USA.
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