Thomas-Fermi ground state of dipolar fermions in a circular storage ring

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This is the published version of a paper published in Physical Review A. Atomic, Molecular, and Optical Physics.

Citation for the original published paper (version of record):

Dutta, O., Jääskeläinen, M., Meystre, P. (2006)

Thomas-Fermi ground state of dipolar fermions in a circular storage ring.

Physical Review A. Atomic, Molecular, and Optical Physics, 73(4): 043610 http://dx.doi.org/10.1103/PhysRevA.73.043610

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Thomas-Fermi ground state of dipolar fermions in a circular storage ring

O. Dutta, M. Jääskeläinen,*and P. Meystre

Department of Physics and College of Optical Sciences, The University of Arizona, Tucson, Arizona 85721, USA 共Received 25 December 2005; published 18 April 2006兲

Recent developments in the field of ultracold gases has led to the production of degenerate samples of polar molecules. These have large static electric-dipole moments, which in turn causes the molecules to interact strongly. We investigate the interaction of polar particles in waveguide geometries subject to an applied polarizing field. For circular waveguides, tilting the direction of the polarizing field creates a periodic inhomogeneity of the interparticle interaction. We explore the consequences of geometry and interaction for stability of the ground state within the Thomas-Fermi model. Certain combinations of tilt angles and interaction strengths are found to preclude the existence of a stable Thomas-Fermi ground state. The system is shown to exhibit different behavior for quasi-one-dimensional and three-dimensional trapping geometries.

DOI:10.1103/PhysRevA.73.043610 PACS number共s兲: 03.75.Ss, 03.65.Sq, 05.30.Fk

I. INTRODUCTION

Exciting new opportunities are now emerging in matter- wave optics as a result of the availability of ultracold, possi- bly quantum-degenerate molecular systems 关1–4兴. So far, molecular condensation has only been demonstrated when starting from pairs of identical atoms, but two recent experi- ments have reported the observation of heteronuclear Fesh- bach resonances in Bose-Fermi mixtures of⁶Li and²³Na关5兴 in one case and of⁸⁷Rb and⁴⁰K in the other关6兴. Sweeping through such resonances is expected to lead to the coherent formation of a substantial amount of heteronuclear molecules. Alternatively, photoassociation has resulted in the production of ultracold metastable RbCs molecules in their lowest triplet state when starting from a laser-cooled mixture of⁸⁵Rb and¹³³Cs关7兴.

Much attention has also been given to the trapping and guiding of matter waves in “atom chip” microstructures 关8,9兴, a natural route for applications such as rotation sensors based on the Sagnac effect关10兴. Experimental and theoreti- cal work has focused on various trapping schemes for ring geometries, using both magnetic关11,12兴 and magnetoelectro- static mechanisms 关13兴. In several recent developments, Sauer, Barrett, and Chapman关14兴, Wu et al. 关15兴, and Gupta et al. 关11兴 have realized atomic storage rings in magnetic waveguides. Microtraps for heteronuclear molecules have also been considered by Tscherneck et al.关16兴. These molecules 关17兴 are polar, with a permanent electric-dipole mo- ment of the order of ea₀关18兴. As such, their center-of-mass trajectories can be manipulated by relatively modest static or quasistatic inhomogeneous electric fields 关19–21兴. The dipole-dipole interaction is central to the dynamics of molecular samples关22–24兴, where it is expected to lead to fas- cinating new physics. For example, Bochinski et al. have demonstrated the phase-space manipulation of polar molecules in the presence of a dipole-dipole interaction关25兴 by using pulsed inhomogeneous electric fields, and it would be of considerable interest to extend such studies to the quan-

tum regime. The anisotropy of the dipole-dipole interaction should also permit us to understand how the nature and stability of quantum-degenerate atomic and molecular systems is influenced by the anisotropic character of interparticle in- teraction关26兴. New kinds of quantum phase transitions have been predicted in these systems关27兴, and their ground state and stability are expected to differ fundamentally from the situation in condensates characterized by a contact interac- tion关26,28兴.

The question that is specifically addressed in the present paper is the role of the dipole-dipole interaction in determin- ing the ground state of quantum-degenerate fermionic polar molecules in a ring-shaped waveguide. By applying a static electric field tilted with respect to the waveguide, the dipole- dipole interaction between molecules can be made inhomogeneous along the ring, the amount of inhomogeneity being controlled by the tilt angle. We consider specifically situations where the trapping mechanism is different from the polarizing mechanism, as is possible, for instance, with CrRb which possesses both a large permanent magnetic-dipole moment and a large static electric-dipole moment.

