Zero Sound and First Sound in a Disk-Shaped Normal Fermi gas

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by  Giovanni Mazzarella

Zero Sound and First Sound in a Disk-Shaped Normal Fermi gas
Giovanni Mazzarella1, Luca Salasnich2 and Flavio Toigo1
1Dipartimento di Fisica “Galileo Galilei” and CNISM,
Universit`a di Padova, Via Marzolo 8, 35131 Padova, Italy
2CNR-INFM and CNISM, Unit`a di Padova, Via Marzolo 8, 35131 Padova, Italy
(Dated: January 22, 2009)
We study thezero sound and thefirst sound in a dilute and ultracold disk-shaped normal Fermi
gas with a strong harmonic confinement along the axial direction and uniform in the two planar
directions. Workingatzerotemperaturewecalculatethechemicalpotentialµofthefermionicfluid
9 asafunctionoftheuniformplanardensityρandfindthatµchangesitsslopeincorrespondenceto
0 thefillingofharmonicaxialmodes(shelleffects). Withinthelinearresponsetheory,andunderthe
0 random phase approximation, we calculate the velocity c0 of the zero sound. We find that also c0
s s
2 changes its slope in correspondence of the filling of the harmonic axial modes and that this effect
n dependsontheFermi-FermiscatteringlengthaF. Inthecollisionalregime,wecalculatethevelocity
a cs of first sound showing that cs displays jumps at critical densities fixed by the scattering length
J aF. Finally, we discuss the experimental achievability of these zero sound and first sound waves
2 with ultracold alkali-metal atoms.
2
PACSnumbers: 03.75.Ss,03.75.Hh

]
r
e I. INTRODUCTION the so-called collisionless regime. In presence of a repul-
h
sive interaction, the resulting collective mode is known
t
o as zero sound [23, 24, 25, 27, 28, 29]. In the present
Effects of quantum statistics are observable with both
t. work we approach the problem calculating the density-
a bosonic[1,2]andfermionic[3,4,5,6,7]vaporsofalkali-
density response function in the random-phase approxi-
m metal atoms at ultra-low temperatures. In these quan-
mation. The poles of this response function lead to the
tumdegenerategases,theroleofdimensionalityhasbeen
- zero sound dispersionlaw and to its velocity of propaga-
d experimentally studied with bosons [8, 9, 10], while for
tionc0 inthe radialplane,where the gasisnotconfined.
n fermions, there are only theoretical predictions. In par- s
o ticular,ithasbeensuggestedthatareduceddimensional- We investigatethebehaviorofthe zerosoundvelocityc0s
c as a function of the chemical potential, or, equivalently,
ity strongly modifies density profiles [11, 12, 13, 14], col-
[ of the fermionic density in the transverse radial plane.
lective modes [15, 16] and stability of mixtures [17, 18].
1 Alsosoundvelocitieshavebeentheoreticallyinvestigated When collisions between atoms are characterizedby a
v sufficiently small collision time such that ωτ 1, the
in reduced dimensions for both normal [16, 19] and su- ≪
8 system may be assumed to be at thermal equilibrium.
perfluid Fermi gases [20, 21, 22].
3 Inthis regime,the Fermi fluidis accuratelydescribedby
4 It has been shown that, in the weak coupling limit,
quantum hydrodynamics equations where the fermionic
3 the zero sound velocity c0 and the first sound velocity
s nature appears in the bulk equation of state [23, 25, 27].
