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arXiv:0803.2488v2 [cond-mat.str-el] 6 Aug 2008

BCS-BEC Crossover of a Quasi-two-dimensional Fermi Gas: the Significance of

Dressed Molecules

Wei Zhang, G.-D. Lin, and L.-M. Duan

FOCUS center and MCTP, Department of Physics, University of Michigan, Ann Arbor, MI 48109

(Dated: August 6, 2008)

We study the crossover of a quasi-two-dimensional Fermi gas trapped in the radial plane from

the Bardeen-Cooper-Schrieffer (BCS) regime to the Bose-Einstein condensation (BEC) regime by

crossing a wide Feshbach resonance. We consider two effective two-dimensional Hamiltonians within

the mean-field level, and calculate the zero temperature cloud size and number density distribution.

For model 1 Hamiltonian with renormalized atom-atom interaction, we observe a constant cloud size

for arbitrary detunings. For model 2 Hamiltonian with renormalized interactions between atoms

and dressed molecules, the cloud size deceases from BCS to BEC side, which is consistent with the

picture of BCS-BEC crossover. This qualitative discrepancy between the two models indicates that

the inclusion of dressed molecules is essential for a mean-field description of quasi-two-dimensional

Fermi systems, especially on the BEC side of the Feshbach resonance.

PACS numbers: 03.75.Ss, 05.30.Fk, 34.50.-s

I.INTRODUCTION

The interest on low-dimensional Fermi systems has

been recently reinvoked by the experimental develop-

ments of cooling and trapping atoms in optical lat-

tices [1, 2, 3] and on atom chips [4]. With the aid of

tuning an external magnetic field through a Feshbach

resonance, these techniques provide a fascinating pos-

sibility of creating quasi-low-dimensional Fermi systems

with a controllable fermion-fermion interaction. In par-

ticular, the interaction between fermions can be tuned

from a Bardeen-Cooper-Schrieffer (BCS) limit to a Bose-

Einstein condensation (BEC) limit, such that the BCS-

BEC crossover can be studied in quasi low dimensions.

The BCS-BEC crossover has been extensively studied in

three-dimensional (3D) Fermi systems, where a single-

channel model [5] and a two-channel model [6] are both

applied to give a consistent description around a wide

Feshbach resonance.This agreement between single-

and two-channel models is rooted from the fact that the

closed-channel (Feshbach molecule) population is negli-

gible near a wide resonance, so it will not cause any sig-

nificant difference by taking the molecules into account

(as in the two-channel model) or completely neglecting

them (as in the single-channel model). The BCS-BEC

crossover of a uniform two-dimensional (2D) Fermi sys-

tem has also been considered in connection with high-

Tc superconductors [7], where an effective 2D Hamil-

tonian with renormalized fermion-fermion interaction is

employed.

In this paper, we study the BCS-BEC crossover in a

quasi-2D Fermi gas, first using an effective 2D Hamilto-

nian with renormalized atom-atom interaction (model 1)

[8, 9], and then a more general model with renormalized

interaction between atoms and dressed molecules (model

2) [10]. The dressed molecules mainly come from popula-

tion of atoms in the excited levels along the strongly con-

fined axial direction near a Feshbach resonance [10, 11].

When considering the effect of a weak harmonic trap

in the two loosely confined dimensions under the local

density approximation (LDA), we adapt the mean-field

(MF) treatment to calculate the zero temperature cloud

size and number density distribution in the radial plane.

We find a significant difference between the two models.

By using model 2, we show that the cloud size decreases

from the limiting value of a weakly interacting Fermi gas

as one moves from the BCS to the BEC side of the Fes-

hbach resonance, and approaches to the limiting value of

a weakly interacting Bose gas in the BEC limit. This be-

havior is a signature of the BCS-BEC crossover in quasi

two dimensions. On the contrary, model 1 fails to de-

scribe this crossover behavior, but predicts a constant

cloud size and identical density profile for all magnetic

field detunings. This discrepancy implies that the MF

results given by model 1 is unreliable, even at a quali-

tative level. Given this qualitative discrepancy and the

problem associated with model 1 for description of the

two-body ground state of the system [12], it is likely that

the oversimplification is rooted in the model itself instead

of the mean-field approximation.