The paper is organized as follows: Section II discusses the effective dipole-dipole interaction for molecules trapped in a storage ring with circular geometry. In case the transverse trap energy is larger than the mean-field energy associated with the dipole-dipole interaction, the system is quasi one dimensional, while it is truly three dimensional in the opposite limit. Section III concentrates on the quasi-one- dimensional case and determines the ground state of the system in the Thomas-Fermi local density approximation.

Regions where this approach leads to unphysical results are determined and found to become more of a problem in the regime of low densities, a result of the dominance of the mean-field energy over the kinetic energy. Section IV ex- tends these considerations to the three-dimensional limit, where the transverse distribution is no longer given by the trap ground state. Finally Sec. V is a summary and outlook.

II. EFFECTIVE INTERACTION

We consider an ultracold sample of neutral fermionic atoms or molecules with permanent electric- or magnetic-

*Corresponding author. Email address: mrq@optics.arizona.edu

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dipole moment␮ជ and confined to a ring-shaped trap of ra- dius R and length L₀. The interaction between two pointlike dipoles␮ជ1, and␮ជ2 at positions rជ1and rជ2is given by

V_dd共rជ1,rជ2兲 = 1

4␲␧0冋^␮ជ¹^r^·³^␮ជ²⁻^共^␮ជ¹^{· r}^ជ^r^兲共⁵^␮ជ²^{· r}^ជ^兲册^, ^共1兲

where

r⬅ 兩rជ兩 = 兩rជ − r1 ជ 兩₂ 共2兲 is the distance between the two dipoles. We consider specifically the situation where the atomic or molecular sample is polarized along some arbitrary direction, in which case Eq.

共1兲 reduces to

V_dd共rជ1,rជ2兲 = − ␮1␮2

4␲⑀0r³关2 cos²␪− sin²␪兴, 共3兲 where␪is the angle between rជand the direction of polariza- tion of the dipoles, which we take to be tilted by an angle␣ with respect to the normal of the plane of the trap; see Fig. 1.

In the limit of strong fermion confinement in the radial direction, their single-particle wave function of the atoms or molecules can be factorized as

⌿共rជ_⬜,z兲 = ⌿_⬜共rជ_⬜兲⌿共z兲, 共4兲 where rជ_⬜is the radial coordinate vector measured from the center of the ring-shaped trap and z is the longitudinal coor- dinate along the ring. In writing Eq. 共4兲, we implicitly as- sumed that the transverse trap potential is invariant along z.

We further assume that the two-particle wave function

⌿共rជ1, rជ2兲 can be factorized as

⌿共rជ1,rជ2兲 = ⌿_⬜共rជ_1⬜兲⌿_⬜共rជ_2⬜兲⌿共z1,z₂兲, 共5兲 in which case the effective longitudinal dipole-dipole inter- action between two identical dipoles at positions z₁and z₂is given by

V_dd,储共z1,z₂兲

=冕冕^兩⌿^⬜^共r^ជ^1⬜^兲兩²^兩⌿^⬜^共r^ជ^2⬜^兲兩²^V^dd^共r^ជ¹^,r^ជ²^兲dr^ជ^1⬜^dr^ជ^2⬜^.

共6兲 Assuming that the transverse single-particle wave function⌿_⬜共rជ_⬜兲 is a Gaussian of half-width a0and that the trap radius R is large compared to the transverse trap size, we can evaluate Eq.共6兲 to find

V_dd,_储共z1,z₂兲 = ␮²

4␲⑀0a₀³冋^{3 sin}²^␣^sin²^冉²^L^␲⁰^z^c^冊^{− 1}册

⫻

冋

^␨⁻^冑^␲²^{共1 +}^␨²^兲erfcx^冉^冑^␨²^冊

册

^, ^共7兲

where

erfcx冉^冑^␨²冊^{= erfc}冉^冑^␨²冊^exp^冉⁻^␨²²^冊 ^共8兲

and erfc共x兲 is the complementary error function. Here, zc=共z1+ z₂兲/2 and zr= z₁− z₂ are the center-of-mass and relative coordinates of the fermion pair and

␨共zr兲 =2R₀

a₀ 冏^sin冉^␲L^z₀^r冊冏^. ^共9兲

The longitudinal potential can therefore be factorized as V_dd,_储共zc,zr兲 = ␮²

4␲⑀0a₀³Vc共zc兲Vr共zr兲. 共10兲 The center-of-mass and relative components V_c共zc兲 and V_r共zr兲 of the effective potential are plotted in Fig. 2. The short-range behavior of V_r共zr兲 is regularized by the trans- verse integration to yield finite values as zr→0. For large FIG. 1. Storage ring of radius R with dipoles ␮1 and ␮2 at

locations z₁and z₂. The poles are tilted at an angle␣ with respect to the direction e₃.

