Stability and phase coherence of trapped 1D Bose gases

5 Pages·0.17 MB·English

by  D. M. Gangardt

Stability and phase coherence of trapped 1D Bose gases
D.M. Gangardt1 and G.V. Shlyapnikov1,2,3
1Laboratoire Kastler-Brossel, Ecole Normale Sup´erieure, 24 rue Lhomond, 75005 Paris, France
2FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands
3 3Russian Research Center Kurchatov Institute, Kurchatov Square, 123182 Moscow, Russia
0 (Dated: February 1, 2008)
0
We discuss stability and phase coherence of 1D trapped Bose gases and find that inelastic decay
2
processes, such as three-body recombination, are suppressed in the strongly interacting (Tonks-
n Girardeau) and intermediate regimes. This is promising for achieving these regimes with a large
a
number of particles. ”Fermionization” of the system reduces the phase coherence length, and at
J
T =0thegasisfullyphasecoherentonlydeeplyintheweaklyinteracting(Gross-Pitaevskii)regime.
4
1 PACSnumbers: 03.75Fi,05.30Jp

]
h
Recent success in creating quantum degenerate 1D finite-temperature 1D Bose gases follows from the Bo-
c
e trapped atomic gases [1, 2, 3] has stimulated an interest goliubovk−2 theoremashasbeenexpoundedin[9]. The
m incorrelationpropertiesofthesesystems. The1Dregime long-range order is destroyed by long-wave fluctuations
- isreachedbytightlyconfiningtheradialmotionofatoms of the phase leading to an exponential decay of the one-
t
a inacylindricaltraptozeropointoscillations. Thediffer- particle density matrix at large distances [10, 11]. A
t ence between such a kinematically 1D gas and a purely similar picture is found at T =0 [12], where the density
s
. 1D gas is related only to the value of the interparticle matrix undergoes a power-law decay [11, 13, 14]. In the
t
a interaction, which depends on the radial confinement. pastfewdecades,ageneralapproachhasbeendeveloped
m
The 1D Bose gas with repulsive short-range interac- forexactcalculationsofone-andtwo-particlecorrelation
- tions characterized by the coupling constant g > 0 ex- functions at an arbitrary γ (see [15, 16] for review).
d
n hibits remarkable properties. Counterintuitively, it be- Realization of 1D trapped Bose gases raises the ques-
o comesmorenon-idealwithdecreasing1Ddensityn[4,5]. tion of their phase coherence and stability. The phase
c The equation of state and correlation functions depend coherencepropertiesarestronglyinfluenced bythe trap-
[
crucially on the parameter pingpotential,whichintroducesafinitesizeofthesystem

3 and provides a low-momentum cut-off of the phase fluc-
γ =mg/¯h2n, (1)
v tuations. In the GP regime at sufficiently low T, the
8
where m is the atom mass. For comparatively large phase fluctuations are suppressed and the equilibrium
3
3 n, the parameter γ 1 and √γ represents the ratio state is a true condensate. At higher temperatures, it
≪
7 of the mean interparticle separation 1/n to the corre- is transformed into a phase-fluctuating condensate [17].
0 lation length lc = h¯/√mng. In this case the gas is in Thedynamics,interferenceeffects,andexcitationsof1D
2 the weakly interacting or Gross-Pitaevskii (GP) regime, trappedBosegasesarecurrentlyasubjectofactivestud-
0
and the amplitude of the boson field obeys the familiar ies[18,19,20,21]. Ofparticularinterestisthechangein
/
t Gross-Pitaevskii equation. For sufficiently low densities phase coherence properties while going from the GP to
a
m (or large g), one has γ 1 and the strongly interact- the TG regime, where the phase coherence is completely
≫
ing or Tonks-Girardeau (TG) regime is reached. In this lost.
-
d regime the 1D gas acquires fermionic properties in the Thestrongtransverseconfinementrequiredforthe1D
n sense that the wave function strongly decreases as par- regimecanleadtohigh3Ddensitiesofatrappedgas. At
o
ticles approach each other (see [4, 5] and cf. [6]). For a large number of particles, the 3D density can exceed
c
: γ = any correlation function of the density can be 1015 cm−3, and one expects a fast decay due to three-
v ∞
calculated straightforwardly [7] by using the exact map- body recombination. It is then crucial to understand
i
X pingontothesystemoffreefermionsfoundbyGirardeau how the correlation properties of the gas influence the
r [4]. decay rate.
a
Auniform1Dsystemofbosonsinteractingviaadelta- InthisLetter,wediscussstabilityof1DBosegasesand
functional potential is integrable by using the Bethe calculate local density correlators,as those are responsi-
ansatz and has been a subject of extensive theoretical ble for inelastic decay processes [22]. We find that the
studies. Lieb and Liniger [5] have calculated the ground decay rates are suppressed in the TG and intermediate
stateenergyandexcitationspectrumforanyvalueofthe regimes, which is promising for achieving these regimes
parameter γ. Yang and Yang [8] have studied thermo- with a largenumber of particles. We then analyze phase
dynamic properties of this system at finite temperatures coherence of a trapped 1D Bose gas and show that vac-
and found no phase transition for T > 0. The absence uumfluctuationsofthephasemakethezero-temperature
oflong-rangeorderandtrueBose-Einsteincondensatein coherence length smaller than the Thomas-Fermi size of
2
1
the sample, unless the gas is deeply in the GP regime.
