Table Of ContentEfficiency for preforming molecules from mixtures of light Fermi and heavy Bose
atoms in optical lattices: the strong-coupling-expansion method
Anzi Hu1, J. K. Freericks2, M. M. Maśka3, C. J. Williams1
1Joint Quantum Institute, University of Maryland and National Institute
of Standards and Technology, Gaithersburg, Maryland 20899, USA
2 Department of Physics, Georgetown University, Washington, D.C. 20057, USA and
3Department of Theoretical Physics, Institute of Physics,
University of Silesia, PL-40007 Katowice, Poland
1
1
We discuss the application of a strong-coupling expansion (perturbation theory in the hopping)
0
forstudyinglight-Fermi-heavy-Bose(like40K-87Rb)mixturesinopticallattices. Weusethestrong-
2
couplingmethodtoevaluatetheefficiencyforpre-formingmolecules,theentropyperparticleandthe
n thermalfluctuations. Weshowthatwithinthestronginteractionregime(andathightemperature),
a thestrong-couplingexpansionisaneconomicalwaytostudythisproblem. Insomecases,itremains
J
valid even down to low temperatures. Because the computational effort is minimal, the strong-
3 coupling approach allows us to work with much larger system sizes, where boundary effects can
1 be eliminated, which is particularly important at higher temperatures. Since the strong-coupling
approachissoefficientandaccurate,itallowsonetorapidlyscanthroughparameterspaceinorder
]
tooptimizethepre-formingofmoleculesonalattice(bychoosingthelatticedepthandinterspecies
s
a attraction). Based on the strong-coupling calculations, we test the thermometry scheme based on
g thefluctuation-dissipationtheoremandfindtheschemegivesaccuratetemperatureestimationeven
- at very low temperature. We believe this approach and thecalculation results will beuseful in the
t
n design of the next generation of experiments, and will hopefully lead to the ability to form dipolar
a matterin thequantumdegenerate regime.
u
q
. I. INTRODUCTION low at low temperature, and never reaches appreciable
t
a sizes at higher temperatures, as the clouds become more
m
diffuse. On the other hand, if the mixture is first loaded
In recent years, there has been much interest in ultra-
- onto an optical lattice, the motion of the atoms can be
d cold polar molecules [1], as they have the promise for
morestronglyconfined,anditispossibletocreatealarge
n being a new state of quantum degenerate matter, with
o unique properties. In order to have a large dipole mo- area where exactly one atom of each species sits at the
c same lattice site, leading to a reduced three body loss
ment, the polar molecules must be in their rovibrational
[ [12] and almost unit efficiency [17] for pre-forming the
ground state, where further cooling can ultimately lead
molecules.
1 to quantum degenerate dipolar matter [2]. Such polar
v moleculescanhavelong-range,anisotropicorthree-body Whenmixturesof40Kand87Rbareloadedintoanop-
6 interactions[3],whichmayleadtonovelquantumphases tical lattice, the atoms of each species are influenced by
6
[4, 5] and new applications in quantum information sci- the optical lattice differently[18]. With the same optical
6
ence [6]. In most ultra-cold polar molecule experiments, lattice depth, the heavy atoms usually have much lower
2
. one starts with a mixture of ultra-cold gases of atoms of tunneling rate than the light atoms because of their sig-
1
different species, for example various isotopic combina- nificantly larger mass. In Ref. [17], it was shown that
0
tionsofKandRb[7–11]. Theseatomscanformaweakly for sufficient lattice depths, the hopping rate of Rb is
1
1 boundstatethroughamagneticfieldsweepovertheFes- more than an order of magnitude less than that of K. It
: hbachresonance[11,12]. Tocreatemoleculeswithsignif- is therefore reasonable to ignore the quantum effects of
v
icantly higher dipolar moments, the loosely bound Fes- the tunneling of the heavy bosonic atoms while allowing
i
X hbach molecules are coherently transferred to a ground the light fermionic atoms to hop between nearest neigh-
r statewithveryhighefficiencythroughstimulatedRaman bors (a classical effect of the motion of the Rb atoms is
a adiabatic passage (STIRAP) [13–16]. takeninto accountby averagingoverallenergetically fa-
vorabledistributions ofRbatoms). Suchsystemscanbe
Although the rate of transferring a Feshbach molecule
describedby the Fermi-Bose Falicov-Kimballmodel [19–
to the groundstate is veryhigh,the overallefficiency for
21]. Using this model, we quantitatively determine the
forming dipolar molecules is still low due to the low effi-
probabilityofhavingexactlyoneatomofeachspeciesper
ciency of forming the loosely bound Feshbach molecules
duringthefieldsweep. InRef.[15],thefermionic40Kand lattice site in order to optimize the formation of dipolar
the bosonic87Rb atoms are trapped by an optical trap. molecules.