FIG. 2. 共a兲 Vr共zr兲 as a function of interparticle distance z_r= z₁− z₂.共b兲 Vc共zc兲 as a function of the center-of-mass coordinate z_c=共z1+ z₂兲/2. In 共a兲 the dashed line shows 共zr/ a₀兲^−1/3, the bare dipole potential, as a reference.

DUTTA, JÄÄSKELÄINEN, AND MEYSTRE PHYSICAL REVIEW A 73, 043610共2006兲

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distances, V_r共zr兲⬇zr

−3, as expected for the dipole-dipole in- teraction. Note also that V_c共zc兲 depends on the tilt angle␣ and can become negative for

␣⬎ arcsin冉^冑¹³冊^, ^共11兲

resulting in the appearance in the storage ring of regions with attractive interaction.

III. QUASI-ONE-DIMENSIONAL REGIME

We first consider a quasi-one-dimensional situation where the dipolar fermions are subject to a strong transverse trapping and thus remain in the transverse ground state of the storage ring potential. Our goal is to determine the ground state of the system within the Thomas-Fermi approximation 关29,30兴.

Our starting point is the one-dimensional energy functional

E₁= T₁+ V₁, 共12兲 which in the Thomas-Fermi theory can be expressed as a functional of the ground-state particle density distribution.

Within that local density approximation, the kinetic energy becomes

T₁=冕0 L

n共z兲t共n,z兲dz, 共13兲

where n共z兲 the particle density. The kinetic energy density is

t共n,z兲 =冕冉^p^2m^储²^共z兲冊P„p储共z兲dp共z兲, 共14兲 where m is the particle mass andP(p_储共z兲)dp_储共z兲 is the prob- ability that a particle at the position z has a momentum be- tween p_储共z兲 and p_储共z兲+dp_储共z兲 and the density dependence is still obscured at this point.

At zero temperature T = 0,P(p_储共z兲)dp_储共z兲 is given explic- itly by

P„p储共z兲…dp储共z兲 =冉2p_储¹_,f共z兲冊^dp^储^共z兲, ^共15兲

where p_储,f共z兲 is the position-dependent Fermi momentum. In- serting this expression into Eq.共14兲 we have

t共n,z兲 =p_储,f² 共z兲

6m . 共16兲

In one dimension the density n共z兲 and the Fermi momentum p_储,f共z兲 are related by

n共z兲 =p_储,f共z兲

␲ប . 共17兲

Inserting this into Eq.共13兲 and carrying out the integration results in the Thomas-Fermi kinetic energy functional

T₁=冉^␲^6m²^ប²冊ⁿ³^共z兲. ^共18兲

Combining finally this expression with the interaction energy contribution gives then the Thomas-Fermi energy functional as

E₁=␲²ប² 6m 冕0

L₀

n³共z兲dz

+1

2冉⁴␲⑀^兩^␮^兩0²a₀³冊冕0 L

dzdz

⬘

^Vdd,储共z/a0,z

⬘

^/a0兲n共z兲n共z

⬘

兲, 共19兲 where we have factorized the absolute squared two-body longitudinal wave function as

兩⌿共z,z

⬘

兲兩²= n共z兲n共z

⬘

兲. 共20兲 Scaling lengths to the characteristic size a₀of the transverse single-particle wave function⌿共rជ_⬜兲 via

x = z

a₀ 共21兲

and normalizing the density for the correct total number of atoms via

␳1共x兲 =冉^aN⁰冊^n共z兲, ^共22兲

with

冕0 L

␳1共x兲dx = 1, 共23兲

yields finally the dimensionless Thomas-Fermi energy functional as

␧1= E₁

␶^N³⁼冕0 L

␳1

3共x兲dx +g₁ 2冕0

L

dxdx

⬘

^Vdd,储共x,x

⬘

兲␳1共x兲␳1共x

⬘

兲.