The1Dregimeinatrappedgasisrealizediftheampli-
tude of transverse zero point oscillations l0 = ¯h/mω0 2n
is much smaller than the longitudinal correlatiopn length /
)
lc =h¯/√mµ,where ω0 is the frequency ofthe transverse γ(
confinementandthechemicalpotentialofthe1Dsystem 2
g
is µ ¯hω . One then has a 1D system of bosons inter-
0
≪ 0
acting with each other via a short-range potential char-
0 5 10 15 20
acterized by an effective coupling constant g > 0. This
γ
constantis expressedthroughthe 3Dscatteringlengtha
[23],assumingthatl greatlyexceedstheradiusofinter- FIG. 1: Local correlation function g2 versusγ.
0
action between atoms. For a positive a l we have
0
≪
following from the Bogoliubov approach:
g =2h¯2a/ml2, (2)
0
g (γ)/n2 =1 2√γ/π, γ 1. (4)
2
and a characteristic distance h¯2/mg related to the inter- − ≪
In the limit of large γ, we have e(γ)= π2/3 (1 4/γ),
action between particles in the described 1D problem is
−
l2/a l . In the GP regime, the chemical potential and the two-particle correlation functio(cid:0)n is gi(cid:1)ven by
∼ 0 ≫ 0
µ gn, and the condition l l leads to the inequal-
ity≈na 1. In the TG recgi≫me0the correlation length g2(γ)/n2 =4π2/3γ2, γ ≫1. (5)
≪
l 1/n,andoneshouldhavenl 1. Wethusseethat
c 0 The results in Fig. 1 and Eq. (5) clearly show that two-
∼ ≪
at any value of γ it is sufficient to satisfy the inequali-
particlecorrelationsand,hence,theratesofpairinelastic
ties a l0 1/n. Then the 1D regime is reached and processes,aresuppressedforγ >1. This providesa pos-
≪ ≪
the system can be analyzed on the basis of the 1D Lieb- sibility for identifying the TG a∼nd intermediate regimes
Liniger model, assuming a delta-functional interatomic
of a trapped 1D Bose gas through the measurement of
potential with the coupling constant g given by Eq. (2).
photoassociationin pair interatomic collisions.
The rate of three-body recombination is proportional
The three-particle local density correlator g cannot
3
to the local three-particle correlation function g =
3 be obtained from the Hellmann-Feynman theorem. In
Ψ†(x)Ψ†(x)Ψ†(x)Ψ(x)Ψ(x)Ψ(x) [22],whereΨ(x)isthe
the weak coupling GP regime (γ 1) one can use the
h i
field operator of atoms and the symbol ... denotes the ≪
Bogoliubov approach, which immediately yields
h i
expectation value. Similarly, the rates of two-body in-
elastic processes involve the correlation function g2 = g3(γ)/n3 1 6√γ/π, γ 1. (6)
Ψ†(x)Ψ†(x)Ψ(x)Ψ(x) . Assuming that local correlation ≃ − ≪
hproperties are insensitiive to the geometry of the system For the TG regime (γ 1), we have developed a
≫
we consider a uniform 1D gas of N bosons on a ring of methodforcalculatingthe leadingbehavioroflocalden-
circumference L. The Hamiltonian of the system reads: sitycorrelators. Detailswillbegivenelsewhere,andhere
we present a compact derivation of g at T = 0. In first
3
H = dx (h¯2/2m)∂ Ψ†∂ Ψ+(g/2)Ψ†Ψ†ΨΨ . (3) quantization the expression for this function reads
x x
Z
(cid:2) (cid:3) N! 2
g (γ)= dX Φ(γ)(0,0,0,x ,...,x ) ,(7)
Forfindingg2atT =0,weusetheHellmann-Feynman 3 3!(N 3)!Z (cid:12) 0 4 N (cid:12)
theorem[24,25]. This is similar to the calculationofthe − (cid:12) (cid:12)
(cid:12) (cid:12)
mean interaction energy in ref. [5]. Namely, one shows where dX = dx ...dx , and Φ(γ) is the ground state
4 N 0
that the expectation value of the four-operator term in wave function given in the domain 0<x <...<x <
1 N
the Hamiltonian (3) is proportional to the derivative of L by the Bethe ansatz solution:
thegroundstateenergywithrespecttothecouplingcon-
stant: dE /dg = Φ dH/dg Φ = g L/2. The first Φ(γ)(x ,x ,...,x ) a(P)exp i k x , (8)
identity fo0llows frohm0t|he nor|ma0ilization2 of the ground 0 1 2 N ∝XP n X Pj jo
state wave function Φ , and the second one is obtained
0 where P is a permutation of N numbers, quasimomenta
straightforwardlyfromthe Hamiltonian(3). Theground
k are solutions of the Bethe ansatz equations, and
state energy can be written as E = Ne(γ)h¯2n2/2m, j
0
where the quantity e(γ) is a solution of the Lieb-Liniger 1
iγn+k k 2
equations [5] and is calculated numerically for any value a(P)= Pj − Pl .