1. cs of a normal Fermi gas are equal in the strictly one- In this case, the collective mode is called first sound (or
0 dimensional (1D) geometry [16], contraryto the 3D case ordinarysound,orcollisionalsound). Thefirstsoundve-
9 where they differ by a factor √3 if the systemis uniform locity c has been studied previously both in the 1D-3D
0 [23, 24, 25], or by a factor √5 if the system is cigar- and 2Ds-3D crossovers [30, 31], and in a 3D cylindrical
:
v shaped due to a isotropic harmonic confinement in two geometry [32] for an ultracold Bose gas. It is important
Xi directions [19]. Very recently we have studied the first to stress that the propagation of ordinary sound in spin
sound velocity cs of a normal cigar-shaped Fermi gas in unpolarized Fermi gases may be studied also from the
r
a the 1D to 3D crossover, showing that cs exhibits jumps experimentalpointofview. Forinstance,Josephandco-
in correspondenceof the filling ofplanar harmonic shells workers [33] were able to excite first sound waves in an
[26]. optically trapped Fermi degenerate gas of spin-up and
In this paper we investigate, at zero temperature, the spin-down atoms by using magnetically tunable interac-
zero sound velocity c0 and the first sound velocity c of tions. In this paper we predict that, as in the 1D-3D
s s
a normal Fermi gas in the dimensional crossover from crossover[26],the firstsoundvelocityc displaysobserv-
s
2D to 3D by considering a harmonic confinement in the able shelleffects inthe 2D-3Dcrossoverofa disk-shaped
axial direction. We show that by increasing the planar ultracold Fermi gas. By increasing the planar density ρ,
density ρ of fermions (or equivalently the chemical po- thatis byinducing the crossoverfromtwo-dimensionsto
tential µ) one induces the dimensional crossover in the three-dimensions, c shows jumps in correspondence of
s
system. In general, collective sound modes in a normal the filling of harmonic modes. Such jumps of the sound
Fermi system are of two kinds. When the collision time velocity, even if less pronouncedthan the ones predicted
τ betweenquasiparticlesis muchgreaterthantheperiod in the 1D-3D crossover [26], can be experimentally ob-
(ω−1) of the mode itself, i.e. if ωτ 1, the system is in served.
≫
2
z
In the last part of the paper we establish the experi- H ( )isthej-thHermitepolynomialofargumentz/a ,
j z
a
mentalconditionstobeachievedfordetectingzerosound z
andfirstsound. Inparticular,byconsidering40Katoms, az = ¯h/(mωz) being the characteristic length of the
axial harmonic potential.
we estimate the critical mode frequency ω which dis-
c p
The two dimensional wave vector K (K ,K ) em-
criminates between the collisional and the collisionless x y
≡
beds the translational invariance in the radial plane. By
regimes.
imposing periodic boundary conditions in the box of
length L along x and y directions, the components K
x
II. CONFINED INTERACTING FERMI GAS 2π
and K of K are quantized according to: K = i
y x x
L
2π
We consider an ultracold normal Fermi gas with two- and Ky = iy, where ix and iy are integer quantum
L
spin components confined by a harmonic trap in the ax- numbers. The index α reads (j,K ,K ), or in a more
x y