TheefficiencytoformtheFeshbachmoleculedependson Forthe Falicov-Kimballmodel, the phenomenaofpre-
the phase-spacedensity of the two species. But, because forming molecules has been discussed for Fermi-Fermi
theFermicloudstopsshrinkingonceitreachesthequan- mixtures or Fermi-hard-core-Bosemixtures [22] on a ho-
tum degenerate regime, and the Bose cloud continues to mogeneous lattice and Fermi-Fermi mixtures in a har-
shrink as it Bose condenses, this phase space density is monictrap[23]. Inpreviouswork[17],weconsideredthe
2
Fermi-soft-core-Bosemixturesinaharmonictrapandde- peratureis further studied by applyingthe thermometry
termined the efficiency for pre-forming molecules as the proposal in Ref. [31] to the Fermi-Bose mixture. To
probability to have exactly one atom of each species per benchmark the accuracy,we compare the SC calculation
site. We used inhomogeneous dynamical mean-field the- with the IDMFT and MC calculations for all parame-
ory (IDMFT) and Monte Carlo (MC) techniques to cal- ters considered. Overall,we findexcellentagreementbe-
culate the efficiency as well as the density profile and tween the three methods. Such agreement even extends
the entropy per particle. Both of these methods have to the low temperature region when the interaction is
advantages and disadvantages. The IDMFT approach strong enough. This is particularlyuseful, given the fact
is approximate for two-dimensional systems, but it can that the SC expansion calculation is significantly faster
calculate both the efficiency and the entropy per parti- than the IDMFT and MC calculations. Such a speedup
cle. TheMCmethodisnumericallyexactafteritreaches makes it possible to consider much larger lattice sizes to
thermal equilibrium, but it can not calculate the contri- eliminate the boundary effects, to scan the large param-
butionstoentropycomingfromtheheavyparticles. Both eter space for optimal parameter regions for pre-formed
methods require large computational times to calculate molecules and to estimate the density fluctuations and
properties of a trapped system of reasonable size. Using other properties.
these methods, we have shown that the efficiency is sig- The paper is organized as follows: in Sec. II, we dis-
nificantlyincreasedbyfirstloadingontoanopticallattice cuss the Fermi-Bose Falicov-Kimball model and define
beforeformingthemoleculesandnearunitefficiencycan the efficiency for pre-forming molecules; in Sec. III, we
beachievedwithparametersthatarerealisticforcurrent discuss the formalism for evaluating the efficiency, the
experiments. entropy,and other related quantities; in Sec. IV, we dis-
The efficiency of pre-formed molecules is also likely to cuss our result for various parameters and benchmark
be affectedby the heating(the temperature increase)in- the SC expansion calculation with the IDMFT and MC
duced by loading onto an optical lattice [24–26]. Con- calculations; in Sec. V, we discuss the application of the
sidering that thermal fluctuations generally destroy the fluctuation-dissipationtheoremfordetermining the tem-
ordering and the localization of the particles, it is rea- perature and we present our conclusions in Sec. VI.
sonable to expect that the efficiency of having exactly
one Rb atom and one K atom per site should be re-
ducedifthetemperaturebecomestoohigh. Ontheother
II. FERMI-BOSE FALICOV-KIMBALL MODEL
hand, if the temperature is low enough, the presence of
the lattice significantly increases the efficiency, almost
to unity in the case of deep lattices. The temperature For mixtures of heavy bosons and light fermions, such
of the lattice system, however, remains difficult to mea- as 87Rb/40K mixtures, the hopping parameter for the
sure in experiment [27–31]. Instead, it is often assumed heavy bosons (87Rb) is usually more than an order of
that the process of loading atoms onto optical lattices magnitude less than the hopping parameter for the light
is adiabatic and therefore the total entropy of the sys- fermions (40K) when one takes reasonable lattice depths
tem is conserved [24–26, 32, 33]. Determined based on (greaterthan15Rbrecoilenergies)[17]. Inthis case,we
the thermal properties of the gas before adding lattices, canignorethequantum-mechanicaleffectsofthehopping
the entropy per particle is then used as an effective tem- ofthe heavy bosonsanddescribe suchmixtures with the
perature scale for the lattice system [25, 34]. There are Fermi-Bose Falicov Kimball model in the presence of a
also several proposals for directly determining the tem- trappotential. The Hamiltonianofthis modelis written
perature for systems of bosonic atoms [35–37], fermionic as
atoms [38] or the magnetic systems [39]. In Ref. [31],
a general thermometry scheme is derived based on the H =H0+Hh = H0j +Hh, (1)
fluctuation-dissipation theorem. Through quantum MC j
X
simulation, this proposal is shown to be applicable to
with
the non-interacting fermionic systems [31] and interact-
ing bosonic systems [40]. H = (V −µ )f†f +U f†f b†b
0j j f j j bf j j j j
In our paper, we discuss light-Fermi-heavy-Bose mix-
1
turesinopticallatticesbasedonthestrong-coupling(SC) + (V −µ )b†b + U b†b (b†b −1), (2)
j b j j 2 bb j j j j
expansionmethod (perturbationtheory in the hopping).