The quasi-2D geometry can be realized by arranging a

one-dimensional (1D) optical lattice along the axial (z)

direction and a weak harmonic trapping potential in the

radial (x-y) plane, such that fermions are strongly con-

fined along the z direction and form a series of quasi-2D

pancake-shaped clouds [3]. Each such pancake-shaped

cloud can be considered as a quasi-2D Fermi gas when

the axial confinement is strong enough to turn off inter-

cloud tunneling. The strong anisotropy of trapping po-

tentials introduces two different orders of energy scales,

with one characterized by ?ωz and the other by ?ω⊥,

where ωz (ω⊥) are the trapping frequencies in the axial

(radial) directions. The separation of these two energy

scales (ωz ≫ ω⊥) allows us to first deal with the axial

degrees of freedom and derive an effective 2D Hamilto-

nian, and leave the radial degrees of freedom for later

treatment.

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2

II.MODEL 1 WITH RENORMALIZED

ATOM-ATOM INTERACTION

The effective 2D Hamiltonian for model 1 is obtained

by assuming that the renormailied atom-atom interac-

tion can be characterized with an effective 2D scattering

length, with the latter derived from the exact two-body

scattering physics [8, 9]. Thus, for a wide Feshbach res-

onance where the Feshbach-molecule population is neg-

ligible, we can write down an effective Hamiltonian only

in terms of 2D fermionic operators ak,σ and a†

(pseudo) spin σ and transverse momentum k = (kx,ky).

The model 1 Hamiltonian thus takes the form [7, 8, 9]

k,σ, with

H1 =

?

Veff

1

L2

k,σ

(ǫk− µ)a†

k,σak,σ

+

?

k,k′,q

a†

k,↑a†

−k+q,↓ak′,↓a−k′+q,↑,(1)

where ǫk = ?2k2/(2m) is the 2D dispersion relation

of fermions with mass m, µ is the chemical potential,

and L2is the quantization area. The bare parameter

Veff

1

is connected with the physical one Veff

the 2D renormalization relation

L−2?

tering length as and the characteristic length scale for

axial motion az≡

in Ref. [8, 9, 10]. Notice that the chemical potential µ

can be a function of the radial coordinate r = (x,y) un-

der LDA. In the following discussion, we choose ?ωz as

the energy unit so that µ, Veff

dimensionless.

By introducing a BCS order parameter (also dimen-

sionless in unit of ?ωz) ∆ ≡ (Veff

we get the zero temperature thermodynamic potential

density

1p through

?Veff

?Veff

1

?−1=

1p

?−1−

k(2ǫk+ ?ωz)−1(?ωz is from the zero-point en-

ergy), and Veff

1p(as,az) depends on the 3D scat-

1p= Veff

??/(mωz) with the expression given

1, and ǫk= a2

zk2/2 become

1/L2)?

k?ak,↓a−k,↑?,

Ω = −∆2

Veff

1

+

1

L2

?

k

(ǫk− µ − Ek),(2)

where Ek=

tation spectrum. The ultraviolet divergence of the sum-

mation over k cancels with the renormalization term in

?Veff

(n = N/L2is the density of particles), leading to

?(ǫk− µ)2+ ∆2is the quasi-particle exci-

1

?−1.The gap and number equations can be ob-

tained respectively from ∂Ω/∂∆2= 0 and n = −∂Ω/∂µ

1

Veff

1p(as,az)

=

ln

?

−µ +

?µ2+ ∆2?

4πa2

z

?µ2+ ∆2

z

,(3)

n =

µ +

2πa2

.(4)

Notice that Eq.

−µ +

(3) can be rewritten as F(as,az) =

?µ2+ ∆2, where the function F absorbs all the

dependence on asand az. Thus, by substituting this ex-

pression into Eq. (4), we get a closed form for the number

equation,

n =

1

πa2

z

?F(as,az)

2

+ µ

?

. (5)

Now we take into account the harmonic trapping po-

tential U(r) = (ω⊥/ωz)2r2/(2a2

writing down the position dependent chemical poten-

tial µ(r) = µ0− U(r), where µ0 is the chemical po-

tential at the trap center. It can be easily shown that

the spacial density profile is now a parabola, n(r) =

(ω⊥/ωz)2(R2

cloud size RTF =

condition that the total number of particles in the trap is

fixed by N =?n(r)d2r, the cloud size takes the constant

pendent on the 3D scattering length as. In fact, as one

varies the scattering length as, the chemical potential at

the trap center µ0is adjusted accordingly such that the

identical density profile is maintained.

