Weakly Interacting Bose Mixtures at Finite Temperature

0.12 MB·English

by  Bert Van Schaeybroeck

Weakly Interacting Bose Mixtures at Finite Temperature
Bert Van Schaeybroeck
Instituut voor Theoretische Fysica,
Katholieke Universiteit Leuven,
Celestijnenlaan 200 D, B-3001 Leuven, Belgium.
(Dated: January20,2009)
Motivated by the recent experiments on Bose-Einstein mixtures with tunable interactions we study
repulsiveweaklyinteractingBosemixturesatfinitetemperature. WeobtainphasediagramsusingHartree-
Focktheorywhicharedirectlyapplicabletoexperimentally trappedsystems. Almostallfeatures ofthe
9
diagrams can be characterized using simple physical insights. Our work reveals two surprising effects
0
whicharedissimilartoasystematzerotemperature. Firstofall,nopurephasesexist,thatis,ateachpoint
0
inthetrap,particlesofbothspeciesarealwayspresent.Second,evenforveryweakinterspeciesrepulsion
2
whenfullmixingisexpected,condensateparticlesofbothspeciesmaybepresentinatrapwithoutthem
n
beingmixed.
a
J
PACSnumbers:03.75.Hh,03.75.Mn,67.60.Bc,67.85.Fg
0
2
Introduction – Since the early realization of mixtures trapped situations, and yet, independent of the specific
]
r of Bose-Einstein condensates (BECs) ten years ago [1], trapping parameters. We realize this by drawing the dia-
e
a host of experiments revealed the rich physics of two- grams as a function of the chemical potentials of the two
h
component Bose [2, 3] and Fermi [4] systems. Whereas specieswhiletakingthetemperaturetobefixed. Thesedi-
t
o
most experiments use mixtures of hyperfine states of like agrams are apt for gases confined by a smoothly varying
.
t atomic species, it is also possible for unlike species to be potentialforthenthe useofalocaldensityapproximation
a
m trappedsimultaneously. Moreoverboth systems allow the is appropriate and an effective chemical potential can be
use of Feshbach resonances by means of which the inter- takenaslocallyconstantateachpositioninthetrap. Rather
-
d atomic interactions can be tuned to arbitrary values and thangivingexactdiagramsusingspecificparameters[17],
n which revolutionizedthe physicsof ultracoldgases. Such wefocushereontheirbasicunderstandingbyhighlighting
o
resonances were observed for binary Bose gases consist- thegenericfeatures;moreelaborateworkwillbepresented
c ingof87Rb 41K[5],87Rb-133Cs[6]and85Rb-87Rb[7,8] in a forthcoming paper. We argue that our diagrams can
[
mixtures;cl−earphasesegregationwasachievedforthelast be almost fully characterized, based on the knowledge of

1
mixture by changing the intraspecies interactions of the single-species Bose systems at finite temperature and the
v 85Rbparticles[8]. BECmixtureatzerotemperature.
8
4 To date theoretical works on BEC mixtures predomi-
0 nantly focussed on static and dynamic properties at zero Equation of state — Consider two Bose species, la-
3 temperature [9]. Interesting questions, however, can be belled 1 and 2, at temperature T and at chemical poten-
1. posed concerning the finite temperature regime: for in- tials µ1 and µ2. The particles of species i and j inter-
0 stance, will, upon increasing the interspecies repulsion, act weakly via s-wave scattering, quantified by a posi-
9 phasesegregationforthermallydepletedparticlessetinat tive scattering length a and a coupling constant G =
0 thesametimeasforcondensateparticles? Doesincreasing 2πℏ2a (m−1 + m−1)ij(henceforth i,j = 1,2). Weijuse
: ij i j
v the temperature induce an increased tendency to mix the theself-consistentHartree-Fock(HF)equationsofstateas
i species? Can pure phases still exist? The issue of phase first introduced by Huang et al. [11]. This very model is
X
segregationatnonzerotemperaturewasalreadyaddressed knownto givegoodagreementwith experiments[12]and
r
a in Ref. 10 where the authors calculated the conditions for evenwithquantumMonteCarlosimulations[13]. TheHF