The calculation is oriented to develop an efficient way and
of estimating the efficiency of pre-forming molecules for
a given experimental system. With the SC expansion Hh =− tjj′fj†fj′. (3)
method, we obtain analytical expressions for the effi- jj′
X
ciency of pre-forming molecules, the entropy per parti-
cle and the local charge compressibilities. The behavior Here, j, j′ label the sites of a two-dimensional square
of the efficiency is studied both as a function of entropy lattice, with a lattice constant, a. The symbols f† and
j
perparticleandtemperature. Thedeterminationoftem- f denote the creationandannihilationoperatorsforthe
j
3
fermions at lattice site j, respectively. The symbols b† and
j
andb denotethecreationandannihilationoperatorsfor
j
the bosons at lattice site j, respectively. The fermionic ρf,j =hfj†fji= Wj(nb,j)nf,j(nb,j). (8)
operatorssatisfythe canonicalanticommutationrelation Xnb,j
{fj,fj†′} = δj,j′ and the bosonic operators satisfy the Here n is the occupation number oef bosons on site j,
canonical commutation relation [bj,b†j′] = δj,j′. The nb,j =0b,,j1,.... The coefficient Wj(nb,j) is the probability
quantity V is the trap potential, which is assumed to
j ofhavingexactly n bosons atsite j andthe coefficient
b,j
be a simple harmonic-oscillatorpotentialcenteredatthe
n (n )istheprobabilityforhavingonefermiononsite
f,j b,j
center of the finite lattice. We assume that the jth site
j for the occupation number n . The joint probability
b,j
has the coordinate (x ,y ), so that V can be written as
j j j ofhavingexactlyonebosonandonefermionatsitej can
e
be written as
~Ω 2
V =t x2+y2 , (4)
j 2ta j j Ej =Wj(nb,j =1)nf,j(nb,j =1), (9)
(cid:20) (cid:21)
(cid:0) (cid:1)
where Ω is the trap frequency. The quantity µ is the and the efficiency E is the averageof E over all sites,
f e j
chemical potential for fermions and µ is the chemical
b
potential for bosons. Combining the trap potential and jEj jWj(nb,j =1)nf,j(nb,j =1)
E = = . (10)
the chemical potentials, we can define an effective posi- N N
P P
tiondependent localchemicalpotentialfor the fermions, e
µ = µ −V , and for the bosons, µ = µ −V . U Itcanbe shownthatthe expressionfor theefficiency ob-
f,j f j b,j b j bf
tainedin this wayis the sameas fromEq. (5). Now,the
is the interaction energy between fermions and bosons
and U is the interaction energy between the soft-core efficiency is obtained directly from the density of bosons
bb
andfermions,whichcanbeeasilyderivedfromtheparti-
bosons. The symbol −tjj′ is the hopping energy for
fermions to hop from site j′ to site j. We consider a tionfunctionZ by taking derivativeswithrespectto the
appropriate chemical potentials,
general tjj′ for the formal developments in the earlier
part of the next section, but later specialize to the case
1∂ln(Z)
of nearest-neighbor hopping with amplitude t, which we ρ = , (11)
b,j
β ∂µ
will take to be the energy unit. We also set the lattice b,j
constant, a equal to one.
and
The efficiency for pre-forming molecules is defined as 1∂ln(Z)
ρ = . (12)
theaveragedjointprobabilityofhavingexactlyoneboson f,j β ∂µ
f,j
and exactly one fermion on a lattice site,
1 1 To study the behavior of the efficiency as a function
E = N hPˆ1j,1i= N Tr Pˆ1j,1e−βH , (5) of the entropy per particle, we evaluate the entropy per
Xj Xj (cid:16) (cid:17) particlebydividingthetotalentropybythetotalnumber
of particles,
with β = (k T)−1 the inverse temperature. We define
B
theprojectionoperatorPˆj forhavingexactlyoneboson s = S/(N +N )
1,1 b f
and one fermion at site j,
∂ln(Z)
= k ln(Z)−βk /(N +N ). (13)
B B b f
Pˆ1j,1 =|nb,j =1,nf,j =1ihnb,j =1,nf,j =1|, (6) (cid:18) ∂β (cid:19)
andN isthesmallervalueinthetotalnumbersofbosons It is worthwhile noticing that the formalism develop-
and fermions, N and N . In our case, we assume equal ment in this section is based on the grand-canonical en-
b f
numberofbosonsandfermions,thereforeN =N =N . semble. Thisensembleisappropriatebecauseweassume
b f
that in the lattice system both the energy and the num-
In general, one can obtain E directly from Eq. (5) berofparticlesfluctuate. Thismayseemincontradiction
for a readily diagonalized Hamiltonian. In our case, the with the use of the entropy per particle as an effective
efficiency E is derived by distinguishing the contribution temperature scale, because strictly speaking entropy is
from terms corresponding to n = 1 in the expression used as a parameteronly for the micro-canonicalensem-
b,j
for the density of fermions. We assume that the density ble. This contradiction is resolved because the entropy
of bosons and fermions at site j, ρ and ρ , can both per particle is assumed to be conserved during the pro-
b,j f,j
be written as a series in terms of the occupationnumber cess of turning on the optical lattice. It is a conserved
of bosons at site j in the following way, quantity when comparing the systems before and after
turning on the optical lattice, which is particularly use-
ρ =hb†b i= W (n )n , (7) ful from the experimentalpoint of view, since the exper-
b,j j j j b,j b,j
Xnb,j iments often start without the optical lattices. For the
4
lattice system itself, assuming it is in thermal equilib- expansion of U divided by Z(0),
rium,itismorereasonabletoconsideritwiththegrand-
canonicalensemble,allowingtheenergyandnumberfluc- 1 β β
tuations. The difference between the different ensembles Z(2) = 2Z(0)Tr"e−βH0ˆ0 ˆ0 dτ1dτ2TτHh(τ1)Hh(τ2)#.
of course is not problematic if we assume the system is
(18)
large enough to be in the thermodynamical limit, where
all three ensembles are equivalent. To simplify the notation, we introduce µ¯f,j(nb,j) to
representthenegativeofthefermionicpartoftheHamil-
tonian H [Eq. (2)] when there is a fermion at site j,
0j
µ¯ (n )≡µ −V −U n , (19)
f,j b,j f j bf b,j
III. STRONG-COUPLING EXPANSION
FORMALISM and µ¯ (n ) for the negative of the bosonic part of the
b,j b,j
Hamiltonian H ,
0j
Inthissection,weexplaintheSCexpansionformalism.
