VOLUME 85, NUMBER 22
PHYSICAL REVIEW LETTERS
27 NOVEMBER 2000
Rapidly Rotating Fermi Gases
Tin-Lun Ho and C. V. Ciobanu
Department of Physics, The Ohio State University, Columbus, Ohio 43210
(Received 25 May 2000; revised manuscript received 5 September 2000)
We show that the density profile of a Fermi gas in rapidly rotating potential will develop prominent
features reflecting the underlying Landau-level –like energy spectrum. Depending on the aspect ratio of
the trap, these features can be a sequence of ellipsoidal volumes or a sequence of quantized steps.
PACS numbers: 05.30.Fk, 03.75.Fi
Currently, there is an intense effort to cool trapped Fermi
gases down to the degenerate limit. Recent experiments at
JILA on 40 K have reached one-half of their Fermi temperature [1]. One of the motivations of cooling fermions
is to reveal their possible superfluid ground states, which
can be quite novel in the case of multicomponent fermion
systems [2]. Current theoretical estimates, however, indicate that the interactions between different spin states
of 40 K are all positive [3], implying a normal instead of
superfluid ground state. The absence of superfluid ground
states, however, does not mean that the system cannot have
novel macroscopic quantum phenomena. Quantum Hall
effect is an excellent example. In a strong magnetic field,
the energy levels of a two-dimensional electron system organize into highly degenerate Landau levels, leading to a
whole host of dramatic effects. In this paper, we show
that similar organizations will take place in a fast rotating three-dimensional trapped Fermi gas, leading to many
macroscopic quantum phenomena.
For neutral atoms in rotating harmonic traps, we shall
see that Landau-level–like energy spectrum will appear
when the rotation frequency V approaches the transverse
confining frequency v⬜ . This “fast rotating” limit might
appear hard to achieve as the system is at the verge
of flying apart due to centrifugal instability [4]. Such
instability, however, can be prevented by imposing an
additional repulsive potential which dominates over the
centrifugal force beyond a certain radius. With centrifugal
instability eliminated, the Landau-like levels will show up
in many ways. We will see that for cylindrical harmonic
traps with vz ø v⬜ , where v⬜ and vz are the transverse
and longitudinal trapping frequencies, the density is a sum
of one-dimensional– like density distributions residing in
different “Landau volumes.” For traps with vz comparable
or smaller than v⬜ , the density consists of a set of disks
along z, each of which is made up of a sequence of density
steps quantized in units of Mv⬜ 兾共p h̄兲.
2D case: We first consider 2D rotating Fermi gases in
harmonic potentials since they illustrate the basic physics.
The Hamiltonian in the rotating frame is
H 2 VLz 苷
4648
1 2
1
2 2
p 1
Mv⬜
r 2 V ẑ ? r 3 p⬜ ,
2M ⬜
2
(1)
0031-9007兾00兾85(22)兾4648(4)$15.00
where p⬜ 苷 共px , py 兲 and r 苷 共x, y兲. The eigenfunctions
and eigenvalues of Eq. (1) are
≠n e2jwj
ejwj 兾2 ≠m
p 1 2
un,m 共r, u兲 苷
,
2
pa⬜
n! m!
2
2
en,m 苷 h̄共v⬜ 1 V兲n 1 h̄共v⬜ 2 V兲m 1 v⬜ ,
(2)
(3)
where n, m are non-negative
integers 0, 1, 2, . . . , w ⬅
p
共x 1 iy兲兾a⬜ , a⬜ 苷 h̄兾Mv⬜ , and ≠6 ⬅ 共a⬜ 兾2兲 共≠x 6
i≠y 兲. To derive Eqs. (2) and (3), we note that Eq. (1)
can be written as P2 兾2M 1 共v⬜ 2 V兲Lz with P 苷
p⬜ 2 Mv⬜ ẑ 3 r, which is precisely the canonical moeB
mentum P 苷 p⬜ 2 2c ẑ 3 r of an electron in a magnetic
field B in the symmetric gauge, with eB兾Mc 苷 2v⬜ .