aninstabilitytooccurinavolumewithfixedparticlenum- modeltreatsnoncondensedparticlesasfreeparticleswitha
bers. AlsothecaseofaBosemixturesinatrapatnonzero meanfieldchemicalpotentialshift,andtakesonlythermal
temperaturewasbrieflydiscussed. and no quantum fluctuations into account. The HF grand
In this Letter, we aim at clarifying the possible phase potential per unit volume Ω is expressed in terms of the
structures in traps of Bose mixtures at finite temperature. condensate densities n and thermally depleted densities
ci
The presented phase diagrams are directly applicable to n [3,11]:
di
Ω = g5/2 eβµ0i kBT +µ0n +G n2ci +2n n +n2 µ (n +n ) +G (n +n )(n +n ),
i=1,2"− (cid:0) λ3i (cid:1) i di ii(cid:18) 2 ci di di(cid:19)− i ci di # 12 c1 d1 c2 d2
X
2
whereλ = 2πℏ2/(m k T),β = 1/k T andg (x) =
i i B B l
∞ s−lxl. Thefirsttwotermsareduetotheentropyand
s=1 p
the kinetic energy of the depleted particles. The very last
Ptermcouplesthetwospeciesbyrepulsiveinteractions. At
fixedchemicalpotential,minimizationofΩwithrespectto
thedensitiesn andn yieldstheself-consistentHartree-
ci di
Fockequationsofstate:
µ = µ0 +2G (n +n )+G (n +n ), (1a)
1 1 11 c1 d1 12 c2 d2
µ = µ0 +2G (n +n )+G (n +n ), (1b)
2 2 22 c2 d2 12 c1 d1
and, µ0 = G n whenever n = 0. The depleted
i − ii ci ci 6
densities on the other hand are governedby minimization
withrespecttoµ0:
i
n = g (eβµ0)/λ3. (2)
di 3/2 i i FIG. 1: (Color online) Generic phase diagram for two Bose
For most systems of ultracold gases only few length- speciesatfixedtemperatureincaseofstronginterspeciesrepul-
sion (∆ < 0) as a function of the chemical potentials of both
scales are relevant. These include the particle de Broglie
wavelength λ , the interparticle distance n−1/3 and lastly species. TheBEC1(green) phaseconsistsofcondensate parti-
i i clesofspecies1andthermallydepletedparticlesofspecies1and
the scattering lengths a . The HF theory as introduced
ij 2andviceversafortheBEC2(blue)phase.The(grey)phaseNO
above is valid in case of weak interactions i.e. for small
BECconsistsofdepletedparticlesofbothspecies.Thechemical
values of a /λ and when single-particle excitations pre- potentialµC isgivenbyEq.(6)andthelocusofthetriplepoint
vail i.e. whijen in a λ2 1 [2]. Also, the theory fails TPisgiveniinEq.(7)whilethefirst-orderphaseboundaries are
i ij i ≪
veryclosetothetransitionpointwherefluctuationsbecome describedbyEqs.(8),(9)and(10).
dominant. In nowadaysexperiments,thescatteringlength
takesvaluesoforderoftennanometersandbelow,whereas
Single-species at finite T — Consider now a gas of
theinterparticledistancesandtheBrogliewavelengthsare
speciesiatfixedtemperature. Onecanthinkofthechem-
typicallyoftheorderofmicronsandhigher. Itfollowsthat
ical potentialµ as a measurefor the particle density: the
the necessaryconditionsfor the HF theory to be valid are i
higher µ , the higher the density of species i. For large
fulfilled[18]. i
and negativevaluesof µ , the system is dilute and the in-
We proceed by first discussing BEC mixtures at zero i
teraction energy small. On the other hand, above a cer-
temperatureandasinglespeciesBosegasatfinitetemper-
tain “critical” positive chemical potential a transition to a
ature,sincethesesystemsprovideallingredientsnecessary
phase with nonzero condensatedensity sets in; by the HF
fortheunderstandingoftheresultsobtainedfurther.
approach,thiscriticalvalueisattainedwhenµ0 = 0such
BEC mixture at T = 0 — At zero temperature, the HF i
that[3][19]:
theory reduces to Gross-Pitaevskii (GP) theory for a spa-
3
tially homogeneoussystem. Accordingto GP theory, two µC = 2G ζ(3/2)/λ , (6)
i ii i
regimescanbedistinguished,dependingontheparameter with ζ the Riemann zeta function. Note that µC de-
i
∆definedas[9]: pends on temperature via λ and from the scaling µC
i i ≃
k T(a /λ )weconcludethatµC k T.