共24兲 Here␧1 is the dimensionless energy of the trapped fermions and the dimensionless interaction constant g₁, given by

g₁=1

␶冉⁴␲⑀^␮0²Na₀³冊^, ^共25兲

is the ratio of the characteristic transverse dipole-dipole energy to the characteristic transverse kinetic energy per particle:

␶⁼ ␲²ប²

6ma₀². 共26兲

The kinetic energy term, which has a cubic dependence on density, dominates over the two-body interaction at high densities. In this quasi-one-dimensional regime the dimen- sionless interaction strength g₁⬀N⁻¹scales in the same way as the dimensionless coupling constant in the Tonks-

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Girardeau gas关31兴. Likewise, the scaling of the dimensionless energy 共24兲 is similar to that of the impenetrable Bose gas in the thermodynamic limit关32兴.

The ground-state density of fermions is obtained by mini- mizing the energy␧ under the constraint that the total num- ber of particles be equal to N. In the variational equation we have

␦

冋

^␧¹⁻^␮¹^冉^冕⁰^L^␳¹^{共x兲dx − 1}^冊

册

^{= 0,} ^共27兲

␮1 being the chemical potential. It should be noted that assuming␮1to be a variational parameter in the energy functional trivially leads to the conservation of particle number and adds no information about the value of the chemical potential. The resulting integral equation for the variational ground state is

␳12共x兲 +g₁ 3冕0

L

V_dd,储共x,x

⬘

兲␳1共x

⬘

兲dx

⬘

⁼␮1. 共28兲 Solving Eq. 共28兲 numerically requires knowledge of the chemical potential␮1. In the noninteracting case, it is equal to the Fermi energy, which is given by 1 / L²in dimensionless units. For weak interaction strengths, this same value can be taken as a first approximation. The exact value of the chemical potential, however, depends on the interaction strength and, hence, on the degree of inhomogeneity of the system as we shall see.

To determine the longitudinal density profile corresponding to the semiclassical many-body ground state, we invoke the periodicity of the system to express ␳1共x兲 as a Fourier series in the dimensionless coordinate x. It follows from the normalization condition 共23兲 that the first term in such a series must equal 1 / L, giving

␳1共x兲 =1

L冋^{1 −}^␩¹^cos^冉⁴^L^␲^x^冊⁻^␩²^cos^冉⁸^L^␲^x^冊⁺ ^¯册^.

共29兲 In the following we truncate the series after two terms and treat the first- and second-harmonic amplitudes␩1and␩2of the density as variational parameters that minimize the en- ergy共24兲,

⳵␧1

⳵␩1

= 0, 共30兲

⳵␧1

⳵␩2

= 0. 共31兲

This truncation, which considerably simplifies the numerics, reproduces the density profile␳1共x兲 and variational energy ␧1

with the chemical potential taken as␮1= 1 / L², in the case of weak interaction strengths of experimental relevance, as further discussed later on.

Figure 3 shows the resulting density profile for a tilt angle

␣= 3␲/ 20 and as a function of the effective interaction strength g₁. For g₁= 0 the density distribution is of course uniform, but as g₁ is increased, the inhomogeneity in the

interaction forces the molecules to increasingly bunch to- gether in the two regions of lower interaction energy. This can be seen by comparing the density distribution of Fig. 3 with the spatial dependence of Vc共zc兲 in Fig. 2共b兲. The presence of inhomogenous ground states in ring geometries has previously been found for bosons with homogenous interaction strength关33,34兴.

In addition to the tilt angle␣, the ground-state properties of the system depend on two dimensionless parameters only:

the aspect ratio L = L₀/ a₀ between the characteristic size of the transverse wave function and the storage ring length and the interaction constant g₁, as defined in Eq.共25兲. We found numerically the existence of a region in parameter space with no stable solution of Eq.共30兲. This appears to be due to the attractive contribution of the dipole interaction in some regions along the ring, which for strong enough interaction strengths may destabilize the Thomas-Fermi ground state.