The eigenfunctions of P2 兾2M are those in Eq. (2) [5],
L
with eigenvalues en,m
苷 h̄v⬜ 共2n 1 1兲, where n is the
Landau-level index, and m is an “angular momentum” index labeling the degeneracy in each level. Since Lz un,m 苷
h̄共m 2 n兲un,m , Eq. (2) is also an eigenstate of Eq. (1)
with eigenvalues Eq. (3). Note that the function un,m in
Eq. (2) peaks at
2
rn,m ⬅ 具r 2 典n,m 苷 a⬜
共n 1 m 1 1兲 ,
(4)
and decays away as a Gaussian over a distance a⬜ .
Equation (3) shows that the system is unbounded when
V . v⬜ unless an additional repulsive potential Vwall 共r兲
(say, introduced by an additional optical trap) is present.
We shall in particular consider potentials Vwall 共r兲 which
are zero for r , R but become strongly repulsive for
r . R, with R ¿ a⬜ . The specific form of Vwall is not
important for the key features discussed below, as long as
it is smooth over length scale a⬜ . The condition R ¿ a⬜ ,
however, allows us to fit many m states inside r , R and
is a necessary feature for many effects discussed below.
Since Vwall 共r兲 is cylindrically symmetric, the eigenstates
are still labeled by quantum numbers 共n, m兲. For states
originally with rn,m , R, Eqs. (2) and (3) remain valid
because Vwall 苷 0 for r , R. For states 共n, m兲 originally peaked beyond R, their energies increase rapidly because Vwall is strongly repulsive. (For 40 K in a tight trap
v⬜ 苷 4000 and 105 Hz, we have a⬜ 艐 2.5 3 1025 and
5 3 1026 cm, respectively. The condition R ¿ a⬜ is satisfied for R . 5 3 1024 cm.)
© 2000 The American Physical Society
VOLUME 85, NUMBER 22
PHYSICAL REVIEW LETTERS
Let us first consider the case V , v⬜ with
V
density in the ground state is r共r兲 苷
Pwall 苷 0. The
2
ju
共r兲j
Q共m
2 en,m 兲, where Q共x兲 苷 1 or 0 if
n,m n,m
x . 0 or ,0, and m is the chemical
P potential related
to the particle number N as N 苷 n,m Q共m 2 en,m 兲.
P ⴱ
We can write r共r兲 苷 nn苷0 rn 共r; mnⴱ 兲, where rn 共r; L兲 is
density contribution of the nth Landau level with angular
momentum states filled up to m 苷 L;
L
X
jun,m 共r兲j2 Q共m 2 en,m 兲 ,
(5)
rn 共r; L兲 苷
m苷0
mnⴱ is the highest angular momentum state in the nth Landau level with energy less than m, and nⴱ is the highest
Landau level below m,
∏
∑
m兾h̄ 2 v⬜ 2 共v⬜ 1 V兲n
,
(6)
mnⴱ 苷 Int
共v⬜ 2 V兲
m兾 h̄2V
nⴱ 苷 Int关 v⬜ 1V 兴, where Int关x兴 denotes the integer part of
x, and x is understood to be positive. Since 共n, mnⴱ 兲 is the
state in rn farthest
from the origin, its peak location 共rn 苷
p
rn,mnⴱ 苷 a⬜ n 1 mnⴱ 1 1 兲 gives the size of rn . When V
is very close to v⬜ , we have mnⴱ ¿ 1 and
m 2 h̄V共2n 1 1兲
rn2 苷
.
Mv⬜ 共v⬜ 2 V兲
(7)
Note that the difference in area between successive Landau
disks is a constant
∂
µ
2V
2
2
2
p共rn21 2 rn 兲 苷 共pa⬜ 兲
.