This result of a constant cloud size is obviously incon-

sistent with the picture of a BCS-BEC crossover in quasi

two dimensions. In fact, in a typical experiment with

az (∼ µm) much greater than the interatomic interac-

tion potential Re(∼nm), the scattering of atoms in this

quasi-2D geometry is still 3D in nature. In particular,

fermions will form tightly bound pairs on the BEC side

of the Feshbach resonance as they do in 3D. Thus, in the

BEC limit when fermion pair size apair≪ azand binding

energy |Eb| ≫ ?ωz, the system essentially behaves like a

weakly interacting gas of point-like bosons, for which one

would expect a vanishing small cloud size in the loosely

confined radial plane [8, 13].

The MF result of a finite cloud size in the BEC limit

from model 1 indicates a finite interaction strength be-

tween paired fermions, no matter how small they are in

size. This statement can be extracted directly from the

number equation (5), which can be written in the form

µ = nπa2

z− F(as,az)/2. In the BEC limit, the second

term on the right-hand side denotes one half of the bind-

ing energy, while the first term indicates a finite inter-

action energy per fermion pair since it is proportional to

the number density. As a comparison, the actual equa-

tion of state for fermion pairs one should expect must

take the form as for a quasi-2D Bose gas in the weakly

interacting limit [8]

z) in the radial plane by

TF− r2)/(2πa4

√2µ0az(ωz/ω⊥).

z), with the Thomas-Fermi

By assigning the

value RTF= RBCS≡

?2ωz/ω⊥(N)1/4az, which is inde-

µB≈ 3nazas,(6)

in which case the quasi-2D gas is treated as a 3D con-

densate with the ground state harmonic oscillator wave

function in the z-direction.

The interaction strength between paired fermions can

also be derived by writing down a Bose representation for

this system, where the fermionic degrees of freedom are

integrated out in the BEC limit [14]. This Bose represen-

tation leads to a two-dimensional effective Hamiltonian

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3

for bosonic field φ(r),

Heff=

?

dr

?

φ†(r)

?

−?2∇2

4m

+ 2U(r)

?

φ(r) −g2

2|φ(r)|4

?

,

(7)

where the quartic term characterizes the bosonic interac-

tion. Within the stationary phase approximation, the in-

teraction strength g2is calculated by the leading diagram

of a four-fermion process with four external boson lines

and four internal fermion propagators, leading to [14]

g2= 2

?

p,ω

Λ4

0(p)G2

0(p,ω)G2

0(−p,−ω).(8)

Here, Λ0 = (−p2/m + |Eb|)χ0(p) is the boson-fermion

vertex, χ0(p) is the Fourier transform of the relative wave

function χ0(r) of two colliding fermions in the s-wave

channel, and G0(p,ω) = (iω − p2/2m − |Eb|/2)−1is the

free propagator for fermions. After summing over mo-

mentum p and Matsubara frequency ω, we can directly

show that g2indeed takes a constant value, being inde-

pendent of the binding energy |Eb| of paired fermions

and hence the 3D scattering parameter as. Thus, we

conclude that the MF theory based on model 1 fails to

recover the picture of a weakly interacting Bose gas of

paired fermions in the BEC limit, and can not be directly

applied to describe the BCS-BEC crossover in quasi two

dimensions.

III.MODEL 2 WITH INCLUSION OF DRESSED

MOLECULES

Having shown the problem associated with model 1,

next we consider model 2 by taking into account the axi-

ally excited states via inclusion of dressed molecules. As

derived in Ref. [10], the effective 2D Hamiltonian takes

the form (also in unit of ?ωz),

H2 =

?

αb

L

k,σ

(ǫk− µ)a†

k,σak,σ+

?

q

?ǫq

2+ λb− 2µ

?

d†

qdq

+

?

?

k,q

?

a†

k,↑a†

−k+q,↓dq+ h.c.

?

