Semiclassical theory of trapped fermionic dipoles

8 Pages·0.21 MB·English

by  K. Goral

Semiclassical theory of trapped fermionic dipoles
Krzysztof G´oral,1 Berthold-Georg Englert,2,3 and Kazimierz Rza¸z˙ewski1
1Center for Theoretical Physics and College of Science,
Polish Academy of Sciences, Aleja Lotnik´ow 32/46, 02-668 Warsaw, Poland
2Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany
3Atominstitut, Technische Universit¨at Wien, Stadionallee 2, 1020 Wien, Austria
(e.g. molecular) dipoles, when supposedly the scatter-
We investigate the properties of a degenerate dilute gas
ing length can be neglected, it is the trap geometry that
of neutral fermionic particles in a harmonic trap that inter-
plays a crucial role [10] – the system is stable provided
act via dipole-dipole forces. We employ the semiclassical
the trapassures the domination of repulsive interactions
1 Thomas-Fermi method and discuss the Dirac correction to
in the gas.
0 the interaction energy. A nearly analytic as well as an exact
A full quantum mechanical description of the system
0 numerical minimization of the Thomas-Fermi-Dirac energy
2 of many interacting fermions is of course very complex.
functionalareperformedinordertoobtainthedensitydistri-
Butthelessonsofsemiclassicalatomicphysicscanbeap-
n bution. Wedeterminethestabilityofthesystemasafunction
plied, in particular the Thomas-Fermi approach [14,15]
a oftheinteractionstrength,theparticlenumber,andthetrap
J geometry. We find that there are interaction strengths and and its refinements (see, e.g., [16]). Its success in de-
9 particle numbers for which the gas cannot be trapped stably scribingstatic properties ofatoms is well known,and we
1 in a spherically symmetric trap, but both prolate and oblate note that, in recent years, these methods were also used
traps will work successfully. successfully for studying dynamical processes of atoms
2 and molecules in superstrong light pulses [17].
v PACS numbers: 03.75.Fi, 05.30.Jp Our paper is organized as follows: in Section II, the
3
Thomas-Fermi model is revisited with an eye on the
9
dipole-dipole forces. The Dirac correction to the inter-
1
0 action energy term is discussed and scaling properties
I. INTRODUCTION
1 derived with the help of the virial theorem. In Section
0 III the results of approximate (nearly analytic) and nu-
0 The experimental achievement of quantum degener- mericalminimizationsoftheThomas-Fermi-Diracenergy
t/ acy in a dilute trapped gas of cold fermionic atoms [1] functional are presented. Unexpectedly, we find that the
a has stimulated theoretical interest in the properties of systemmaywithstandlargerdipolarforcesbothforvery
m
this fundamental system. Attention has focused on the flat and for highly elongated traps.
- critical temperature [2] and the detection [3] of Cooper
d
pairing as well as on the properties of mixtures of vari-
n
o ous fermionic and bosonic species [4,5]. Another impor- II. THOMAS-FERMI MODEL
c tantproblemconcernstheinteractionsbetweenultracold
v: fermions [5–7]. Owing to the exclusion principle, spin- A. General considerations
i polarized Fermi atoms do not interact via s-wave col-
X
lisions, whereas they dominate in the low-energy regime
Let’s begin by recalling some basic things, mainly col-
r forbosonsandhavepronouncedeffectsonthestaticsand
a lected from Refs. [16,18,19], with due attention to the
dynamical properties of cold boson gases [8]. Hence, in
changed situation: here fully spin-polarized fermions —
the absence of low-energy collisions other types of forces
therefermionswithnonetspin. The spatialone-particle
come into play.
density is denoted by n(~r), it is normalized to the total
A good candidate is a dipole-dipole interaction be-
particle number N,
tweenatomsormolecules,notanalyzedsofarinthecon-
text of cold trapped fermions. Some atoms possess per-
manent magnetic dipole moments of considerable mag- N = (d~r)n(~r), (1)
nitude (chromium, for instance, has µ = 6µ ). It was Z
B
also proposed to induce electric dipoles in atoms [9,10]. where (d~r) dxdydz denotes the volume element. The
Hugepermanentelectricdipolemomentsoccurnaturally spatial one-≡particle density matrix n(1)(~r′;~r′′) and the
indiatomicpolarmolecules[11]. Thebehaviorofatomic one-particle Wigner function ν(~r,~p) are related by
bosonic dipoles in traps has been investigated in [12,13],
whichaddressedthequestionofinstabilitiesinthesystem n(1)(~r′;~r′′)= (d~p) ν 1(~r′+~r′′),~p ei~p·(~r′−~r′′)/¯h .