µ¯ (n )≡(µ −V )n −U n (n −1)/2. (20)
We first discuss the evaluation of the partition function b,j b,j b j b,j bb b,j b,j
Z,approximatedbythesecond-orderexpansioninterms
Both µ¯ and µ¯ depend on the number of bosons at
f,j b,j
of the hopping, H . From the partition function, we de-
h site j. The effective fugacities for bosonic and fermionic
rive the expressions for the density of fermions shown in
particles can then be written as the exponential of µ¯
f,j
Eqs. (32) to (34), the density of bosons in Eqs. (35) to
and µ¯ respectively,
b,j
(37), the efficiency in Eqs. (38) to (40) and the entropy
per particle in Eqs. (44) to (47). For readers who pre-
fer to see the final expressions, we suggest skipping the
φ (n )=exp[βµ¯ (n )], (21)
following derivation and referring to the equations listed f,j b f,j b,j
above for the corresponding quantities.
and
The evaluation of the partition function in the SC
approach starts with the exact solution of the atomic φ (n )=exp[βµ¯ (n )]. (22)
b,j b b,j b,j
Hamiltonian H . Hence, we use an interaction picture
0
with respect to H0, where for any operator A, we de- The atomic partition function Z(0) can then be written
fine the (imaginary) time-dependent operator A(τ) = in terms of the effective fugacities as,
eτH0Ae−τH0. The partitionfunctionis writtenusingthe
standard relation, Z(0) = Π Z(0), (23)
j j
Z =Tr e−βH =Tr e−βH0U(β,0) . (14) where Z(0) is the atomic partition function at site j,
j
(cid:0) (cid:1) (cid:0) (cid:1)
β
eHrearteo,rUw(iβth,0T)τ=beTiτngextpheh´t0imHe-ho(rτd)edrτinigisotpheeraetvoorlufotrioinmoapg-- Zj(0) =Xnb,jφb,j(nb,j)(1+φf,j(nb,j)). (24)
inary times. Expanding the exponential in U(β,0) up to
secondorderinH (τ)andevaluatingtheresultingtraces
h Nowweevaluatethesecondterminthepartitionfunc-
with respect to equilibrium ensembles of H , we have
0 tion, Z(2) of Eq. (31). To satisfy the total number
conservation, only terms with j = k′ and j′ = k in
U(β,0) ≃ 1+
H (τ )H (τ ) are non-zero after the trace and Z(2) is
1 β β rehduc1ed ihnto2a sum of products of the fermionic annihi-
+ dτ dτ T H (τ )H (τ ). (15)
2ˆ 1ˆ 2 τ h 1 h 2 lation and creation operators at the same site,
0 0
Here, the first order correction to the partition function 1 β β
vanishes because the hopping connects different sites. Z(2) = 2ˆ ˆ dτ1dτ2 tjktkj
Substituting Eq. (15) into Eq. (14), we can write the 0 0 jk
X
partition function as, ×Tr Tτe−βH0jfj†(τ1)fj(τ2) /Zj(0)
Z =Z(0)(1+Z(2)), (16) ×TrhTτe−βH0kfk(τ1)fk†(τ2)i/Zk(0). (25)
where Z(0) is the partition function in the atomic limit h i
Using the cyclic permutation relationship of the trace,
(t=0),
the products can be represented by the local atomic
Z(0) =Tr e−βH0 , (17) Green’s function,
and Z(2) corresponds to th(cid:0)e secon(cid:1)d-order term in the Gjj(τ)=−Tr Tτe−βH0jfj(τ)fj†(0) /Zj(0), (26)
h i
5
and Z(2) is expressed as integrations of the atomic SubstitutingEq.(29)intoEq.(26),weobtaintheatomic
Green’s functions in terms of their relative times, Green’s function in terms of the effective fugacities as,
1 β β
Z(2) =− dτ dτ t t G (τ −τ )G (τ −τ ).
2ˆ ˆ 1 2 jk kj kk 1 2 jj 2 1
0 0 jk
X
(27)
Solving the Heisenberg equation of motion for the an- − φb,j(nb)eτµ¯f,j, τ >0
nihilation operator fj(τ), G (τ)= nb Zj(0) (30)
jj Pφb,j(nb)φf,j(nb)eτµ¯f,j. τ <0
∂fj(τ) =eH0τ[H0,fj]e−H0τ (28) Pnb Zj(0)
∂τ
one easily finds the expression for the annihilation oper-
ator f (τ) in the interaction picture,
j
We nowperformthe integrationoverτ andτ inZ(2)
1 2
fj(τ)=eµ¯f,j(nb,j)τfj(0), (29) and obtain the final expression for Z(2),
1 φ (n )φ (n )β[φ (n )−φ (n )]
Z(2) = t t b,j b,j b,k b,k f,j b,j f,k b,k . (31)
jk kj
2 Z(0)Z(0) µ¯f,j(nb,j)−µ¯f,k(nb,j)
Xjk nbX,jnb,k j k
Note that the partition function we derived here is not where ρ(0) is the density of fermions in the atomic limit,
f,j
limited to the case of nearest-neighbor hopping with a
utonidfoersmcribheophpoipnpginpgarbaemtweteeernta.rbEiqtr.ar(y31s)itceasnjbaendapkpalinedd ρ(0) = 1 ∂In Zj(0) = nb,jφf,j(nb,j)φb,j(nb,j), (33)
thehoppingparametertjk canvaryoverdifferentsitesof f,j β ∂(cid:16)µf (cid:17) P Zj(0)
the lattice.