Figure 4 shows the domain of stability of the Thomas- Fermi ground state共a兲 for a fixed aspect ratio L and varying tilt angle␣^and共b兲 for a fixed tilt angle and varying effective length L. Increasing the tilt angle causes the ground state to become unstable for lower values of the interaction strength g₁, which is inversely proportional to the number of particles. Higher values of␣cause the attractive contribution to V_c共z兲 to become stronger, thus localizing the fermionic atoms or molecules into a region of decreasing size. This bunching gives rise in turn to a density profile with a high gradient, eventually causing the mean-field treatment to break down.

Figure 4共b兲 shows that larger values of the effective length result in the Thomas-Fermi ground state becoming unstable for lower values of g₁. Equivalently, for a given transverse trap frequency, a larger ring size increases the number of molecules that can be stored, as would be intuitively expected.

One important question is to determine whether the instability of the Thomas-Fermi ground state is simply a result of the truncation scheme共29兲 and whether increasing the number of terms in the expansion would improve things in case the density gradients along the ring become significant.

While increasing the number of Fourier components does FIG. 3. One-dimensional density distribution as a function of z and g₁for␣=3␲/20.

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help to better determine the stability domain, the instability appears to be real, as can be seen by considering the simplified model of a local two-body interaction

V_dd,储共zc,zr兲 = Vc共zc兲␦共zr兲. 共32兲 In that case Eq.共28兲 reduces to

␳1 2共x兲 +g₁

3Vc共x兲␳1共x兲 =␮1. 共33兲 Equation共28兲 is quadratic in ␳1 and can be solved with the chemical potential␮1 as a parameter. We find that, for

␮1艌0, the normalization constraint 共23兲 can only be satis- fied for certain values of the parameters␣^{, L, and g}1.

The electric-dipole moment of a typical heteronuclear dimer is of the order of 1 D, and its mass is of the order of

⬃100 g/mol. Tight transverse trapping yields a0⬃1␮^m.

For a storage ring length of 500␮m these figures imply that a critical interaction strength is g₁⯝2⫻10⁻³, a value found in Fig. 5 at L⬃500. As a consequence of the scaling of g1

with 1 / N, it follows that the Thomas-Fermi ground state is stable for particle numbers around N⬃10²– 10⁴and unstable for smaller samples.

The strength of the magnetic-dipole moment for paramag- netic molecules such as CrRb or atoms with a permanent magnetic moment is of the order of ⬃10␮B. This results in an effective one-dimensional interaction strength g₁

⬃10⁻⁶– 10⁻⁸, yielding a similar stability regime to the case of permanent electric dipoles treated above.

IV. THREE-DIMENSIONAL CASE

When the transverse trapping potential of the ring is weak compared to the two-body interaction energy, the system is truly three dimensional and the wave function of the fermi-

ons can no longer be taken as the single-particle ground state. We instead use a Gaussian profile whose width is treated as a variational parameter.

The three-dimensional Thomas-Fermi energy functional is E = T₃+ V_trap+ V_int, 共34兲 where the kinetic energy T₃ is given by

T₃=␥冕ⁿ^5/3^共r^ជ^兲d³^r, ^共35兲

n共rជ兲 is the single-particle density obeying the normalization condition

冕^n共r^ជ^兲d³^{r = N,} ^共36兲

N is the total number of particles, and

␥⁼冉␲³冊^2/3³10m^␲²^ប². 共37兲 A key difference between Eq.共35兲 and the one-dimensional expression共13兲 is the scaling of the kinetic energy with density, a property with well-known consequences such as the appearance of a Tonk-Girardeau gas in dilute quantum- degenerate bosonic systems.

The second term in the Thomas-Fermi energy functional E is the transverse trapping potential of the storage ring,

V_trap=1

2m␻_⬜² 冕^共x²^{+ y}²^兲n共r^ជ^兲d³^r, ^共38兲

and the third term is the effective dipole-dipole interaction V_int=1

2 冕^V^dd^共r^ជ^,r^{ជ 兲n共r}

^⬘

^ជ^兲n共r^{ជ 兲d}

^⬘

³^rd³^r

^⬘

^. ^共39兲

As in the one-dimensional case, we factorize the density ac- cording to

FIG. 4. Thomas-Fermi stability diagram for the one-dimensional case. Critical dimensionless interaction strength as a function of共a兲 L for␣=2␲/5 and 共b兲 as a function of ␣ for L=100. The dashed line shows the critical interaction strength as a function of共a兲 L for

␣=2␲/5 and 共b兲 ␣ for L=100 for a local interaction potential V_r共zr兲=␦共zr兲.