caused by an attractive component of dipolar interac- (2π¯h)3 2
Z
tions. Theconclusiondrawnwasthatalargeenoughpos- (cid:0) (cid:1)
(2)
itive scattering length (providing repulsive interactions)
can stabilize a system of bosonic dipoles. For strong
1
The spatial and momental one-particle densities are ob- manner—istwo-fold: Weapproximaten(2) byproducts
tained by integrating ν(~r,~p) over the other variable, of n(1) factors (Dirac),
n(~r)=n(1)(~r;~r)= (d~p) ν(~r,~p), n(2)(~r′1,~r′2;~r′1′,~r′2′)=n(1)(~r′1;~r′1′)n(1)(~r′2;~r′2′)
Z (2(πd~r¯h))3 −n(1)(~r′1;~r′2′)n(1)(~r′2;~r′1′), (10)
ρ(~p)= ν(~r,~p). (3)
(2π¯h)3 and n(1) by a brutally simple Wigner function (Thomas
Z
and Fermi),
Theyareneededforthecalculationofthekineticenergy,
ν(~r,~p)=η ¯h2[6π2n(~r)]23 ~p2 , (11)
~p2 −
E = (d~p) ρ(~p), (4)
kin 2M where η( ) is Heavisi(cid:0)de’s unit step func(cid:1)tion. This gives
Z
the density functional of the kinetic energy as
andtheexternalpotentialenergy(oftheharmonictrap),
¯h2 1 5
E [n]= (d~r) 6π2n(~r) 3 , (12)
Etrap = (d~r)21Mω2 x2+y2+(βz)2 n(~r), (5) kin Z M 20π2
Z (cid:2) (cid:3)
(cid:2) (cid:3) andthedensityfunctionalforthepotentialenergyinthe
whereM isthe massoftheatomspeciesconsidered,ω is
trap is E of (5).
the (transverse)trapfrequency,andβ is the aspectratio trap
The dipole-dipole interaction energy consists of two
ofthecylindricaltrap. Thetrapissphericallysymmetric
parts, corresponding to the two summands in (10),
forβ =1;forβ <1,theequipotentialsurfacesareprolate
(“cigar shaped”) ellipsoids; for β > 1, they are oblate
E [n]=E(dir)[n]+E(ex)[n] (13)
(“lentil shaped”) ellipsoids. dd dd dd
For the dipole-dipole interaction energy, E , we need
dd with the direct term
(the diagonal part of) the two-particle density matrix
n(2)(~r′1,~r′2;~r′1′,~r′2′), E(dir)[n]= 1 (d~r′)(d~r′′)n(~r′)V (~r′ ~r′′)n(~r′′) (14)
dd 2 dd −
E = 1 (d~r′)(d~r′′)V (~r′ ~r′′)n(2)(~r′,~r′′;~r′,~r′′), (6) Z
dd dd
2 − and the exchange term
Z
where E(ex)[n]= 1 (d~r′)(d~r′′)n(1)(~r′;~r′′)n(1)(~r′′;~r′)
Vdd(~r)= 4µπ0 "~µr32 −3(~µr·5~r)2 − 83π~µ2δ(~r)# . (7) dd −2Z ×Vdd(~r′−~r′′)
1
= (d~r) (d~s)V (~s)
dd
We note that the contact term, proportional to δ(~r), is −2
Z Z
requiredbytheconditionthatthemagneticfieldmadeby n(1)(~r+ 1~s;~r 1~s)n(1)(~r 1~s;~r+ 1~s).