(8)
v⬜ 2 V
By using Eq. (2), it is straightforward to show that
∑
µ
∂
∏
s
1
ⴱ
p
r0 共r; m0 兲 苷
1 2 erf
共1 1 关. . .兴兲 , (9)
2
2pa⬜
2m0ⴱ
Rx
p
2
where s 苷 共r兾a⬜ 兲2 2 m0ⴱ , erf共x兲 苷 共2兾 p 兲 0 e2z dz,
ⴱ 21
and the term 关. . .兴 in Eq. (9) is of order 共12m0 兲 and
smaller. The densities rn 共r兲 of the higher Landau levels
共n . 0兲 can be generated from r0 as [6] rn 共r; mnⴱ 兲 苷
1 1 2 n
៬ ⴱ l៬ 2 ៬ . From the properties
n! 共 4 =l 兲 关r0 共r៬ 2 a⬜ l; mn 兲e 兴l苷0
2
兲 within
of erf共x兲, it is clear thatpr0 is a constant 1兾共pa⬜
ⴱ
a disk of radius r0 苷 m0 1 1 a⬜ and has an edge of
3
width D0 艐 p2 a⬜ [7]. If m0ⴱ ¿ 1, D0 ø r0 , and r0 can
be approximated as a step function on scales larger than
D0 . Likewise, rn can be approximated as a step function
with somewhat larger width within the same approximation. Thus, when mnⴱ ¿ 1, we have
2 21
兲 Q共rn2 2 r 2 兲 .
rn 共r兲 苷 共pa⬜
Mv⬜
p h̄ I共r兲,
(10)
with
Equations
苷
P (10) and (8) then imply r共r兲
2
I共r兲 苷 n Q关m 2 Mv⬜ 共v⬜ 2 V兲r
2
h̄V共2n
1
1兲兴.
P
By using the identity Int关x 1 1兴 苷 n Q关a共x 2 n兲兴, for
all x . 0 and a . 0, r共r兲 can be simplified to
∑
∏
m 2 Mv⬜ 共v⬜ 2 V兲r 2 1 h̄V
Mv⬜
Int
.
r共r兲 苷
p h̄
2h̄V
(11)
27 NOVEMBER 2000
It is, however, instructive to rederive Eq. (11) in a way
generalizable to arbitrary potentials. We rewrite Eq. (1) as
H 2 VLz 苷
共p⬜ 2 MV ẑ 3 r兲2
2M
1
2
1
M共v⬜
2 V 2 兲r 2 .
2
(12)
The first term gives a set of Landau levels with spacing
2h̄V, each of which contributes 共pA2 兲21 to the density,
where A2 苷 h̄兾共MV兲. If the second term in Eq. (12) is
absent, the density is given by r 苷 I兾共pA2 兲, where I
is the total number of Landau levels below the chemical
m1 h̄V
potential, I 苷 Int关 2 h̄V 兴. When V is close to v⬜ , the
second term in Eq. (12) is slowly varying over the scale
of A ⬃ a⬜ and can be absorbed in the chemical potential. The density profile within local density approximation (LDA) is then
∑
∏
m共r兲 1 h̄V
MV
I共r兲,
I共r兲 苷 Int
, (13)
r共r兲 苷
p h̄
2h̄V
2
2 V 2 兲r 2 . Clearly, Eq. (13)
where m共r兲 苷 m 2 12 M共v⬜
is equivalent to (11) up to correction 共1 2 V兾v⬜ 兲 ø 1
as V ! v⬜ . Equation (13), once established, is easily
generalized to other potentials. In the presence of Vwall ,
one simply replaces m共r兲 in Eq. (13) by
m共r兲 苷 m 2
1
2
M共v⬜
2 V 2 兲r 2 2 Vwall 共r兲
2
共2D兲 .
(14)
Equations (13) and (14) constitute the LDA solution for the
2D rotating Fermi gas for both V , v⬜ and V . v⬜ .
The schematics of LDA is shown in Figs. 1(a) and 1(b).