+

Vb

L2

k,k′,q

a†

k,↑a†

−k+q,↓a−k′,↓ak′+q,↑,(9)

where d†

erator for dressed molecules with radial momentum q,

and λb, αb, and Vb are the 2D effective bare detuning,

atom-molecule coupling rate, and background interac-

tion, respectively. These parameters can be related to

the corresponding 3D parameters by matching the two-

body physics [10]. By introducing the order parameter

∆ ≡ αb?d0?/L + (Vb/L2)?

q(dq) denotes the creation (annihilation) op-

k?ak,↓a−k,↑?, we obtain the

mean-field gap and number equations,

1

Veff

2p(2µ)=

ln

?

−µ +

?µ2+ ∆2?

4πa2

z

,(10)

n =µ +

?µ2+ ∆2

z

2πa2

+ 2∆2∂[Veff

2,p(x)]−1

∂x

?????

x=2µ

,(11)

where the inverse of effective interaction is connected

with the 3D physical parameters through [10]

?Veff

2p(x)?−1

√2π

a2

=

?

Vb+

α2

b

x − λb

g2

p

x − γp

?−1

+

1

L2

?

k

1

2ǫk+ ?ωz

=

z

?

Up+

?−1

− Sp(x) + σp(x)

. (12)

Here, Up = 4πabg/az, g2

µco(B−B0)/(?ωz) are 3D dimensionless physical param-

eters, where abgis the background scattering length, µco

is the difference in magnetic moments between the two

channels, W is the resonance width, and B0is the reso-

nance point. The functions in Eq. (12) take the form

p= µcoWUp/(?ωz), and γp =

Sp(x) =

−1

4√2π

ln|x|

4π√2π,

?∞

0

ds

?

Γ(s − x/2)

Γ(s + 1/2 − x/2)−

1

√s

?

, (13)

σp(x) =

(14)

where Γ(x) is the gamma function.

Using this model 2 Hamiltonian, we first consider a

uniform quasi-2D Fermi gas with a fixed number density

n, where the inhomogeneity in the radial plane is ne-

glected. In this case, the gap and number equations (10)

and (11) need to be solved self-consistently for a given

magnetic field. A typical set of results for both6Li and

40K are shown in Fig. 1, indicates a smooth crossover

from the BCS (right) to the BEC (left) regimes. Here,

results obtained from model 2 (black) are compared with

those from model 1 (gray). In this figure and the follow-

ing calculation, we use the parameters abg = −1405a0,

W = 300 G, µco = 2µB for

W = 7.8 G, µco = 1.68µB for40K, where a0 and µB

are Bohr radius and Bohr magneton, respectively.

There are two major points that need to be empha-

sized in Fig. 1. First, when plotted as functions of the

inverse of 3D scattering length az/as, the results for6Li

(solid) and40K (dashed) are very close, manifesting the

near resonance universal behavior. Second, the results

from model 1 and model 2 are significantly different, es-

pecially on the BEC side of the resonance. In particular,

the dressed-molecule fraction in model 2 is already siz-

able (∼ 0.16) at unitarity, and becomes dominant on the

BEC side of the resonance (see Fig. 1c). This result pro-

vides another signature of inadequacy of model 1, where

the dressed-molecule population is always assumed to be

negligible.

6Li, and abg = 174a0,

Page 4

4

?

?

FIG. 1: The BCS-BEC crossover behavior of a uniform quasi-

2D Fermi gas at zero temperature, showing (a) the chemical

potential µ, (b) the gap ∆, both in unit of ?ωz, and (c) the

dressed-molecule fraction nb/n. Notice that the results for6Li

(solid) and those for40K (dashed) almost coincide as plotted

as functions of az/as, indicating a universal behavior around

the resonance point. Furthermore, significant difference be-

tween model 1 (gray) and model 2 (black) can be observed in

(b) and (c), which shows that model 1 is oversimplified at uni-

tarity and on the BEC side of the resonance. The parameters

used in these plots are ωz = 2π × 62 kHz, and na2

z= 0.001.