the point dipole be divergence-free [20]. An alternative × 2 − 2 − 2 2
(15)
way of presenting V is
dd
Now we note that
µ 1 ↔
V (~r)= 0~µ ~ ~ 4π1δ(~r) ~µ, (8)
dd
4π ·(cid:20)−∇∇r − (cid:21)· n(1)(~r+ 1~s;~r 1~s)n(1)(~r 1~s;~r+ 1~s)
2 − 2 − 2 2
′ ′′
whichisaparticularlyconvenientstartingpointforeval- = (d~p)(d~p )ν(~r,~p′)ν(~r,~p′′)ei(~p′−~p′′)·~s/¯h
uating the Fourier transform (2π¯h)6
Z
(16)
(d~r) ei~k·~rV (~r)= µ0~µ 4π~k~k 4π↔1 ~µ. (9)
dd 4π ·" k2 − #· depends only on the length s = ~s of vector~s, not on
Z
its direction~s/s, because — in the TF approximation
The vanishing divergence just mentioned is here imme- (11)—theproductν(~r,~p′)ν(~r,~p′′)(cid:12)(cid:12)i(cid:12)(cid:12)nvolvesonlyp′ = ~p′
diately recognized, inasmuch as~k =0. and p′′ = ~p′′ . As a consequence, it is permissible to
· ··· (cid:12) (cid:12)
replace,in(15),V (~s)byitsaverageoverthesolidan(cid:12)gle(cid:12)
(cid:2) (cid:3) (cid:12) (cid:12) dd
associated w(cid:12) ith(cid:12) ~s,
B. Thomas-Fermi-Dirac functionals
µ 8π
V (~s) 0 ~µ2δ(~s) . (17)
dd
The semiclassical approximation now employed — in → 4π − 3
(cid:20) (cid:21)
thespiritofwhattheTFDtrio(Thomas[14],Fermi[15],
and Dirac [21]) did, although in a technically different We thus arrive at
2
1 µ 8π
E(ex)[n]= (d~r)[n(~r)]2 0 ~µ2 are consistent with the constraint (1). They affect the
dd 2 4π 3 various pieces of E(TFD) in accordance with
Z
= 21 (d~r)(d~r′)n(~r)4µπ0 83π~µ2δ(~r−~r′) n(~r′), Ekin λ2+35αEkin,
Z (cid:20) (cid:21) E →λ−2+αE ,
(18) trap trap
→
E λ3+2αE , (26)
dd dd
and accordingly →
so that (E E(TFD))
1 ′ ′ ′ ≡
E [n]= (d~r)(d~r )n(~r)V (~r ~r )n(~r ) (19)
dd 2 dd − E = Ekin+Etrap+Edd
Z
with λ2+53αEkin+λ−2+αEtrap+λ3+2αEdd. (27)
→
µ ~µ2 (~µ ~r)2 In the infinitesimal vicinity of λ = 1, all first-order
0
V (~r)= 3 · , (20) changes of E originate in the explicit change of N,
dd 4π "r3 − r5 # δN =δλαN, and therefore
which is V (~r) of (7) with the contact term removed. ∂E
dd αN =(2+ 5α)E +( 2+α)E +(3+2α)E
In (8) and (9) this corresponds to multiplying the unit ∂N 3 kin − trap dd
↔
dyadic 1 by 1 [13].
3 (28)
The rotational symmetry of the trap potential in (5)
distinguishesthez axis,andwetakeforgrantedthatthis must hold irrespective of the value of parameter α. In
is also the direction of spin polarization, viewofthe linearαdependence, wegettwoindependent
statements,
~µ=µ~e . (21)
z
α=0 : 2E 2E +3E =0,
kin trap dd
Then −
∂E
α= 3 : E +7E =3N . (29)
µ 1 3(z/r)2 −2 kin trap ∂N
V (~r)= 0µ2 − , (22)
dd 4π r3
They enable us to express E , E , and E in terms
kin trap dd
of E and N∂E/∂N,
andthewholesystemisinvariantunderrotationsaround
the z axis.