and ρ(2) is the total contribution to the density at site j
f,j
from particles hopping from all possible sites,
Observables are evaluated by taking appropriate
φ (n )φ (n )
derivatives of the partition function. In calculating the ρ(2) = t t b,j b,j b,k b,k
derivatives, we truncate all final expressions to include f,j jk kj ( Z(0)Z(0)
only terms through the order of t2 . Also note that be- Xk nbX,jnb,k j k
cause sites j and k are differentjskites, we do not nor- (1−ρ(j0,k))φf,j(nb,j)+ρ(j0,k)φf,k(nb,k)
× β
mally have denominators equal to zero in Eq. (31), but µ¯ (n )−µ¯ (n )
" f,j b,j f,k b,j
inanycase,the formulasarealwaysfiniteascanbe veri-
fied by l’Hôpital’s rule. During numericalcalculations of φf,k(nb,j)−φf,j(nb,k)
+ . (34)
the observables,the denominator, µ¯f,j(nb,j)−µ¯f,k(nb,j), [µ¯f,j(nb,j)−µ¯f,k(nb,j)]2#)
can become too small and cause numerical errors. In
our calculations, we use the Taylor expansion in terms Similarly, the density of bosons at site j is written as a
of µ¯f,j(nb,j)−µ¯f,k(nb,j) around zero when the absolute sum of the atomic density and the hopping contribution
value of µ¯f,j(nb,j)−µ¯f,k(nb,j) is less than 10−5. as,
1∂ln(Z)
ρ = =ρ(0)+ρ(2), (35)
b,j β ∂µ b,j b,j
b,j
The density distribution is evaluated by taking the
derivative of the partition function with respect to the where
appropriatelocalchemicalpotential[Eqs.(12)and(11)].
Fstoitruttheestdwenostietyrmosfcfeorrmreisopnosnadtinsgitteojd,etrhievaetxivpersesfrsioomnZco(n0)- ρ(0) = 1 ∂In Zj(0) = nb,jnb,jφb,j(nb,j)[1+φf,j(nb,j)],
and Z(2), b,j β ∂(cid:16)µb,j (cid:17) P Zj(0)
(36)
1 ∂In(Z)
ρ = =ρ(0)+ρ(2), (32)
f,j β ∂µ f,j f,j
f,j
6
and the change of the global chemical potentials,
φ (n )φ (n ) ∂2lnZ
ρ(2) = t t (n −ρ(0)) b,j b,j b,k b,k κg =
b,j jk kj " b,j b,j Z(0)Z(0) j ∂(µf,j +µb,j)∂(µb+µf)
Xk nbX,jnb,k j k
β[φf,j(nb,j)−φf,k(nb,k)] = β h fj†fj +b†jbj Nˆf +Nˆb i
× . (37)
µ¯f,j(nb,j)−µ¯f,k(nb,j) (cid:21) −hhf(cid:16)†f +b†b ihN(cid:17)ˆ(cid:16)+Nˆ i .(cid:17) (42)
j j j j f b
i
And the local compressibility, or the onsite number fluc-
The expression for the efficiency is obtained from the
tuation, is determined from the derivatives with regard
densitydistributionsofthefermionsandbosons. Similar
to the local chemical potential,
to the expressionfor the densities, the efficiency consists
of two terms, one corresponding to the atomic limit and ∂2lnZ
onecorrespondingtothecontributionsfromthehopping, κl =
j ∂2(µ +µ )
b,j f,j
Ej =Ej(0)+Ej(2), (38) = β h fj†fj +b†jbj 2i−hfj†fj +b†jbji2 . (43)
where (cid:20) (cid:16) (cid:17) (cid:21)
Boththeglobalandlocalcompressibilitiesarederivatives
E(0) = φb,j(nb,j)φf,j(nb,j)|nb,j=1, (39) ofthedensitydistributionsandcanbeobtainedfromthe
j Z(0) density expressions above.
j
Finally, we obtain the expression for the entropy per
and particle defined in Eq. (13) by averaging the total en-
tropy of the system and we again write the entropy per
φ (n )φ (n )
E(2) = b,j b,j b,k b,k × particle in terms of the atomic limit expression and the
j Z(0)Z(0) contributions from the hopping,
Xk nb,Xj,nb,k j k
φb,j(n′b,j)φf,j(n′b,j) s = 1 S(0)+ 1 S(2). (44)
− N j N j
Zj(0) !n′ =1 Xj Xj
b,j
×β[φf,j(nb,j)−φf,k(nb,k)] Here Sj(0) is the entropy at site j in the atomic limit,
µ¯ (n )−µ¯ (n )
f,j b,j f,k b,j
+δ β φf,j(nb,j) Sj(0)/kB = ln Zj(0) −βǫj, (45)
nb,j,1(cid:20) µ¯f,j(nb,j)−µ¯f,k(nb,j) (cid:16) (cid:17)
where the parameterǫ correspondsto the onsite energy
φ (n )−φ (n ) j
+ f,k b,j f,j b,k . (40) at site j in the atomic limit,
[µ¯f,j(nb,j)−µ¯f,k(nb,j)]2#)
∂ln(Z(0))
ǫ =
j
∂β
For the trapped system, the local chemical potential 1
= {µ¯ (n )φ (n )[1+φ (n )]
µj includes both the globalchemicalpotential µ and the Z(0) b,j b,j b,j b,j f,j b,j
trappingpotentialVj. Thederivativeswithregardtothe j Xnb,j
local chemical potential or the chemical potential leads +µ¯f,j(nb,j)φb,j(nb,j)φf,j(nb,j)}. (46)
to different physicalquantities. For the Fermi-Bose mix-
The averagedcontributions from the hopping at site j is
ture consideredhere,the cross-derivativesshouldalsobe
evaluated. Specifically, the derivative with regardto the S(2),
j
global chemical potential (µ +µ ) corresponds to the
b f
total number fluctuations, ∂ln(1+Z(2))
S(2)/k = ln(1+Z(2))−β
j B ∂β
∂2lnZ
κ = β2 φb,j(nb,j)φb,k(nb,k)
∂2(µ +µ ) = −
b f 2 Z(0)Z(0)[µ¯ (n )−µ¯ (n )]
= β h Nˆ +Nˆ 2i−hNˆ +Nˆ i2 . (41) Xk nbX,jnb,k j k f,j b,j f,k b,j
f b f b ×{[φ (n )−φ (n )]
f,j b,j f,k b,k
(cid:20) (cid:16) (cid:17) (cid:21) ×[µ¯ (n )+µ¯ (n )−ǫ −ǫ ]
b,j b,j b,k b,k j k
Here we define the total number operators, Nˆ =