∆ = G G /G2 1. (3) B ii i i ≪ B
11 22 12 − TheintroductionofasecondBosegaswillshiftthecriti-
When ∆ < 0, the inter-species repulsion is sufficiently calchemicalpotentialtoahighervaluethanµC;thiscanbe
i
ascribedtotheinterspeciesrepulsionwhichtendstorarify
strong to induces phase segregation and solely BEC 1 or
thegasofspeciesi.
BEC2 mayexistasgroundstates, thatis, species1and2
PhaseDiagramfor ∆ < 0 andfiniteT — Uponwork-
areimmiscible. Onereadilyfindsthefollowingconditions:
ing at fixed particle numbers as in Ref. 10, the onset of
phasesegregationcanbenaturallyprobedbythepresence
BEC1whenµ > µ G /G , (4a) ofadensityinstability. Atfixedchemicalpotentialsonthe
1 2 11 22
other hand, this method is inadequate as an instability in-
BEC2whenµ < µ pG /G . (4b)
1 2 11 22 dicates a spinodal (line) rather than the transition itself in
Ontheotherhand,when∆ > 0,thpeinterspeciesrepulsion caseitisoffirstorder. Instead,wehavecomparednumer-
is weak and a mixed phase of nonzero density of BEC 1 icallythegrandpotentialsfortheeightpossiblethermody-
andBEC2,heredenotedasBECMIX,mayappearasthe namicalphasesandtheirassociatedmultiplesolutions. We
groundstate: arrive at the generic phase diagrams for Bose mixtures at
finitetemperatureinFigs.1and2forthecases∆ < 0and
BECMIXwhenG /G < µ /µ < G /G . (5)
12 22 1 2 11 12 ∆ > 0respectively.
3
Identicaltothe behavioratT = 0, nomixingofunlike
condensate particles is possible at finite T when ∆ < 0.
OnlythreephasesappearinFig.1,noneofwhicharepure;
that is, even for extremely strong interspecies repulsion,
depletedparticlesofbothspeciesarepresentinallphases.
Accordingly, BEC 1 denotes the phase composed of con-
densateparticlesofspecies1anddepletedparticlesofboth
species,andviceversaforBEC2. TheNOBECphasede-
notesthephasewiththermalparticlesofspecies1and2.
A triple point TP is present in the phase diagram of
Fig. 1. As checked numerically, its locus (µTP,µTP) can
1 2
befoundbysettingµ0 = µ0 = 0inEq.(1):
1 2
µTP = µC +µCG /2G , (7a)
1 1 2 12 22
µTP = µC +µCG /2G . (7b)
2 2 1 12 11 FIG.2: (Coloronline)ThesameappliesasinFig.1exceptthat
NotethatinunitsofµC andµC,thepositionofpointTPis weakinterspeciesinteractionsareconsidered(∆>0).The(dark
temperatureindepende1nt. 2 grey) BEC MIX phase consists of condensed and depleted par-
ticlesofbothspecies. Theredarrowed lineindicatesapossible
Consider now the phase boundary between BEC 1 and
tracingofthediagramasafunctionofthe(radial)positioninthe
2;forlargeandpositivevaluesofbothchemicalpotentials,
trap. Thepoint Bindicatesthechemical potentialsatthecenter
the interaction energy per particle will eventuallybecome
ofthetrap.
muchlargerthanthethermalenergyandsoasymptotically
the T = 0 behavior is expected. Indeed, we find that in
accordwithEq.(4)theBEC1-2phaseboundarysatisfies:
Eq.(5)impliesthattheBECMIXphaseisdelimitedbythe
phaseboundaries
µ /µ = G /G whenbothµ ,µ . (8)
1 2 11 22 1 2 → ∞
µ /µ = G /G
Moving alongpthe NO BEC-BEC 1 boundary away from 1 2 11 12 whenbothµ ,µ . (11)
TP, the density of species 2 decreases; in the limit of µ /µ = G /G 1 2 → ∞
(cid:26) 1 2 12 22
µ ,onlyparticlesofspecies1remainsuchthatthe
2 → −∞ ThelocationoftriplepointTP2altersuponvaryingtem-
single-componentvalueµ µC isattained. Theassoci-
1 → 1 perature and ∆. For small and positive values of ∆, TP2
atedasymptoticdecayisexponentialastheNOBEC-BEC
enters at large and positive values of the chemical po-
1phaseboundaryisdescribedby:
tentials while TP2 coalesces with TP1 only in the limit
µ = µC +G eµ2/kBT/λ3 whenµ . (9) ∆ . This leads us to a remarkableconclusion: even
1 1 12 2 2 → −∞ → ∞
whenthezerotemperatureconditionforspeciesmixingis
BecauseµC k T,itfollowsthattheasymptoticconver- satisfied, at finite temperature, two-phase equilibrium be-
2 ≪ B
genceis slow ifµ2 is expressedin units of µC2. The same tweenBEC1and2isstillallowed. Changingthetemper-
lineofargumentwithinterchangedindices1and2applies
ature also induces a displacement of TP2 with respect to
tothedemarcationlinebetweenBEC2andNOBEC:
TP1. Intuitively one expects that increasing the tempera-
µ = µC +G eµ1/kBT/λ3 whenµ . (10) ture favors mixing. Yet, condensateparticles carry no en-
2 2 12 1 1 → −∞ tropy andwe find rather the opposite: increasingthe tem-
Inoutline, wehaveshownthatonecanqualitativelyes- peraturesuppressesmixing. In particular,thedistancebe-
tablish the phasediagramforthe case∆ < 0 withoutex- tweenTP1andTP2(inunitsofµC)increasesuponincreas-
i
plicitly performing any numerics. Indeed, we found that ingthetemperature.