ial direction and investigate the regime where the tem- compact form (j,K).
perature T of the gas is well below the Fermi temper- By inserting Eq. (4) into (2) the Hamiltonian takes
ature TF so that we may use a zero-temperature ap- the form
proachand treat separately the collisionless and the col-
lisionalregimes. TheHamiltonianofthediluteFermigas Hˆ =Hˆ +Hˆ , (6)
0 int
trapped by the external harmonic potential
where
1
U(r)= 2mωz2z2 , (1) Hˆ0 = ǫ0j,Kcˆ†σ,j,Kcˆσ,j,K , (7)
σ,j,K
and free to move in the (x,y) plane in a square box of X
size L, is given by and
Hˆ =σ=↑,↓Z drΨˆ†σ(r)(cid:18)− 2¯hm2 ∇2+U(r)(cid:19)Ψˆσ(r) Hˆint = 2agzFL2 Xσ,σ′j1,jX2,j3,j4K1X,K2,QWj1,j2,j3,j4
X
+ g2F drΨˆ†σ(r)Ψˆ†σ′(r)Ψˆσ′(r)Ψˆσ(r)(1−δσ,σ′), cˆ†σ,j1,K1+Qcˆ†σ′,j2,K2−Qcˆσ′,j3,K2cˆσ,j4,K1(1−δσ,σ′),(8)
σ,σ′=↑,↓Z
X
with
whereσ = , denotesthespinvariable,andtheinter-
{↑ ↓}
atomic coupling is ¯h2K2 1
ǫ0 = +(j+ )h¯ω , (9)
g =4π¯h2a /m, (2) j,K 2m 2 z
F F
and
with a > 0 being the s-wave scattering length. Ψˆ (r)
F σ
and Ψˆ†σ(r) are the usual operators annihilating or cre- 1 4 1
oabtienyginag faenrtmicioomnmofutsaptiinonσrealattitohnesp[2o3s,it2io4n, 2r5,,t2h7e].reTfohree Wj1,j2,j3,j4 =(cid:18)π jiY,i=1r2jiji!(cid:19)
field operator can be written as +∞
dξe−2ξ2H∗(ξ)H∗(ξ)H (ξ)H (ξ). (10)
j1 j2 j3 j4
Ψˆ (r)= φ (r)cˆ (3) Z−∞
σ α σ,α
Xα Eq. (9) gives the single-particle energies of noninter-
in terms of the single-particle operators cˆ (cˆ† ) de- acting fermions. To obtain the single-particle energies
σ,α σ,α of interacting fermions we calculate the two-spin Green
stroying (creating) a fermion of spin σ with the single-
function
particle wave function φ (r). Because of the symmetry
α
of the problem, φα(r) can be factorized into the prod- Gσ,σ′ (t t′)= i T cˆ (t)cˆ† (t′) (11)
uct of eigenfunctions of the harmonic oscillator in the z j1,j2,K − −¯hh σ,j1,K σ′,j2,K i
direction and plane waves describing free motion in the
(cid:0) (cid:1)
(x,y) plane. The field operator,then, may be written as where T denotes the time ordering operator. Here and
in the following, we will represent the operators in the
Ψˆ (r)= 1 ψ (z)exp(iK R)cˆ (4) interaction picture (with Hˆ0 the unperturbed Hamilto-
σ L2 j · σ,j,K nian), and will evaluate the averages with respect to
j,K
X the ground state of the Hamiltonian (6). In the cal-
with R (x,y), and where ψ (z) is the real function culation of the Green function we use a mean field ap-
m
≡
proach and replace the four operators terms involved
ψ (z)= 1 H ( z )exp( z2/2a2). (5) in Hˆint with hcˆ†σ,j1,K1+Qcˆσ,j4,K1icˆ†σ′,j2,K2−Qcˆσ′,j3,K2 +
j s2jj!π1/2az j az − z cˆ†σ,j1,K1+Qcˆσ,j4,K1hcˆ†σ′,j2,K2−Qcˆσ′,j3,K2i. Thepolesofthe
3
5 (l,j) u √2π
l,j
(0,0) 1
4 (0,1) 1/2
(0,2) 3/8
3 (0,3) 5/16
µ (0,4) 35/128

(1,1) 3/4
2
(1,2) 7/16
(1,3) 11/32
1
(1,4) 75/256
(2,2) 41/64
0
0 2 4 6 8 (2,3) 51/128
ρ π
(2,4) 329/1024
(3,3) 147/256
FIG. 1: (Color online). Chemical potential µ vs. planar (3,4) 759/2048
densityρ. Threevaluesofthescaledinteractionstrength: g=
10−5 (dotted curve), g = 0.35 (dashed curve), g =0.6 (solid (4,4) 8649/16384
cuunritvse)o.fT1/hae2zc,haenmdicleanlgptohtseinntiualniitssionfuanzi.ts of ¯hωz, density in Table 1. Coefficients ul,j. Notethat ul,j =uj,l.