f +µ¯ (n )φ (n )−µ¯ (n )φ (n )}(.47)
f,j b,j f,j b,j f,k b,k f,k b,j
f†f and Nˆ = b†b . The global compressibil-
j j j b j j j
ity is introduced as the response of the local density to This ends the discussion on the derivation of the SC
P P
7
expansion method formulas. In general, the expressions fermions are completely localized and the only density
obtained above are accurate in the case when the hop- fluctuations are due to thermalfluctuations. For the low
ping is much smaller than interaction strength and the density case considered here, the bosons always form a
temperatureis veryhigh(βt is small). Inthis parameter plateau of unit filling at the center of trap at low tem-
region, the SC method can evaluate physical quantities, perature and the fermions are attracted by the bosons
like the density distribution, efficiency, compressibility one by one and form an almost identical plateau. The
and entropy, very efficiently. The total number of parti- efficiency therefore always convergesto unity as temper-
cles is fixed by varying the chemical potentials, µ and ature deceases. In Fig. 1, we indeed find the efficiency
b
µ . To maximize the efficiency and reduce three body from the SC calculation always goes to one at low tem-
f
loss, we consider the low density region with attractive peratures. The convergence to unity is also true for the
interspecies interactions and repulsive bosonic interac- IDMFT and MC calculations for all the parameters ex-
tions. For other strong-coupling regions, the formulas cept for U /t = −2 and U /t = 5.7. That’s where the
bf bb
developed above are equally applicable but not further SC calculationdiffers fromthe IDMFT and MC calcula-
discussed in this paper. tion. It is reasonable to assume that the SC calculation
can be applied to the region where the ground state of
the system is a localized, Mott insulator like state.
IV. RESULTS The SC calculation of the entropy per particle is also
compared with the IDMFT and MC calculations for all
the parameters using Eqs. (44-47). The conclusion of
A. Comparison with the IDMFT and MC
calculations the comparisonis similar with the efficiency calculation,
thattheSCcalculationisaccurateexceptforU =−2t.
bf
In Fig. 2 we use one example, U = 11.5t and U =
For a perturbative method like the SC expansion bb bf
−16t,to representallthe caseswherethe SC calculation
method, it is always necessary to determine the param-
agreeswith the IDMFT calculation. As the temperature
eter regions where the approximation is valid. Here, we
increases, the entropy per particle starts to saturate at
use the previous results obtained from IDMFT and MC
around ∼ 2.3k . In the next section, we will show that
methods [17] as a reference to determine the accuracyof B
this saturation is actually the result of finite-size effects.
the SC calculation. It is also worthwhile to notice that
In Fig. 3, we show the behavior of the efficiency as a
the three methods require substantiallydifferent compu-
function of the entropy per particle. This figure can be
tationaltimes. TheSCcalculationusuallytakeslessthan
compared with Fig. 2 in Ref. [17], where the IDMFT
1CPUhourwhileforthesamesystemtheIDMFTcalcu-
calculation is discussed. We verify the findings from the
lationtakesonthe orderof105 CPU hours. We consider
previous work that for strongly attractive inter-species
all the parameters used in the previous work [17]. The
interactions, an efficiency of 100% can be achieved at
latticeis50×50squarelatticewiththetrapfrequencyΩ
forbothspeciesfixedat~Ω/2ta=1/11. Theparameters low temperature (low entropy) region. For an entropy
per particle around 1k , a 80% efficiency can still be
U and U are chosen based on a typical experimen- B
bf bb
reached. This efficiency is much higher than what has
tal setup : U /t = −8, −12, −16 for U /t = 11.5 and
bf bb
been achieved in experiment [15].
U /t=−2, −6, −10forU =5.7. Thetotalnumberof
bf bb
In the following discussion on the SC calculation re-
bosons and fermions are set to be 625. We consider the
sult, we no longer consider the case of U = −2t. This
temperature range 0.05t/k to 20t/k . bf
B B
is also based on the considerationthat the interactionof
In Fig. 1, we show the efficiency as a function tem-
U = −2t is too weak to achieve the desired high effi-
perature calculated with the three methods. Overall,we bf
ciencyofpre-formedmoleculesandthereforeisnotinthe
find excellent agreement between the SC result and the
parameter region of the main interest in this paper.