21 15 ∂E
The total TFD energy functional is then given by the E = E N ,
kin
2 − 2 ∂N
sumofthekineticenergy(12),thepotentialenergyinthe
3 3 ∂E
trap (5), and the dipole-dipole interaction energy (19), E = E+ N ,
trap
−2 2 ∂N
¯h2 1 5 ∂E
E(TFD)[n]= (d~r) 6π2n(~r) 3 + 1Mω2r2n(~r) Edd = 8E+6N . (30)
M 20π2 2 − ∂N
Z (cid:20) (cid:21)
1 ′ (cid:2) (cid:3) ′ ′ In conjunction with (µ= ~µ )
+ (d~r)(d~r )n(~r)V (~r ~r )n(~r ). (23)
dd
2 −
Z ∂ (cid:12) (cid:12)
(cid:12) (cid:12)
µ E =2E ,
The density that minimizes E(TFD) under the constraint ∂µ dd
(1) obeys the nonlinear integral equation
∂
ω E =2E ,
trap
¯h2 2 ∂ω
6π2n(~r) 3 + 1Mω2 x2+y2+(βz)2 ∂
2M 2 M E =Etrap Ekin, (31)
∂M −
(cid:2) + (cid:3)(d~r′)V (~r(cid:2) ~r′)n(~r′)= 1M(cid:3)ω2R2, (24)
dd − 2 they imply that the ground state energy E(M,ω,µ,N)
Z
is of the form (we leave the β dependence implicit)
where 1Mω2R2 is a convenient way of writing the La-
2 4 1
grange multiplier for the constraint. E(M,ω,µ,N)=h¯ωN3e(N6ε) (32)
with
C. Virial theorems
1 µ
ε= ωM3/¯h5 2 0µ2. (33)
4π
Scaling transformations of the form
(cid:0) (cid:1)
Clearly,thedimensionlessnumberεmeasurestherelative
n(~r) λ3+αn(λ~r), N λαN (25) strength of the dipole-dipole interaction. The universal
→ →
3
functione()is tobe foundnumerically(for eachβ value III. RESULTS
of interest). For ε=0, we can solve (24) immediately,
n(~r)= 1 Mω/¯h 3 R2 x2 y2 (βz)2 32 , (34) Owing to its intrinsic semiclassical approximations,
6π2 − − − one expects a few-percent deviation of the TFD energy
from the true ground-state energy. It is, therefore, not
(cid:0) (cid:1) (cid:2) 3 (cid:3)
where, by convention, [ ]2 =0 for negative arguments, really necessary to solve the nonlinear integral equation
···
and
(43). A reasonable variational estimate, in conjunction
R= 48βN 16 Mω/¯h −21 (35) withafewfull-blownnumericalsolutionsforcomparison,
will do.
as a consequence o(cid:0)f (1), a(cid:1)nd(cid:0)so we(cid:1)find
3 1 1
e0 e(0)= (6β)3 =1.363β3 . (36) A. Gaussian variational ansatz
≡ 4
For the spherically symmetric case β = 1, E vanishes
dd
in first-order perturbation theory, so that The Gaussian ansatz
β =1 : e(N61ε)=2−53334 +e2N13ε2+O(N21ε3) (37) g(~x)=(2π)−32κ3γe−12κ2(x21+x22+γ2x23) (44)
with e <0. isconvenient. Inadditiontoitsaspectratioγ,theshape
2
parameter,it containsthe scale parameterκ, sothat the
virialtheoremsofSec.IICwillbeobeyedfortheoptimal
D. Dimensionless variables choice of κ. For this scaled density, the scaled kinetic
energy is given by
These scaling laws invite the use of correspondingly
chosen dimensionless variables, such as Ekin =2−4331695−52π13κ2γ23 , (45)
4
¯hωN3
1 ¯h
~x=~r/a with a=N6 (38) and the scaled value of Etrap is
Mω
r
for the position and Etrap 1 β2
= 1+ . (46)
a3 N ¯hωN43 κ2 (cid:18) 2γ2(cid:19)
g(~x)= n(a~x) or n(~r)= g(~r/a) (39)
N a3 Theyexhibittheanticipateddependenceonthescalepa-
rameter κ, and so does the dipole-dipole interaction en-
for the density. The constraint (1) then appears as
ergy,
and the TFD enerZgy(da~xcq)ugi(r~xe)s=the1,form (40) ¯hEωNdd34 = 4κ√3πN16εγ(γ2−1)Z01dζ 1−ζζ22−+ζγ42ζ2 .