theexpressionsforthetriplepoint(Eq.(7)),andthephase In the ideal case of fully suppressed interspecies inter-
boundaries(Eqs.(8)-(10))agreewellwiththenumerically actions(∆ )the triple pointsTP1 andTP2 coincide
→ ∞
exactresults. and, as expected, the phase boundaries are the horizontal
Phase Diagram for ∆ > 0 at finite T— At zero tem- andverticallinesµ = µC andµ = µC.
1 1 2 2
perature mixing of unlike condensate particles is possible To summarize, the ∆ > 0 phase diagram is similar to
when ∆ is positive. We find the same behavior at finite the∆ < 0diagramsbutismarkedbytheappearanceofan
T. This is shown in Fig. 2 where BEC MIX consists of additionalphaseconsistingofmixedBECspecies, andan
condensedand depleted particles ofboth species. The bi- additionaltriplepointTP2. Despitetheknownasymptotic
furcationoftheBEC1-2linebringsalongtheappearance behavior of the phase boundaries around the BEC MIX
ofanewtriplepointTP2(TP1isidenticaltoTPofFig.1). phase (see Eq. (10)), the locus of TP2 depends in a non-
For the aforementionedreasons, the T = 0 physics is re- trivial way on temperature and ∆ and must therefore be
coveredforlarge and positivechemicalpotentials. Hence determinednumerically.
4
Discussion — In order to extract from Figs. 1 and 2 thespeciessuchthat,spatiallytracingthetrapisequivalent
thepossiblephasestructuresappearinginanexperimental to exploring the phase diagram. We highlight the impor-
trap, the trapping potentials U are required. In most ex- tanceofthetriplepointsinourdiagramsandfindthat,sur-
i
perimentsalocaldensityapproximationorThomas-Fermi prisingly,nopurephasesexist,and,increasingthetemper-
approximation is justified such that at each position r an aturetendstosuppressthemixingofcondensateparticles.
effective chemical potential µ (r) = µ U (r) may be
assumedconstant;µ (r)ismaiximalatthie−ceniterandmin-
i
imal at the edge of a trap. In case both species are har- Acknowledgement— The authorthanksLev Pitaevskii,
monically confined or U (r) = m ω2r2/2, the chemical JosephIndekeuandAchilleasLazaridesforusefulsugges-
i i i
potentials probe our phase diagrams following a straight tions. B.V.S.isaPostdoctoralFellowofFWO.
line:
m ω2(µ (r) µ ) =m ω2(µ (r) µ ). (12)
2 2 1 − 1 1 1 2 − 2
Asanexample,atrapconfigurationwhichisdescribedby
the red path in Fig. 2 contains a BEC 2 core surrounded [1] J.Stengeretal.,Nature(London)396,345(1998);D.S.Hall
byshellsofBEC1andNOBEC.Notethatourphasedia- etal.,Phys.Rev.Lett.81,1539(1998).
[2] C.J. Pethick and H. Smith, Bose-Einstein Condensation in
gramsdonotexplicitlydependontrappingparametersnor
DiluteGases(CambridgePress,Cambridge,2002).
gravity.