If only only the lowest (j = 0) single-particle mode
Green function with σ = σ′ and j1 = j2 = j give di- along the z axis is occupied, then the system may be
rectly the single-particle energies ǫj,K of the interacting considered as strictly 2D. After inserting Eq. (12) into
fermions Eq. (15) an easy integration gives
¯h2K2 1 g lmax
ǫj,K = 2m +(j+ 2)h¯ωz+ 2aFL2 uj,lnl ρ = m (µ ¯hω j gF lmaxu ρ ), (16)
z Xl=0 j π¯h2 − z − 2az j,l l
l=0
(12) X
where
where
nl = hcˆ†σ,l,Kcˆσ,l,Ki (13) µ=µ¯− ¯hω2z (17)
K,σ
X
is the chemical potential measured with respect to the
is the total number of fermions in the harmonic shell l
groundstateenergyoftheaxialharmonicpotential. The
and u =u =W is obtained from Eq. (10).
j,l l,j l,l,j,j
expression (16) may be rewritten as
In Tab. 1 we reportthe numerical values of the coeffi-
cients u for l and j up to 4.
The ml,jaximum positive integer l involved in the lmax m
max δ +gu ρ = (µ ¯hω j), (18)
summation at the right hand side of Eq. (12) clearly j,l j,l l π¯h2 − z
depends on the single-particle state (j,K) and on the Xl=0 (cid:0) (cid:1)
interaction strength gF. where
If spin-up and spin-down states are equally populated
g 2a
we may write the total number of fermions in the l-th g = F = F (19)
harmonic state as 2π¯hωza3z az
n =2 Θ(µ¯ ǫ ). (14) is the scaled inter-atomic strength.
l l,K
−
The total planar density is then given by
K
X
with µ¯ the chemical potential, i.e. the Fermi energy.
lmax
Fromhere we maycalculate the 2Dfermiondensity in ρ= ρ , (20)
l
the (x,y) plane as a function of the chemical potential.
l=0
Under the condition L a on the planar box size, the X
z
summation over K may≫be repacedby an integralto get where the only nonzero densities ρ0,ρ1,...,ρlmax−1,ρlmax
the 2D density associated to the jth axial mode, which areobtained by solvingthe lmax+1 Eqs. (18) at fixedµ
is uniform in the (x,y) plane and is given by andg. Thevalue oflmax is determinedby requiringthat
all the l +1 solutions are positive. Note that when
max
n 1
ρj = Lj2 =2(2π)2 d2KΘ(µ¯−ǫj,K). (15) Fonigly.1ρw0eipslontotnhezecrhoe,mwicealhpaovteenatisatlriµctalsya2fDunscytsiotenmo.fthIne
Z
4
planardensityρforthreevaluesofthescaledinteraction The function χ(0)(Q,ω) χ(0),σ,σ(Q,ω), appearing in
j,j ≡ j,j
strength g. The figure shows the changes in the slope both equations, is given by
of these curves which occur when the planar density is
n n
sufficiently large so that a new excited level of the axial χ(0)(Q,ω)= σ,j,K− σ,j,K+Q , (24)
j,j ¯hω (ǫ0 ǫ0 )+i¯hη
harmonic well (a new shell) begins to be populated. K − j,K+Q− j,K
X
where the positive imaginary term in the denominator
III. ZERO SOUND ensurescausalityandthelimitη 0+ willeventuallybe
→
taken. The totally symmetric density-density response
function defined as
In the present section we analyze the collective mode
of the fermionic gas in the collisionless regime, i.e zero 1
χ(Q,ω)=
sound. To this end we calculate the density-density re- 4
sponse function: χ↑,↑(Q,ω)+χ↓,↑(Q,ω)+χ↑,↓(Q,ω)+χ↓,↓(Q,ω)
χσ,σ(Q,t)= (cid:2) (2(cid:3)5)
iΘ(t)<[nˆ (Q,t),nˆ†(Q,0)]> (21) finally reads
−¯h σ σ
χ(0)(Q,ω)
since its poles provide the dispersion relation and the χ (Q,ω) = j,j
j,j
velocity of propagation of this mode. 2
InEq. (21)wehaveintroducedthe2Ddensitynumber g lmax
operator nˆσ(Q) according to + a FL2χ(j0,j)(Q,ω) uj,lχl,l(Q,ω).
z
l=0
X
nˆ (Q) nˆ (Q)= cˆ† cˆ . (22) (26)
σ ≡ σ,j,K σ,j,K σ,j,K+Q
j,K j,K
X X This equation is one of the main results of the paper.