E(TFD)[g]= 3 (6π2)23 (d~x)[g(~x)]53 (47)
¯hωN43 10 This integral can, of course, be evaluated in terms of
Z
1 elementaryfunctions,butasitstandsweseeimmediately
+ (d~x) x2+x2+β2x2 g(~x)
2 1 2 3 that the integrand, and thus the integral, is positive for
Z
1 1 (cid:0) ′ 1 (cid:1) 3cos2θ ′ 0<γ <∞, so that
+ N6ε (d~x)(d~x)g(~x) − ′ g(~x),
2 ~x ~x 3 E >0 for γ >1 (oblate Gaussian),
Z | − | dd
(41) E =0 for γ =1 (spherical Gaussian),
dd
where θ is the angle between the polarization direction E <0 for γ <1 (prolate Gaussian), (48)
′ dd
(the x direction)andthe relativepositionvector~x ~x,
3
− consistent with the expectation that an oblate density
′ ′ ′
(~x−~x)·~µ= ~x−~x µcosθ =(x3−x3)µ. (42) has a larger magnetic interaction energy than a prolate
1 density. More explicitly, then, we have
The minimum of t(cid:12)he sca(cid:12)led TFD energy is e(N6ε) of
(cid:12) (cid:12)
(32) (with its implicit β dependence); it is obtained for Edd κ3 1
the g(~x) that obeys the dimensionless analog of (24), ¯hωN34 = 4√πN6εγ
2
1 6π2g(~x) 3 + 1 x2+x2+β2x2 1 ϑcotϑ 1 1
2 2 1 2 3 − for γ = >1,
(cid:2) +(cid:3)N16εZ(cid:0)(d~x′)1|−~x−3c~xo′s|23(cid:1)θg(~x′)= 12X2, (43) ×ϑcostinh2ϑϑ−1− 31 for γ = cos1ϑ <1.
where the value of X = R/a is determined by the con- sinh2ϑ − 3 coshϑ
straint (40).  (49)
4
30 2.9
7
25 6
5 2.4
4
20
3
1 6" 15 012 =5:2 (cid:20) 1.9
N 0.0 0.1 0.2 0.3 0.4 0.5 (cid:12)
10
unstable 1.4
5
stable
0 0.9
0 1 2 3 4 5 6
0 1 2 3 4 5
(cid:12) (cid:12)
FIG.1. Stability diagram. The dots, connected by a solid
linetoguidetheeye,definetheborderbetweensystemparam- 5
1
eters (β: aspect ratio of the trap; N6ε: effective interaction
strength) for which E(TFD) is bounded from below (stable) 4
ornotbounded(unstable). Theinsetshows ablow-upof the
region of pronouncedly prolate traps (β < 0.5). For oblate
3
traps with β = 5.2 or larger, the system is always stable, ir- (cid:13)
1
respective of the value of N6ε. This figure presents results
2
obtained in the Gaussian approximation, and so do all other
figures.
1
0
The Gaussiandensity (44) cannotmimic the ε=0 so- 0 1 2 3 4 5
lution (34) very well. But nevertheless, the resulting es-
(cid:12)
timate of the ε=0 energy, FIG.2. Dependenceofcloudparametersκ(cloudsize;top)
andγ(cloudshape;bottom)onβ,theaspectratioofthetrap.
Gaussian: e0 =2−16312255−45π16β31 =1.42β13 (50) The lines refer to different values of theinteraction strength:
1 1 1
N6ε = 0 (solid lines), N6ε = 1 (dashed lines), N6ε = 2
is only 4.4% in excess of the correct value (36). This (dash-dottedlines), N61ε=2.4 (dotted lines).
accuracy is sufficient for our purposes; and, in any case,
anerrorofafewpercentisasmallpricefortheenormous
simplification that the Gaussian ansatz brings about.