[3] L.P.PitaevskiiandS.Stringari,Bose-EinsteinCondensation
The most striking conclusion of this work is that no
(ClarendonPress,Oxford,2003).
purephasesexistatfinitetemperature;evenforlargeinter-
[4] S.Giorginietal.,Rev.Mod.Phys.80,1215(2008).
speciesinteractions, the depletedparticles are notentirely
[5] G.Thalhammeretal.,Phys.Rev.Lett.100,210402(2008).
expelled [20]. However, the fact that upon increasing the [6] K.Pilchetal.,arXiv:0812.3287v1.
interspecies repulsion G in the BEC MIX phase, con- [7] S.B.Pappetal.,Phys.Rev.Lett.97,180404(2006).
12
densateparticleswillbeexpelledfirst, maybeunderstood [8] S.B.Pappetal.,Phys.Rev.Lett.101,040402(2008).
from the following simplistic argument: in the expression [9] P. Ao and S.T. Chui, Phys. Rev. A 58, 4836 (1998); P.
forΩ,itisseenthatduetodifferentexchangetermsthein- O¨hberg and S. Stenholm, ibid. 57, 1272 (1998); F. Riboli
and M. Modugno, ibid. 65, 063614 (2002); D.M. Jezek
traspeciesinteractions for depletedparticles are a factor2
and P. Capuzzi, ibid. 66, 015602 (2002); S. Ronen et al.,
higherthanthoseofcondensateparticles. Anaiveapplica-
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Rev.Lett.77, 3276(1996); C.K.Lawetal.,ibid.79, 3105
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12 11 22 (1997); B.D.Esryetal.,ibid.78,3594(1997); E.Timmer-
tionforvaluesofG higherthan 2G G . mans,ibid.81,5718(1998);S.G.Bhongaleetal.,ibid.100,
p 12 11 22
Asa directextensionofthis work,the interfacephysics 185301 (2008); M.Trippenbach etal.,J.Phys.B33, 4017
and its impact on the phase structpures can be explored in (2000).
[10] H.Shietal.,Phys.Rev.A61,063613(2000).
caseoftwo-phaseequilibrium. AgeneralizationofourHF
[11] K. Huang et al., Phys. Rev. 105, 776 (1957); K. Huang,
theory to spatially inhomogeneous systems is thereby re-
Statistical Mechanics, 2nd Ed. (Wiley, NY, 1987); N.P.
quired; within a first approximation, this can be effectu-
ated by introducing the terms ∇2√n into the grand po- ProukakisandB.Jackson,J.Phys.B41,203002(2008).
ci [12] F.Gerbieretal.,Phys.Rev.A70,013607(2004).
tential[3]. Suchtheorywouldthenallowthecalculationof
[13] M.Holzmannetal.,Phys.Rev.A59,2956(1999).
surface excessquantities which are useful for tightly con- [14] B.VanSchaeybroeck,Phys.Rev.A.78,023624(2008).
finedgaseswheretheireffectonthegroundstateconfigu- [15] G.B.Partridgeetal.,Phys.Rev.Lett.97,190407(2006).
rationturnsouttobesubstantial[8, 14, 15]. Furtherchal- [16] J.O.Indekeuetal.,Phys.Rev.Lett.93,210402(2004).
lengesincludeageneralizationofthetheoryofanomalous [17] Our diagrams differ for different values of (the later intro-
wettingphasetransitionstofinitetemperatures[16]. duced)sixparameters: m1,m2,a11,a22,a12 andT.Since
Conclusion — We have elaborated generic phase dia- eventhescatteringlengthsvaryinexperiments,welimitour
discussiontogenericfeatures.
gramsforBosemixturesatfinitetemperaturebasedonthe
[18] The scatteringlength is known todiverge nearaFeshbach
Hartree-Fockmodel. Ascorroboratedbynumericalanaly-
resonance. Nevertheless, our theory still applies to the ex-
sis,almostallpropertiesofourphasediagramscanbeex- 2
periments ofRef.8sincethelargestvalues ofn a λ are
pressedin terms ofknownresults forsingle-speciesgases around0.1. i ij i
at finite temperature and for binary mixtures at zero tem- [19] Small corrections apply near the critical points, due to the
perature. This means that it is possible to establish the first-ordernatureofthetransition;asmentionedbefore,the
phase diagrams (aside from the triple point TP2 in case theoryisnotaccurateveryclosetothetransitionpoint.
∆ > 0)withoutperforminganynumerics. Thephasedia- [20] Thisveryeffectthereforevanishesatzerotemperaturewhen
gramsaredrawnasafunctionofthechemicalpotentialsof thedepleteddensitiesarequenched.