IDMFT and MC calculations and it is clear that high
(unit)efficiencycanbeachievedwhenthetemperatureis
low (T ∼0.1t/k ) and the interactionis large compared
B
with t. In the case of U = −2t and U = 5.7t, the B. Finite-size effects
bf bb
SCcalculationstartstodeviategreatlyfromtheIDMFT
and MC calculation when T ≤1t/kB. It is worth noting In our calculations, we always assume a hard-wall
thatforT >1t/kB,theSCcalculationsagreenicelywith boundary condition at the edge of the lattice. In ex-
theothermethodsevenforarelativelyweakcross-species periments, however, the atoms are confined only by the
interaction, Ubf =−2t. trappingpotential. Thisadditionalconfinementimposed
The difference between the SC calculation and the by the boundary condition can potentially affect the ac-
other twomethods canbe understoodfromthe factthat curacy of our calculation. This finite-size effect can be
the SC method is a perturbative method based on the neglectedifthesystemissolargethattheatomstrapped
atomiclimitoftheHamiltonian,t=0andthattheprop- by the trapping potential almost never reach the edge
erties derived from the SC expansion are dominated by the system. This, however,is notalwaysthe casefor the
the atomic-limit behavior with relatively small correc- 50×50 lattice. This problem is difficult to address with
tions from the hopping. In the atomic limit, bosons and the IDMFT and MC methods, because of the high com-
8
SC MC IDMFT
1 1 1
0.8 0.8 0.8
y y y
nc0.6 nc0.6 nc0.6
e e e
Effici0.4 Effici0.4 Effici0.4
0.2 (a) Ubb=5.7t,Ubf=−2t 0.2 (b) Ubb=5.7t,Ubf=−6t 0.2 (c) Ubb=5.7t,Ubf=−10t
0 0 0
10−1 100 101 10−1 100 101 10−1 100 101
k T/t k T/t k T/t
B B B
1 1 1
0.8 0.8 0.8
y y y
nc0.6 nc0.6 nc0.6
e e e
Effici0.4 Effici0.4 Effici0.4 (f) U =11.5t,U =−16t
0.2 (d) U =11.5t,U =−8t 0.2 (e) U =11.5t,U =−12t 0.2 bb bf
bb bf bb bf
0 0 0
10−1 100 101 10−1 100 101 10−1 100 101
k T/t k T/t k T/t
B B B
Figure1: (Coloron-line)EfficiencyE asafunctionoftemperaturecalculatedbytheSC(redcross),IDMFT(bluetriangle)and
MC(greencircle) methods. Theinteraction parameters,U andU ,areshownineachplot. In(a),theSCcalculation differs
bb bf
from theothertwomethodsforT <1t/kB. Forthisregion,theSCexpansionformulasderivedherearenolongeraccurate. In
(b)-(f),all threemethodsgivealmost identical results. Thesecalculations also showthat almost 100% efficiency isreached for
relatively strong attraction, Ubf ≥−6t, at low temperature, T <t/kB.
3 sizeattwotemperatures,T =1t/kB (a)andT =20t/kB
)B U =11.5t,U =−16t (b). Here Fig. 4 (a) represents the scaling behavior in
e (k bb bf the low-temperatureregion,where there is no significant
cl 2 difference between different lattice sizes and Fig. 4 (b)
arti represents the scaling behavior in the high-temperature
p
er SC region, where the system of small lattice size is highly
p
y 1 IDMFT affectedbytheboundaryeffect. Notethatthehorizontal
p
o axes are different scales in the two panels. The param-
ntr eters used in the plots are U = −16t and U = 11.5t.
e bf bb
0 We find similar behavior of the density profile for all the
10−1 100 101 other parameters.
k T/t
B In Fig. 5 (a), we show entropy per particle as a func-
tionoftemperatureatdifferentlatticesizes. Inthisplot,
Figure 2: (Color on-line) (a) Entropy per particle as a func- we find that for small lattices, the entropy per particle
tion of temperature T. The SC calculation is marked with becomes saturated at high temperature, while for large
red crosses and the IDMFT calculation by the blue line. We lattices it keep increasing as the temperature increases.
find excellent agreement between the SC calculation and the
The saturation is understood as the result of the finite-
IDMFT calculation.
size effects. When the temperature is high, atoms tend
to expand to a larger area in the trap, which leads to
a large cloud size and higher entropy. When atoms ex-
putational costs. The SC method, on the other hand,
pandtotheedgeofthelattice,thepossibleoccupiedsites
can calculate much larger systems for a fraction of the
are now constrained and the entropy stays similar even
cost.
though the temperature increases, hence the saturation.
In this section, we discuss our calculation for different
Whenthelatticeissufficientlylarge,atomscanfreelyex-
latticesizesanddiscussfinite-sizeeffectsfordifferentlat-
pandasthe temperatureincreasesandthe entropykeeps
tice sizes. To benchmark the SC calculations, the trap
increasing.
frequency and the total number of particles are fixed for
all different lattice sizes. We assume the largest lattice The confinement of the atomic cloud in high tempera-
sizesaresufficienttoneglectthefinite-sizeeffects. InFig. ture also affects the efficiency calculation. In Fig. 5 (b),
4, we show the density profile as a function ofthe lattice we find that the efficiency saturates to a higher value
9
1 N=50 N=100 N=200 N=300
(a) U =11.5t
bb 1
0.8 (a) T=1 t/k
0.8 B
y
c0.6 0.6
cien Ubf=−16t (SC) ρ(r)f0.4
Effi0.4 Ubf=−8t (SC)
0.2
U =−16t (IDFMT)
0.2 bf 0
U =−8t (IDMFT) 0 5 10 15 20 25 30
bf radial distance
00 0.5 1 1.5 2 2.5 1
(b) T=20 t/k
Entropy per particle (k ) 0.8 B
1 B
0.6
0.8 (b) Ubb=5.7t ρ(r)f0.4
0.2
cy0.6
en U =−10t (SC) 0
ci bf 0 25 50 75 100 125 150
Effi0.4 Ubb=−6t (SC) radial distance
U =−10t (IDMFT)
0.2 bf Figure4: (Coloron-line)Finite-sizeeffectsontheradialden-
U =−6t (IDFMT)
bf sity profile. We assume a two-dimensional N × N square
0 lattice with hard-wall boundaryconditions. The dotted lines
0 0.5 1 1.5 2 2.5 indicate the boundaries of different lattices.The interaction
Entropy per particle (k ) parameters are U =−16t, U =11.5t. We use the density
B bf bb
distribution of thefermionic particleto representthegeneral
dependenceof density on thelattice size. In (a), we consider
Figure 3: (Color on-line) Efficiency as a function of entropy
per particle for different interaction parameters. Note here thecaseoflowtemperature,T =t/kB. Here,thedensitydis-
that we didn’t include the case of U = −2t, because it is tribution is concentrated at the center of the trap and there
bf
isnodifferencebetweendifferentlatticesizes. In(b),wecon-
alreadyshowninFig. 1thattheSCcalculationisnotaccurate
forlowtemperaturesinthiscase. In(a)and(b),weconsider sider the case of high temperature at T = 20t/kB. Here,
two different bosonic interaction strengths and five different the density is confined mainly by the size of the lattice. For
N = 50, the density is confined at the edge of the lattice,
inter-species interaction strengths. For all parameters, the
r = 25. For N = 100, the density is again confined at the
efficiency reaches 100% when the entropy is very low. For
edge, r=50. For both N =200 and 300, thedensitygoes to
anintermediateentropy,withanentropyperparticlearound
zerobeforereachingtheedgeofthelatticeandthetwodistri-
1kB, theefficiency is around 80%.