Dipolar interactions are partially attractive and par- On the other hand, one might naively expect an exactly
tially repulsive, depending on the configuration of the opposite effect in prolate traps (β < 1) where attractive
dipoles. One shouldkeepinmind the simple situationof interactions should dominate. This is not quite true –
two dipoles in the plane perpendicular to their polariza- indeed for moderate trap aspect ratios (1>β >0.5) the
tion, which repel each other, as opposed to the situation critical value of the dipole parameter is smaller, but in
of two attracting dipolar particles placed along the di- trapsthatareverysoft(β <0.5)inthez directionofro-
rection of their polarization. Extending this picture to a tational symmetry we observe an increase of the critical
1
cloud of trapped dipoles one would expect that attrac- valueofN6ε(seetheinsetinFig.1). Thiscanbeunder-
tion dominates in prolate traps, and repulsion in oblate stood with the help of the following argument. We note
traps (provided the dipoles are polarized along the trap thatthedipole-dipoleenergytermvanishesforauniform
axis as is the case here). In the case of predominant densitydistribution. Asthetrapismadesofterinthepo-
attraction one may surmise that instabilities occur. By larizationdirection,theshape ofthe cloudalongthe soft
varyingthetwosystemparameters(N61εandβ)wehave axis becomes more and more uniform, contributing to
investigated the issue of stability, see Fig. 1. the interaction energy to a lesser extent (this argument
From this stability diagram one concludes that, for was also used to interpret our earlier results for bosons
oblate traps (β > 1), the bigger the trap aspect ratio, interactingviacontactanddipole-dipoleforces,see[13]).
1 The dependence of γ and κ on β, shown in Fig. 2, is
the bigger values of the dipole parameter N6ε can be
consistent with this argument. We see that γ decreases
stabilized. In fact, we found numerically that fermions
with decreasing β whereas κ increases. Accordingly, the
form stable configurations for β >5.2 irrespective of the
cloudgetsstretchedalongthezaxisofsymmetry,andthe
strength of their dipole interaction (an analogous effect
was observed for dipolar bosons by Santos et al. [10]). diameterofthecircularcrosssectioninthe x,y plane(
∝
5
µ,butalso,tosomeextent,byvaryingN. Onecouldalso
exploittheωdependenceofε √ω,whichisparticularly
5 ∝
relevant for optical traps with tight confinement [22].
Fortheparametersofthefermionicchromiumisotope,
4 ω = 300 Hz and N = 106, one obtains N61ε = 0.012,
which is a very small value. Therefore, the conclusion
3
foratomspossessingevenrelativelylargemagneticdipole
(cid:13) moments is that irrespective of their number and the
2 stable
unstable trap frequency they will always remain stable against
collapse. The following amusing analogy offers a good
1
reasonfor this observation[6]. Let us compare the char-
acteristic sizes of the noninteracting Fermi gas and the
0
0 5 10 15 20 25 30 Bose-condensed atomic gas interacting via a repulsive
contact potential. In the Thomas-Fermi approximation
1
N6" the appropriate quantities, in units of ¯h/(Mω), read:
FIG.3. Aspectratioγ ofthecloudasafunctionofthein- 1 1
teraction strengthN16ε,forvariousvaluesoftheaspectratio RFermi =(48N)6 forfermionsandRBosep=(15Na0)5 for
β ofthetrap. Notethatγ =β forN16ε=0. Thedashedline bosons , where a0 is the scattering length. By equating
the two sizes one can calculate the effective, N depen-
is for β = 5.2, for which the cloud is stable for all values of
1 dent, scattering length due to the exclusion-principle–
N6ε. −1
induced repulsion: a0 1.68N 6. For typical num-
≈
bers of atoms, N = 103 106, this a is huge on the
0
···
scale set by typical scattering lengths for bosonic atoms
κ−1)isreduced. Atthecenterofthecloud,wethushave (a 10−3). Once we realize how strong is the repul-
0
≈
a relatively large volume of (almost) constant density, sion that originates in the Fermi statistics, we under-
and the inhomogeneous parts of the cloud are relatively standthatsmallatomicmagneticdipolescanhardlyhave
far apart. Taken together, these geometric features lead a noticeable effect on the behavior of dipolar fermionic
to a rather small dipole-dipole interaction energy. gases. However, for polar molecules the situation is dif-
Whereas the stability of the system considered can be ferent. For a trap frequency of ω =300Hz and N =106
well understood, the spatial behavior of the fermionic molecules of mass m 100 a.m.u. (the typical mass of
≈
cloud,especiallynearthecollapse,seemstobemuchless an alkaline dimer) and a typical electric dipole moment
intuitive. As we approach the critical parameter values, of 1 Debye, one reaches N61ε 11.5, which may very
the aspect ratio γ of the cloud decreases and the cloud wellputthesystemintotheuns≈tableregime–seeFig.1.
becomes elongated in the attractive z direction. This Inthissituation,a sufficientlylargeβ value willstabilize
type of behavior is general in the sense that it does not the system.
depend on the trap aspectratio,see Fig. 3. It is only for In order to assessthe quality of our variationalresults
traps with β > 5.2 that γ reaches an asymptotic value we havecomputed exactnumericalsolutions of Eq.(43).