Next, we impose a radial harmonic trap U(r) and cal-

culate the Thomas-Fermi cloud size for a fixed number

of particles in the trap N =

Fig. 2. The most important feature of Fig. 2 is that the

cloud size given by model 2 (solid) is no longer a constant

as predicted by model 1 (dashed). On the contrary, by

crossing the Feshbach resonance, the cloud size decreases

from the limiting value RBCSof a noninteracting Fermi

gas in the BCS limit, and approaches to the 3D results

(dotted) in the BEC limit. This trend successfully recov-

ers the corresponding physics in both the BCS and the

BEC limits. In addition, we also find that for a given

number of particles in the trap, the curve trend is in-

sensitive to the radial trapping frequency ω⊥within the

experimentally accessible region. (The ω⊥ = 2π × 10

Hz and 2π ×50 Hz results, not shown, coincide with the

2π×20 Hz line and are hardly distinguishable within the

figure resolution.) Considering the fact that there is a

scaling relation between ω⊥and N such that the physics

is only determined by N(ω⊥/ωz)2, this insensitivity with

respect to the radial trapping frequency suggests that

the experimental measurement has a rather wide range

?2πn(r)rdr, as shown in

?

?

?

?

?

FIG. 2: The Thomas-Fermi cloud size of a quasi-2D Fermi gas

of6Li over a wide BCS-BEC crossover region. Here, results

from model 2 (solid) are compared with those from model 1

(dashed). All curves are normalized to the cloud size of a

noninteracting Fermi gas RBCS. Notice that the results of

model 2 recover the correct pictures in the BCS and BEC

limits, in clear contrast to the model 1 prediction of a flat

line. Parameters used for these two plots are ωz = 2π × 62

kHz, ω⊥ = 2π × 20 Hz, and the total particle number N =

104. For reference, the results for an isotropic 3D Fermi gas

with the same total particle number is also plotted (dotted),

where a single-channel model and a two-channel model are

both incorporated to give indistinguishable predictions.

of tolerance on the number of atoms.

In Fig. 3 we show the number density and the dressed-

molecule fraction distribution along the radial direction

for various values of az/as. A typical case in the BCS

regime is shown in the top panel of Fig. 3, where the

dressed-molecule fraction is vanishingly small, and model

1 and model 2 predict similar cloud sizes and number

density distributions. The middle panel shows the case

at unitarity. As compared with model 1, notice that the

cloud is squeezed in model 2 and the dressed-molecule

fraction increases to a sizable value. The bottom panel

shows a typical case in the BEC regime, where the cloud

is squeezed further in model 2 as the dressed-molecule

fraction becomes significant. Notice that the results of

model 2 successfully describes the BCS-BEC crossover,

in clear contrast to the outcome of model 1.

IV.CONCLUSION

In summary, we have considered in this paper the BCS-

BEC crossover of a quasi-2D Fermi gas across a wide

Feshbach resonance. We analyze two effective Hamilto-

nians and compare predictions of zero temperature cloud

size and number density distribution in the radial plane

Page 5

5

?

?

???

?

??

?

??

?

??

?

??

?

?

?

?

???

?

?

?

??

?

?

?

?

?

??

?

?

FIG. 3: (Color online) The in-trap number density (the solid

lines) and dressed-molecule fraction (the dashed lines) distri-

bution along the radial direction of a quasi-2D Fermi gas of

6Li, obtained from model 2 (a-c) and model 1 (d-f). The top

panels correspond to the case of az/as = −1 (BCS side), the

middle panels to the case of az/as = 0 (unitarity), and the

bottom panels to the case of az/as = 1 (BEC side). The

parameters are ωz = 2π × 62 kHz, ω⊥ = 2π × 20 Hz, and

N = 104.

within a mean-field approach and local density approxi-

mation. Using model 1 with renormalizd atom-atom in-

teraction, we show that the cloud size remains a con-

stant value through the entire BCS-BEC crossover re-

gion, which is inconsistent with the picture of a weakly

interacting Bose gas of fermion pairs in the BEC limit.

On the other hand, model 2 with renormalized interac-

tion between atoms and dressed molecules predicts the

correct trend of cloud size variation. Based on this quali-

tative comparison, it is likely that the inclusion of dressed

molecules [10, 11] is essential to describe the BCS-BEC

crossover in quasi low dimensions.

Acknowledgments

We thank Yinmei Liu for many helpful discussions, and

J. P. Kestner for discussion and thoroughly reading of the

manuscript. This work is supported under the MURI

program and under ARO Award W911NF0710576 with

funds from the DARPA OLE Program.

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