FIG. 5. Stability diagram of the Thomas-Fermi ground state for the three-dimensional regime:共a兲 Critical interaction strength g3共⌳兲 for␣=2␲/5; 共b兲 g3共␣兲 for ⌳=100.

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n共rជ兲 = 兩⌽_⬜共rជ_⬜兲兩²n共z兲, 共40兲 but the transverse wave function is now a Gaussian

⌽_⬜共rជ_⬜兲 = 1

冑␲⌬r_⬜exp冉⁻2共⌬r^r^⬜²_⬜兲²冊 ^共41兲

whose width ⌬r_⬜ is taken as a variational parameter. With this ansatz the normalization condition共40兲 becomes

冕0 L₀

n共z兲dz = N. 共42兲

Inserting Eqs.共40兲 and 共41兲 into the kinetic energy 共35兲 gives T₃= ␥

共⌬r_⬜兲^4/3冕ⁿ^5/3^共z兲dz. ^共43兲

Similarly the transverse trapping energy and the interaction energy become

V_trap=N

2m␻_⬜²共⌬r_⬜兲² 共44兲 and

V_int= 兩␮兩² 8␲⑀0共⌬r_⬜兲³冕0

L₀

dzdz

⬘

^Veff冉⌬r^z_⬜, z

⬘

⌬r_⬜冊ⁿ^{共z兲n共z兲.}

共45兲 We introduce for convenience the dimensionless length

x = z

r_eff, 共46兲

where r_eff= N^1/6冑ប/m␻_⬜, the dimensionless density

␳3共x兲 =冉^r^N^eff冊^n共z兲, ^共47兲

with 冕0

⌳

␳3共x兲dx = 1 共48兲

and the dimensionless variational parameter ᐉ = ⌬r_⬜/r_eff.

This scaling results in the dimensionless energy functional

␧3=E₃r_eff²

␥N^5/3= 1 ᐉ^4/3冕0

⌳

␳3共x兲^5/3dx +␤ᐉ²

+1 2

g₃ ᐉ³冕0

⌳

V_eff冉^xᐉ,x

⬘

ᐉ冊^␳³^共x兲^␳³^共x

^⬘

^兲dxdx

^⬘

^, ^共49兲

where ⌳=L0/ r_eff, ␤⁼共␲^{/ 3}兲^2/3共5/3␲²兲, and the dimensionless interaction strength is

g₃= 兩␮兩²

4␲⑀0␥冑^m^␻ប^⬜N^1/6. 共50兲 In marked contrast with the quasi-one-dimensional case, the two-body interaction now dominates over the kinetic en-

ergy in the limit of high densities, with a scaling reminiscent of the corresponding three-dimensional dilute Bose gas with hard-sphere interaction关35兴.

As in the quasi-one-dimensional case we exploit the periodicity of the longitudinal density distribution to justify a variational ansatz of the form

␳3共x兲 = 1

⌳关^{1 −}␩3cos共4␲^x/⌳兲兴^, 共51兲 where we now keep only the first Fourier component for numerical simplicity. The ground-state density is determined by optimizing the energy functional ␧3 with respect to the free parameters,

d␧3

dᐉ = 0, d␧3

d␩3D

= 0. 共52兲

Figure 5 shows the resulting Thomas-Fermi stability diagram as a function of⌳ for constant tilt angle␣^{= 2}␲^{/ 5 and} as a function of␣ for a fixed effective length⌳=100. For a fixed effective length, a larger value of the tilt angle ␣ increases the range of effective strengths, resulting in a valid Thomas-Fermi ground state. The system becomes uncondi- tionally stable for ␣艋arcsin共1/冑3兲⬃35° since the dipole- dipole interaction is then repulsive along the whole ring. The stability diagram for a fixed angle ␣= 2␲/ 5 shown in Fig.

5共a兲 is more complex, with a minimum of g3 around⌳=8.

For the same atomic or molecular parameters as in the quasi-one-dimensional case, but with a weaker trap of width 10␮m, we find that N艋4⫻10⁴ for the case of fermions with permanent electric dipoles and ␣= 2␲^{/ 5 and N}艋2

⫻10⁴ for ␣⁼␲/ 4. These values indicate that it should be fairly easy to investigate physics beyond mean-field theory for ultracold fermions in ring geometries.