1
(dependent on β) for extremely large values of N6ε.
The dependence of the dipolar energy E on the
dd
1
dipoleparameterN6εandthetrapgeometryisalsoofin-
terestasitisthequantityresponsibleforthe(in)stability
5.2
of the system, see Fig. 4. For all prolate traps, Edd re- 1.0 4.5
mains negative approaching some critical value at the
4
collapse point. For β < 5.2, the dipolar energy can be
3.5
positive for moderate dipole parameters,but iftheir val- 0.5
3 stable
cuoelslaaprsee.larFgoerenβou>gh5,.2E,dEd turinssanlwegaaytsivpeoisnitdiivceatainngdtihne- Edd 2.5 unstable
1dd 2
creases as a function of N6ε. 0.0 1.5
Finally, we take a look at the total TFD energy, see 1
0.5
Fig. 5. For β <5.2, the system becomes unstable at the
1
criticalvalue ofN6ε. ConsistentwithFig.1,we observe
1 -0.5
that larger values of N6ε are supported for β =0.1 and 0 1 2 3 4 5 6 7 8 9
β =1 than for β =0.5.
1
LetusnowdiscussthedipoleparameterN16εandgive N6"
FIG.4. Dipole-dipole interaction energyE asafunction
some typical values for it. Owing to the N dependence, dd
1
of the interaction strength N6ε. The solid lines are for trap
onecanlocatetheexperimentalsysteminvariousregions
aspect ratios β = 0.5,1,1.5,...,4.5; the dashed line is for
ofthestabilitydiagramofFig.1notonlybychoosing(or
β=5.2.
inducing,asproposedforbosons[9,10])aspecificvalueof
6
is2.75(>1.96)ascomparedto2.81(>2.55)obtainedan-
1.6 alytically in the Gaussian approximation.
5.2 Nowtworemarksaboutpossibleextensionsofourwork
)
0
) ( 1.4 4 are in order. Firstly, the results presented in this pa-
D per describe the situation of a dipolar fermionic gas at
F
T 3
( the temperature T = 0. An interesting subject of study
Æ E 1.2 2.5 would be extension of our theory to finite temperatures.
) 2
" Secondly,thereexistsaparallelapproachintheThomas-
1 6 1.5
N 1.0 Fermi model, namely the one in the momentum space
) ( 1 [18,19]. As many experiments with cold gasesyieldtheir
D 0.5
F unstable momentalcharacteristics,investigationofthisalternative
T 0.1
( E 0.8 approachalso presents an attractive theoretical task.
0 1 2 3 4 5 6
ACKNOWLEDGMENTS
1
N6"
FIG.5. Normalized TFDenergyasafunctionoftheinter-
B.-G. E. would like to thank W. Schleich for his sup-
action strength for various trap shapes. The universal func-
1 portinUlmwherepartofthisworkwasdone. B.-G.E.is
tion e(N6ε) of (32),normalized toits initial valuee0 =e(0),
isshown for trap aspect ratios β=0.1,0.5,1,1.5,...,4(solid gratefulforthe kindhospitalityextendedtohiminWar-
lines) andβ=5.2 (dashedline). Notethehorizontalslopeof saw. K. G. acknowledges support by Polish KBN grant
theβ =1 line (spherical trap), as required by(37). no2P03B05715. K.R.andK.G.aresupportedbythe
subsidyoftheFoundationforPolishScience. Partofthe
results has been obtained using computers at the Inter-
disciplinaryCentreforMathematicalandComputational
Thisequationwassolvedforthedensitydistributiong(~x) Modeling (ICM) at Warsaw University.
self-consistently starting from the known analytical re-
sult (34) for a non-interacting (ε = 0) Fermi gas in a
1
trap[6]andslowlyincreasingthe dipoleparameterN6ε.