(cid:18)iγn k +k (cid:19)
of γ [19, 26]. For the two-particle local correlation func- Yj<l − Pj Pl
tion we then obtain g (γ) = n2de(γ)/dγ. The function
2
g (γ)/n2 is shown in Fig. 1. For small values of γ, we Forγ 1,weextracttheleadingcontributiontoΦ(γ) at
2 ≫ 0
obtain numerically the result which coincides with that three coinciding points by symmetrizing the amplitudes
3
a(P) over the first three elements of the permutation P: l . Therefore, the equation for the recombination rate
0
is the same as in 3D cylindrical Bose-Einstein conden-
1 ǫ
P sates with the Gaussian radial density profile. There is
a(P ) (k k ), (9)
3!Xp p ≃ (iγn)3 Yj<l Pj − Pl only an extra reduction by a factor of g3/n3. A char-
acteristic decay time τ is then given by the relation
where P = P ,P ,P ,P ,...P , and j,l = 1,2,3. τ−1 =α n2 (g /3n3), where α is the recombination
p p1 p2 p3 4 N 3D 3D 3 3D
The sign of the permutation P is ǫ , and p runs over rate constantfor a 3D condensate, and n =n/(πl2) is
P 3D 0
six permutations of 1,2,3. For large γ, the difference the maximum 3D density. Even for n 1015 cm−3,
3D
∼
of quasi-momenta k from their values at γ = is of thelifetimeτ cangreatlyexceedsecondswhenapproach-
j
∞
order 1/γ and can be neglected. Then, from Eqs. (8) ingtheTGregime. Forexample,thisisthecasefor87Rb
and (9), we conclude that to this level of accuracy the (α 10−29 cm6/s) optically trapped with ω 100
3D 0
∼ ≈
ground state wave function at three coinciding points is kHz (l 200 ˚A), assuming L 100 µm and N = 200.
0
≈ ≈
givenbyderivativesofthewavefunctionoffreefermions Then one has γ 10 and Eq. (11) predicts a reduction
≈
Φ(∞)(x ,x ,x ,x ,...) at x =x =x =0: ofthethree-bodyratebymorethanthreeordersofmag-
0 1 2 3 4 1 2 3
nitude.
Φ(γ)(0,0,0,x ,...) 1 (∂ ∂ ) Φ(∞).(10) We now turn to phase coherence of a 1D Bose gas in
0 4 ≃−(γn)3hYj<l xj − xl i 0 a harmonic potential V(x) = mω2x2/2. We consider
the case of T = 0 and rely on the hydrodynamic ap-
Substituting Eq. (10) into Eq. (7) we express the lo- proach[14]in whichlong-wavepropertiesofthe 1Dfluid
cal correlator g through derivatives of the three-body are described in terms of two conjugate fields, density
3
correlation function of free fermions. Using Wick’s fluctuations δn and phase φ. They satisfy the commuta-
theorem, the latter is given by a sum of products of tion relation [δn(x),exp(iφ(x′))]=δ(x x′)exp(iφ(x)).