Figure 6 shows the relative contributions to the energy functional as functions of the effective length⌳ for fixed tilt angle␣^{= 2}␲/ 5. The interaction energy V_intincreases and the attractive interaction energy decreases with⌳, but the total energy is always dominated by the kinetic energy, and as a result the critical value of the effective strength increases.

Figure 5 shows that the critical interaction strength increases for small values of ⌳, even though the 共attractive兲 interaction energy decreases; see Fig. 6. This behavior can be intuitively understood in terms of a simplified analysis that takes the relative part of the interaction to be short ranged,

VR=␦关共x − x

⬘

^{兲/ ᐉ 兴.} ^共53兲 We consider for illustration the stability of the Thomas- Fermi ground state for a tilt angle␣⬎arcsin共1/冑3兲 and thus take the interaction to be attractive. Inserting the ansatz共53兲 into Eq.共49兲 gives

␧3⬇ 1

ᐉ^4/3冉⌳¹冊^2/3⁺^␤^ᐉ²^{− g}³^␬ᐉ¹², 共54兲 where␬ is a coefficient that includes numerical factors and the value of the interaction energy. For the value ᐉm that

043610-6

(8)

minimizes Eq.共54兲 the effective interaction strength will sat- isfy the condition

冋²³^冉^⌳¹^冊^2/3^{− 2}^␬^ᐉ^m^2/3^g³册³⁼^␤³^ᐉ^m¹⁰^. ^共55兲

Forᐉmto be positive we have then

冉⌳¹冊^2/3^⬎³2

␬1g₃ ᐉm

2/3, 共56兲

where␬1 is a positive constant. Equation 共55兲 can further- more be rearranged to give

冋²³^冉^⌳¹^冊^2/3⁻^␤^ᐉ^m^10/3册³⁼^共2^␬^ᐉ^gm2³^兲³, 共57兲 which gives, forᐉm⬎0,

冉⌳¹冊^2/3^⬎^␬²^ᐉ^m^10/3^, ^共58兲

where␬2is another positive constant. Upon combining Eqs.

共55兲 and 共58兲 we finally have

g₃⬍␬3⌳^−4/5, 共59兲 where the constant ␬3 is also positive. It follows from that simplified model that increasing⌳ causes the critical value of the g₃to decrease, which explains qualitatively its behavior in the regime of small effective lengths. The upper bound

given by Eq.共59兲 is shown as a dashed line in Fig. 4共a兲. For small values of⌳ the critical interaction strength is seen to scale as⌳^−4/5, in agreement with the simplified analysis. This indicates that for small values of⌳ the short-range properties of the interaction dominate the stability properties of the system, which remains stable as long as the different contributions to the energy functional共51兲 can balance each other.

V. CONCLUSIONS

In this paper we have studied the Thomas-Fermi ground state of a system of spin-polarized fermions interacting via the dipole-dipole interaction in a circular waveguide geometry and subjected to an additional static external field. That field results in an inhomogeneous interaction with a periodicity of␲in the center-of-mass frame, and due to the tunable inhomogeneity of the interaction, it can be attractive in certain domains along the ring.

We find that the Thomas-Fermi ground-state density distribution of the atomic or molecular fermions is then likewise periodic, with a large fraction of particles residing in the region of minimum energy. For small rings and large interaction strengths the particles are then localized in diametri- cally opposite regions. However, in the quasi-one- dimensional case there is a lower bound to the number of particles for which that ground state is stable and an upper bound in three-dimensional case. The different scaling in these two situations is a direct consequence of the scaling of the kinetic energy of the fermionic sample with density.

The instability of the Thomas-Fermi ground state, and the concomitant breakdown of the Thomas-Fermi theory, signals the presence of nontrivial quantum correlations in the true many-body ground state, and a treatment beyond mean-field theory is needed to deal with that situation. It is known, for instance, that in quasi-one-dimensional systems and for low particle densities fermionic systems can under appropriate circumstances undergo a Wigner crystallization关36兴, in anal- ogy with the situation for trapped ions; see e.g.关37–39兴.

We finally note that in the case of large rings 关21兴, the system is also far from the Thomas-Fermi situation described in this paper. A quantum dynamical treatment using wave- packet methodologies that is more appropriate in this case will be described in a subsequent paper.

ACKNOWLEDGMENTS

This work is supported in part by the U.S. Office of Naval Research, by the National Science Foundation, by the U.S.

Army Research Office, by the Joint Services Optics Pro- gram, and by the National Aeronautics and Space Adminis- tration.

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