1
For each value of N6ε the solution was iterated until
convergence was reached. Then, the value of the dipole
parameter was slightly increased. In order to compute
the dipole (integral) term we note that it has the form [1] B. DeMarco and D.S. Jin, Science 285, 1703 (1999).
of a convolution. Thus, it can be conveniently evaluated [2] M. Houbiers et al., Phys. Rev. A 56, 4864 (1997); D.
in the Fourier space where it is a simple product of the V.Efremov,M.S.Marenko,M.A.Baranov,M.Yu.Ka-
Fouriertransformsofthe density(computednumerically gan,cond-mat/9911169;M.A.BaranovandD.S.Petrov,
with the aid of an FFT) and the interaction potential, Phys. Rev. A 58, R801 (1998); G. Bruun, Y. Castin, R.
Dum, and K. Burnett,Eur. Phys. J. D 7, 433 (1999).
the latter being known analytically [13]:
[3] J.Ruostekoski,Phys.Rev.A60,R1775(1999)andPhys.
(d~x) ei~q·~x1−3cos2θ = 4π(1 3cos2α), (51) RReevv..LAe6tt1.,8053,3468075((22000000));;GP..MT¨o.rBmr¨auuanndanPd.CZo.lWler.,CPlharyks,.
~x3 − 3 −
Z | | Phys. Rev.Lett. 83, 5415 (1999).
[4] K.Mølmer,Phys.Rev.Lett.80,1804(1998);L.Viverit,
where α is the angle between the Fourier variable ~q and
C. J. Pethick, and H. Smith, Phys. Rev. A 61, 053605
the z direction. In orderto assurethat the integralterm
(2000); M. Amoruso, I.Meccoli, A.Minguzzi, and M. P.
is evaluated accurately we used a Gaussian distribution
Tosi, Eur. Phys. J. D 8, 361 (2000).
for comparison and chose the grid parameters accord-
[5] M. J. Bijlsma, B. A. Heringa, and H. T. C. Stoof, Phys.
ingly.
Rev. A 61, 053601 (2000).
Ournumericalcalculations,performedinthreedimen- [6] D. A. Butts and D. S. Rokhsar, Phys. Rev. A 55, 4346
sions, were quite demanding so we limited their use to
(1997).
a check of the main features of the stability diagram in [7] G. M. Bruun and K. Burnett, Phys. Rev. A 58, 2427
Fig. 1. The solutions obtainedsatisfy the virialrelations
(1998).
(29) very well, and total energies obtained numerically [8] F.Dalfovo,S.Giorgini,L.P.Pitaevskii,andS.Stringari,
arealwaysbelowthe correspondingvaluesfromthe vari- Rev. Mod. Phys. 71, 463 (1999).
ational analysis. The critical value of the dipole param- [9] M. Marinescu and L. You, Phys. Rev. Lett. 81, 4596
1
eter for a spherical trap is N6ε = 1.96 as compared to (1998).
2.55 obtained by the variational calculation. We also [10] L. Santos,G. V.Schlyapnikov,P.Zoller, andM. Lewen-
confirmedthe effect of increaseofthe criticalinteraction stein, Phys.Rev.Lett. 85, 1791 (2000).
strength that we found, in the Gaussian approximation, [11] J.D.Weinsteinetal.,Nature(London)395,148(1998);
1
for prolate traps: for β = 0.07 the critical value of N6ε H. L. Bethlem, G. Berden, and G. Meijer, Phys. Rev.
7
Lett.83,1558(1999);H.L.Bethlemetal.,Nature(Lon-
don) 406, 491 (2000).
[12] S.YiandL.You,Phys.Rev.A61,041604(2000);cond-
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[13] K. G´oral, K. Rza¸z˙ewski, and T. Pfau, Phys. Rev. A 61,
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[15] E. Fermi, Rend.Lincei 6, 602 (1927).
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[17] See for instance M. Brewczyk and K. Rza¸z˙ewski, Phys.
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[18] B.-G. Englert, Phys.Rev. A 45, 127 (1992).
[19] M. Cinal and B.-G. Englert, Phys. Rev. A 48, 1893
(1993).
[20] An analogous argument can be used for electric dipoles
–inthiscasethecontacttermisneededtohavethecurl
of an electric field vanish.
[21] P. A. M. Dirac, Proc. Cambridge Philos. Soc. 26, 376
(1930).
[22] V. Vuleti´c, C. Chin, A. J. Kerman, and S. Chu, Phys.
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8