To understand the validity of LDA [Eq. (13)], we calculated the density numerically using Eq. (2). The result for
a system of 2000 fermions at V兾v⬜ 苷 0.996 is shown in
Fig. 2(a). The system exhibits a sequence of quantized
steps at locations well described by LDA. The evolution of the density within the range 0.99 , V兾v⬜ , 1
of this fermion system is shown in Fig. 2(b). As V decreases, more Landau levels are populated while the steps
near the surfaces are closer together [as expected from
Eqs. (6) and (8)], yet the step structures remain discernable and correctly described by LDA [the LDA construction is not displayed, so as to keep Fig. 2(b) readable].
The behaviors of the densities at lower frequency 0.98 ,
V兾v⬜ , 1 are shown in Figs. 3(a) and 3(b) for a system of 1000 fermions. At the lowest frequency displayed
[Fig. 3(a)], the step structure near the surface is completely
smeared out by the spread of the edges. Nevertheless, the
density of the innermost plateau remained quantized, with
a size correctly described by LDA. This core of quantized density (or “quantized core” for short) is clear evidence for Landau levels. Our studies show that, for about
2000 particles, Landau levels will show up as a sequence
4649
VOLUME 85, NUMBER 22
PHYSICAL REVIEW LETTERS
decreases. On the other hand, the LDA in Fig. 1(b) shows
that, by introducing an additional potential Vwall , Landau
levels (in the form of a sequence of steps near the center
or a quantized core) can still be revealed at frequencies
farther beyond v⬜ , even though the steps near the surface
are smeared out.
3D case: For a 3D harmonic trap, Eq. (1) acquires terms
pz2 兾2M 1 12 Mvz z 2 , which give rise to harmonic oscillator eigenfunctions fnz 共z兲 with P
eigenvalue enz 苷 h̄vz 共nz 1
1
2
2
兲.
The
density
is
r共r,
z兲
苷
nz ,n,m j fnz 共z兲j jun,m 共r兲j 3
2
Q共m 2 enz ,n,m 兲, where enz ,n,m 苷 enz 1 en,m .
When
V兾v⬜ ⬃ 1, we first perform the m sum because it produces the smoothest change in the energy. By repeating
the steps leading to Eq. (10), we have
(µ(r ) +hΩ ) / 2hΩ
8
(a)
6
4
2
0
8
( b)
6
4
2
0
0
10
20
30
40
50
r / a⊥
FIG. 1. (a) and (b) correspond to V , v⬜ and V . v⬜ , respectively. The rapid drop at large r is due to the strongly repulsive potential Vwall . The LDA densities are indicated by the
steps in thick lines. The integer value of 关m共r兲 1 h̄V兴兾共2 h̄V兲
(i.e., I) is related to the index n of the intersected Landau level as
I 苷 n 1 1. The relation 关m共r兲 1 h̄V兴兾共2 h̄V兲 苷 I is equivalent to m共r兲 苷 共2n 1 1兲V.
of discernable steps only when 0.99 , V兾v⬜ , 1, which
is achievable with the current capability to control frequencies, especially for large v⬜ . For lower frequencies, the
existence of Landau levels can only be revealed through the
presence of a quantized core, which shrinks in size as V
( π a ⊥2 ) ρ
5
3
r共r, z兲 苷
(a )
M
1
1
0
0
5
20
30
40
r / a⊥
( π a ⊥2 ) ρ
3
2
2
1
1
0
0
10
20
30
40
r / a⊥
FIG. 2. (a) LDA (dotted line) and numerical calculations (solid
line) of the density for N 苷 2000 fermions at V兾v⬜ 苷 0.996.
(b) Density profiles of N 苷 2000 fermions at V兾v⬜ 苷 0.993
(crosses), 0.996 (dots), and 0.998 (solid line).