At any time, we may define It may be recast in the form of a system of equations
for the response functions χ (Q,ω) pertaining to the
l,l
nˆ (Q,t)= nˆ (Q,t). (23) various axial modes l:
σ,j σ,j,K
XK lmax g χ(0)(Q,ω)
δ F u χ(0)(Q,ω) χ (Q,ω)= j,j
We proceedby calculating the equations of motionfor j,l−a L2 j,l j,j l,l 2
spindiagonal(σ =σ′)andnondiagonal(σ =σ′)density- Xl=0 (cid:18) z (cid:19)
6 (27)
density response functions
in terms of which the full response function is given by
d
i¯h χσ,σ (Q,t) lmax
dt j,j,K χ(Q,ω)= χ (Q,ω) (28)
l,l
= δ(t)<[nˆσ,j,K(Q,t),nˆ†σ,j(Q,0)]> Xl=0
i
Θ(t)<[[nˆ (Q,t),Hˆ(t)],nˆ† (Q,0)]>, The poles of this response function coincide with the
− ¯h σ,j,K σ,j zeros of the determinant of the matrix of coefficients of
i¯hdχσ,σ′ (Q,t) χl,l(Q,ω) in the system of equations (27) in the limit
dt j,j,K η 0+. Since zero sound is appreciably damped for
→
i large Q, [24], we look for zeros of the determinant in the
= Θ(t)<[[nˆ (Q,t),Hˆ(t)],nˆ† (Q,0)]>.
¯h σ,j,K σ′,j limitQ 0. Inthisregime,onemayevaluateχ(0)(Q,ω)
→ l,l
of Eq.(24) under the condition
Afteralengthybutstraightforwardcalculationandus-
ing RPA [24, 28] to decouple the higher order response
function coming from the commutator on the r.h.s. of ω π¯h2
s= > 2 ρ (29)
the equation of motion, the time Fourier transforms Q s m2 l
χ (Q,ω)ofthe spindiagonalandnon-diagonaldensity-
l,l
density response functions read, respectively to find
mL2 s
χσ,σ(Q,ω) = χ(0)(Q,ω) χ(0)(Q,ω)= , (30)
j,j j,j l,l 2π¯h2 s2 2πh¯2ρ
+ gF χ(0)(Q,ω)lmaxu χσ′,σ(Q,ω) (cid:16)q − m2 l(cid:17)
a L2 j,j j,l l,l where use of Eq. (16) has been made. Note that
z
l=0
X
χσ′,σ(Q,ω) = gF χ(0)(Q,ω)lmaxu χσ,σ(Q,ω). π¯h2
j,j azL2 j,j j,l l,l vF,l =s2m2 ρl (31)
l=0
X
5
3
is the planar Fermi velocity in the l-th axial shell (l =
0,1,...,l ). v corresponds to the Fermi velocity of
max F,l
a strictly 2D uniform Fermi gas with density ρ , which
l
at fixed µ is a function of g. Using Eq. (28), we may
2
calculate the response of the system for any value of the
chemical potential, i.e. for any number of excited levels
0
of the axial harmonic trap. We start by considering the cs
case when only the lowest state of the harmonic axial
1
well is populated, i.e. l = 0. Then, from Eq. (27),
max
one gets the density response function χ (Q,ω) as
0,0
χ(0)(Q,ω)/2
χ (Q,ω)= 0,0 , (32) 0
0,0 1 (g /a L2)u χ(0)(Q,ω) 0 2 4 ρ π 6 8
− F z 0,0 0,0
By using Eq. (30) with l = 0, we find easily that the
poles of Eq. (32) satisfy ω = c0Q where the velocity of FIG. 2: (Color online). Zero sound velocity c0s vs. planar
s density ρ. Three values of the scaled interaction strength:
zero sound is given by
g = 10−5 (dotted curve), g = 0.35 (dashed curve), g = 0.6
c0s =vF,0 f(g) (33) (dseonlisditycuinrvuen).itsTohfe1/zaer2zo,asnodunldengvtehlosciintyuinsitisnoufnaizt.s of azωz,
with
1.08
1+g u
f(g)= 0,0 , (34) 1.07 1.0216
1+2 g u0,0 1.06 1.0208
1.02
the term which takes inpto account the effects of interac- 01.05 1.0192
tion. We remind that the chemical potential µ is related vF,1.04 0 1 2 3 4 5