−
one-particle fermionic Green’s functions G(x y) = We assume the Thomas-Fermi regime and use the local
kF dkeik(x−y)/2π, where k = πn is the− Fermi density approximation [19, 20]: the mean density n(x)
−kF F is obtained from the local equation of state µ(n(x)) =
wRavevector. ThecalculationfromEq.(7)isthenstraight-
µ V(x), where µ(n) is the chemical potential for
forward and we obtain (γ 1) 0 −
≫ the Lieb-Liniger problem. The density is non-zero only
g3n(3γ) = γ36n69h(G′′)3−G(4)G′′Gi= 1165πγ66, (11) wthiethninortmhealTizhaotmioans-cFoenrdmitiiroandiu−RsRTRTFTFFn(=x)pdx2µ=0/Nmωg2iv,easnda
relation between µ0 and N. ERquations of motion for the
where G and its derivatives are taken at x y =0. fields δn and φ follow from the quantum Hamiltonian:
−
In fact, in our problem the correlation functions g
2
¯h
and g3 are slightly nonlocal. They are related to inter- Hq = dx vN(πδn)2+vJ(∂xφ)2 =h¯ Ωjb†jbj,
particle distances r l0 lc, since at smaller r the 2π Z (cid:0) (cid:1) Xj
∼ ≪
relative motion of particles is three-dimensional and the
where Ω and b are frequencies and annihilation opera-
localcorrelatorsdo not change. For largeγ,at distances j j
tors of elementary excitations. The quantities v (x) =
l an extra (coordinate-dependent) contribution to g N
i∼s o0f the order of (nl )2 and to g of the order of (nl )62 (π¯h)−1∂µ/∂n and vJ(x) = π¯hn(x)/m determine the lo-
0 3 0
cal sound velocity v (x)v (x) and the local Luttinger
(see, e.g. [7] and references therein). Under the condi- N J
tionl0 a,these contributionscanbe neglectedasthey parameterK(x)=pvJ(x)/vN(x). The HamiltonianHq
are mu≫ch smaller than the results of Eqs. (5) and (11), is a generalizationopf the effective harmonicHamiltonian
respectively. of Ref. [14] to a non-uniform system.
Ourmethodis readilygeneralizedforthe caseoffinite Using the density-phase representation for the field
temperature by considering the temperature-dependent operators, we calculate the one-particle density matrix
Green’sfunctionsoffreefermions. ForT µweobtaina g1(x,x′)= Ψ†(x)Ψ(x′) for x x′ lc. Asthedensity
smallcorrection (T/µ)2tothezero-tem≪peratureresult. fluctuationshare small, tihis m|at−rix|re≫duces to
∼
The same conclusion holds for the GP regime (γ 1).
Thus, from Eq. (11) we conclude that the thre≪e-body g1(x,x′)= n(x)n(x′)exp{−h(φ(x)−φ(x′))2i/2}.
p
decayof1DtrappedBosegasesisstronglysuppressedin The phase operator is given by its expansion in eigen-
theTGregime. Moreover,Eq.(6)showsthateveninthe modes labeled by an integer quantum number j >0:
GP regime with γ 10−2, one has a 20% reduction of
≈ 1/2
the three-body rate. Thus, one also expects a significant πvN(0)
φ(x)= i f (y)b +H.c., (12)
reduction of the three-body decay in the intermediate − (cid:18)2Ω R (cid:19) j j
Xj j TF
regime.
For l a, the recombination process in 1D trapped wherewehaveintroducedadimensionlesscoordinatey =
0
≫
gasesoccursatinterparticledistancesmuchsmallerthan x/R . The eigenfunctions f are normalized by the
TF j
4
1
x R the coherence is already lost for γ 1. Ther-
TF 0
∼ ≈
) mal fluctuations of the phase are readily included in our
x 0.8
( γ0 = 0.1 scheme and will be discussed elsewhere.
n
/ In conclusion, we have found an enhanced stability of
x) 0.6
− a trapped 1D Bose gas in the Tonks-Girardeau and in-
, termediate regimes and describedthe reductionof phase
x 0.4 γ = 1
( 0 coherence in these regimes.
1
g
WeacknowledgefruitfuldiscussionswithE.Br´ezin,R.
0.2
γ = 10 Combescot, and V. Kazakov and express our gratitude
0
to Chiara Menotti for providing us with numerical data.
0
0 0.2 0.4 0.6 0.8 This work was financially supported by the French Min-
y = x/RTF ist`eredesAffairesEtrang`eres,bythe DutchFoundations
FIG.2: Thedensitymatrixg1(x,−x)forN =104andvarious NWO and FOM, by INTAS, and by the Russian Foun-
valuesofγ0. Thesolidcurvesshownumericalresults,andthe dation for Basic Research. Le LKB est UMR 8552 of
dashed curves theresults of thequasiclassical approach. CNRS, of ENS and of Universit´e P. et M. Curie.
condition −11dy(vN(0)/vN(y))fj∗(y)fj′(y)=δjj′. From
the HamiltRonian Hq we obtain the continuity and Euler
equations which lead to the eigenmode equation,
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E.H. Lieb, ibid,130, 1616 (1963).
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≈ { }
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ter case, for x′ = x we obtain (φ(≫x) φ( x))2 (R2e0v0.1)A; K65.K,.0D63a6s,03G.(J2.0L02a)p;eyKr.eK,.anDdaEs,.MM..DW.riGghirta,rPdehayus,.
− h − − i ≈
K−1(x)ln 2x/lc(x) ,whichisclosetoHaldane’sresult and E.M. Wright,cond-mat/0205605.
{| | }
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c
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1
(2002).
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−
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≫
5
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