4650
5
10
15
20
25
r / a⊥
( π a ⊥2 ) ρ
(b)
4
3
0
(a)
0
5
(b)
4
N = 1000
Ω / ω ⊥ = 0.982
3
2
10
( π a ⊥2 ) ρ
4
2
0
X j fn 共z兲j2
z
Q关m共r兲 2 h̄V共2n 1 1兲 2 enz 兴 ,
2
pa⬜
nz ,n
(15)
2
where m共r兲 苷 m 2 Mv⬜ 共v⬜ 2 V兲r 2 艐 m 2 2 共v⬜
2
2 2
V 兲r .
The order of summation of the remaining integers nz
and n depends on the relative strength between vz and v⬜ .
When vz ø v⬜ , we first sum nz . To do that,
p we note that
the density of a 1D Fermi gas is r1D 苷 2Mm兾共p h̄兲,
where m is the chemical potential. In a harmonic trap,
the density within LDA is obtained by changing m
to m共z兲 苷 m 2 12 Mvz2 z 2 . pWe then have r1D 共z兲 苷
P
2Mm共z兲
2
,
which
turns
nz j fnz 共z兲j Q共m 2 enz 兲 苷
p h̄
5
N = 2000
Ω / ω ⊥ = 0 .9 9 6
4
27 NOVEMBER 2000
0
5
10
15
20
25
r / a⊥
FIG. 3. (a) LDA (dotted line) and numerical calculations (solid
line) of the density for N 苷 1000 fermions at V兾v⬜ 苷 0.982.
(b) Density profiles of N 苷 2000 fermions at V兾v⬜ 苷 0.982
(crosses), 0.989 (dots), and 0.995 (solid line).
VOLUME 85, NUMBER 22
PHYSICAL REVIEW LETTERS
( π a ⊥3 ) ρ
N = 2000
Ω / ω ⊥ = 0 .99
ω z / ω ⊥ = 0 .20
1.2
0.8
n=0
0.4
n=1
0
0
4
8
12
16
r / a⊥
FIG. 4. LDA (dotted line) and numerical calculations (solid
line) of the density for N 苷 2000 fermions at V兾v⬜ 苷 0.99
and vz 兾v⬜ 苷 0.2. The topmost curve, which represents the
full density, is the sum of the lower two curves.
Eq. (15) into a useful LDA
1
1
2
p 2 a⬜
az
Xq
1
3
关m共r, z兲 2 h̄V共2n 1 1兲兴兾共 2 h̄vz 兲 , (16)
n
p
M
2
2
where az 苷 h̄兾Mvz , and m共r, z兲 苷 m 2 2 共v⬜
M 2 2
2 2
V 兲r 2 2 vz z 2 Vwall . We have included Vwall in
m for the more general situation as in the 2D case.
Equation (16) shows that r共r兲 is a sum of 1D densities
(labeled by “n”), each of which is distributed over in
a “Landau volume” bounded by the “Landau surface”
m共r, z兲 苷 h̄V共2n 1 1兲. When Vwall 苷 0, V , v⬜ the
Landau surfaces are ellipsoidal surfaces. It is easy to verify
that the surface areas An for successive ellipsoids differ by
h̄
16pV 2
a constant An21 2 An 苷 MV 共 v 2 2V 2 兲. When V $ v⬜ ,
⬜
centrifugal instability against harmonic confinement sets
in and stability can only be established by Vwall .
To demonstrate the validity of the LDA equation (16),
we have evaluated the density numerically for a system of
2000 fermions for vz 兾v⬜ 苷 0.2 at V兾v⬜ 苷 0.99. The
results are shown in Fig. 4. It shows that the LDA (dotted
line) is a good approximation. The Landau volumes can be
clearly identified by the change of slope in the density. The
appearance of a plateau at the center is because vz 兾v⬜ is
only 0.2, revealing the 2D feature of the nz levels. For
smaller ratios of vz 兾v⬜ , the plateau disappears and the
LDA expression [Eq. (16)] is achieved.