butionsoverlap with each other. We estimate that finite-size
effects are eliminated for the 300×300 square lattice for the
trap frequency and numberof particles considered here.
V. THERMOMETRY
for smaller lattices. This is because the confinement in-
A. Temperature and Density fluctuations
creases the density overlap between the two species. In
the low temperature region, the atoms are close to unit
Basedonthefluctuation-dissipationtheorem,thecom-
fillingatthecenterofthetrapandtheefficiencyissimilar
pressibility can be related to the density fluctuations as
for all difference lattice sizes.
[31],
We find that a lattice of 300×300 sites is sufficient ∂ρ(r) 1
κ= = [hρ(r)Ni−hρ(r)ihNi], (48)
to eliminate the finite-size effects for our parameter re- ∂µ k T
B
gions. Hence, we use this lattice size for the efficiency
and entropy per particle calculations. In fig. 6, we show where ρ(r) is the radial density profile, µ is the chemi-
the result for the efficiency as a function of the entropy cal potential and N is the total number of particles. For
per particle. We estimate the calculationresultfromthe a system with a spherically symmetric harmonic trap-
50×50latticeisaccuratewhenthetemperatureisaround ping potential, −V r2, the local chemical potential at a
t
or below T =1.25t/k . radial distance r is µ−V r2. Within the local density
B t
10
N=50 N=100 N=200 N=300 1
(a) U =5.7t
)B5 0.8 bb
k
e ( 4 (a) y
cl c0.6
parti 3 cien
per 2 Effi0.4 U =−10t
opy 1 0.2 Ubf=−6t
Entr 0 0 bf
−1 0 1 0 1 2 3 4 5
10 10 10
k T/t Entropy per particle (k )
B 1 B
1 (b) U =11.5t
bb
0.8
0.8 (b)
ncy0.6 ncy0.6
Efficie0.4 Efficie0.4 UUbf==−−81t2t
bf
0.2 0.2
U =−16t
bf
0 10−1 100 101 00 1 2 3 4 5
k T/t Entropy per particle (k )
B B
Figure 5: (Color on-line) Finite-size effects on the entropy Figure 6: (Color on-line) Efficiency as a function of entropy
perparticle andtheefficiency. Weassumeatwo-dimensional perparticlefora300×300squarelatticesystem. Weconsider
N×N squarelatticewithhard-wallboundaryconditions. The 625 atoms for each species. Compared with Fig. 3, the effi-
interaction parameters are U = −16t, U = 11.5t. In (a), ciencyissignificantlyhigherforthesamevalueoftheentropy
bf bb
weshowthebehavioroftheentropyperparticleasafunction perparticleinthe300×300lattice systemwhentheentropy
of temperaturefordifferent system sizes. Wesee theentropy perparticle is large. Ontheother hand,thebehavioris sim-
is significantly affected by the finite size when the lattice is ilar in both lattice systems when the entropy per particle is
smaller than around 200×200. The finite-size effect is not less than 1kB. The unit efficiency is reached roughly when
noticeable at lower temperature (T <1t/kB). theentropy perparticle is less than 0.5kB.
approximation,the trappingpotentialisinterpretedasa
variance in the chemical potential and the compressibil-
ityinthe trappedsystemcanbe re-writtenasafunction
of the density gradient,
π 1
ρ(0)= hN2i−hNi2 . (51)
V k T
t B
(cid:0) (cid:1)
∂ρ(r) 1 ∂ρ(r) Here,ρ(0)standsforthedensityatthecenterofthetrap.
=− . (49)
∂µ 2V r ∂r
t
With the development of in situ measurements, it is
These two equations lead to a simple relationship be-
now possible to measure the density gradient and the
tween the density gradient and the density fluctuations
fluctuations [38, 41] in experiment and this thermome-
in the trapped system,
try scheme has shown promise to be a reliable way of
1 ∂ρ(r) 1 estimating the temperature [31, 40]. Here we test this
− = [hρ(r)Ni−hρ(r)ihNi]. (50)
method for the Bose-Fermi mixtures and Eqs. (50 and
2V r ∂r k T
t B
51) are extended to mixtures by considering the density
Based on this relationship, one can determine the tem- as the total density of both species and the total num-
perature from the independently measured density gra- ber as the total number of both species. With the SC
dient and density fluctuations. For a two dimensional method, we calculate the density gradient directly from
system, a simplified relationship can be found by inte- the density profile expressions. To simulate the fluctu-
grating the above equation over all the two dimensional ations measured in the experiments, we use a simplified
plane, MC simulation explained in the next section.