When vz . v⬜ , summation of n in Eq. (15) gives
Mv⬜ X
r共r兲 苷
j fnz 共z兲j2
p h̄ " nz
#
1
m共r兲 2 h̄vz 共nz 1 2 兲 1 h̄V
, (17)
3 Int
2h̄V
r共r兲 苷
27 NOVEMBER 2000
2
where m共r兲 苷 m 2 12 M共v⬜
2 V 2 兲r 2 2 Vwall . In this
limit, the density consists of a sequence of disks (labeled
by nz ) in the z direction. Each disk j fnz 共z兲j2 consists of
a sequence of density steps in the xy plane reflecting the
number of filled Landau levels. The behavior of the density within each disk in the xy plane is identical to the
2D case discussed previously. Finally, we note that, as
the temperature increases, the Landau-level structure near
the surface will first melt away, and the melting will proceed toward the center. The temperature below which the
Landau-level effect begins to appear is T 苷 2h̄v⬜ 兾kB ,
which is 3.8 3 1027 and 9.6 3 1026 K for for v⬜ 苷
4000 and 105 Hz, respectively, a temperature range achievable in current experiments [8].
Thus far, we have discussed only the effect of Landau
levels on the density profiles of fast rotating Fermi gases.
If the development of the quantum Hall effect in the past
decade is a guide, one expects many more novel phenomena in Fermi gases in the fast rotating regime.
This work was completed during a workshop at the
Lorentz Center of the University of Leiden. We thank
Professor Henk Stoof and the Lorentz Center for their
generous support. This work is supported by a Grant
from NASA (NAG8-1441) and by NSF Grants No. DMR9705295 and No. DMR-9807284.
[1] D. B. DeMarco and D. S. Jin, Science 285, 1703 (1999).
[2] The pairing states in large spin Fermi gas are discussed in
T. L. Ho and S. K. Yip, Phys. Rev. Lett. 82, 247 (1999).
[3] J. Burke (private communication).
[4] In the recent experiment of F. Chevy, K. W. Madison, and
J. Dalibard (cond-mat/0005221) on rotating Bose gas, the
highest rotational frequency used is quite close to v⬜ .
[5] R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).
P jwj2m
2
[6] Note that r0 共r; L兲 苷 L0 m! ejwj . Since unm 苷 unm;l苷0 ,
2
m n lw ⴱ 2jwj2
where
unm;l 苷 共pn! m!兲21兾2 ejwj 兾2 ≠1
≠l e
苷
ⴱ
2
共pn! m!兲21兾2 ≠ln 关共l 2 w兲m elw 2jwj 兾2 兴 (with a⬜ 苷 1),
PL
PL jl2wj2m
1
2
n
2
3
we have
m苷0 junm;l j 苷 pn! 共=l 兾4兲
n苷0
m!
ⴱ
ⴱ
2
1
៬ L兲el៬ 2 兴. Explicitly,
elw 1l w2jwj 苷 pn! 共=l2 兾4兲n 关r0 共r៬ 2 l;
r1 共r; m1ⴱ 兲 苷 共1 1 14 =2 兲r0 共r; m1ⴱ 兲, r2 共r; m2ⴱ 兲 苷 共1 1 12 =2 1
3 2
3 4
ⴱ
ⴱ
1 4
32 = 兲r0 共r; m2 兲, and r3 共r; m3 兲 苷 共1 1 4 = 1 32 = 1
ⴱ
1
6
384 = 兲r0 共r; m3 兲.
[7] The extent
of
erf共x兲 is about 1.5; we then have
p ⴱ
1
2
D共s兾
2m
兲
苷
1.5.
0
2
p
p ⴱ Since D共s兲 艐 2rD共r兲兾a⬜ , we have
3
ⴱ
m0 D共r兲 艐 p 2 m0 a⬜ .
[8] The studies of finite temperature effects will be presented
elsewhere. At higher temperatures, the density profile is
2
2 2
that of a Boltzman gas, ~e2M共v⬜ 2V 兲r 兾2kB T .
4651