to the 2D density ρ by Eq. (18), i.e. 0/ 1.03
s
c
1.02
π¯h2
1.01
µ= (1+g u ) ρ. (35)
0,0
m 1
The expression(33) forthe zerosoundvelocitycoincides 0.990 1 2 3 4 5
µ
with that of a strictly 2D Fermi gas [29] of density ρ ,
0
with the effect of the axial harmonic trap embodied in
the definition of the dimensionless parameter g u0,0 as FIG. 3: Ratio c0s/vF,0 vs. chemical potential µ. Three values
the effective 2D coupling. We recognize in this effective of the scaled interaction strength: g =0 (dotted curve), g =
2D coupling the familiar Landau parameterF [29] for a 0.35 (dashed curve), g = 0.6 (solid curve). Note that the
0
strictly interacting 2D Fermi gas. Let us denote g u dottedcurveissuperimposed to thesolid one. The velocities
byF . ByusingthisdefinitionandtheEq. (35),wem0a,y0 c0s and vF,0 are in units of azωz, lengths in units of az, and
0
express the zero sound velocity (33) in terms of F thechemical potential in unitsof ¯hωz.
0
2 1+F
c0 = (µ π¯hω a2ρF ) 0 . (36) c0/v does not change when the chemical potential is
s m − z z 0 √1+2 F s F,0
r 0 varied (dashed line). This behavior corresponds to the
one described by Eq. (33). When the chemical potential
Nowweconsideradensityρlargeenoughsothatsome
is greater than a critical value µ , excited axial modes
excitedaxialharmoniclevelsarepopulated. Inthiscase, c
begin to be occupied by the fermions and c0/v grows
the velocity of zero sound is given by the zero of the s F,0
with µ (see the inset of Fig. 3).
determinant of the matrix in the l.h.s. of Eq. (27) with
ω >Q. The calculationis again straightforwardand the
results are reported in Fig. 2, where we display the zero
IV. FIRST SOUND
sound velocity c0 as a function of the planar density ρ.
s
As it happens in the strictly 2D case, for a given value
of the planar density ρ, c0 increases by increasing the In this section we analyze the collisionalcase of a nor-
s
repulsive two body interaction g. mal paramagnetic Fermi gas. To analyze a sound wave
In Fig. 3 we plot the quantity c0/v as a function thattravelsinaplanardirectionitisimportanttodeter-
s F,0
of the chemical potential µ. When g = 0 one finds that minethecollisiontimeτ ofthegas. Accordingto[15]and
c0=v . If g > 0 but µ is sufficiently small so that only [34], if the local Fermi surface is not strongly deformed
s F,0
the lowest axial mode is populated, then the quantity then τ = τ (T/T )2, where τ = 1/(ρ(3D)σv ) with σ
0 F 0 F
6
thescatteringcross-sectionandρ(3D) the3Ddensity. In- The height of the first jump of c is then given by
s
stead, if the local Fermi surface is strongly deformed, as
in our case, then the collision time is simply given by √k+1 √k
∆c =ω a − . (43)
τ = τ0 [34]. It is easy to find that in our 2D geometry s z z √2
τ 1/(ω ρa2). Settingω thefrequencyofoscillationof
th0e∼soundzwaFve, the wave propagates in the collisionless Wheninteractionsarenotnegligibleintheequationof
regimeif ωτ 1 andinthe collisionalregime ifωτ 1. state, one must use Eq. (18) to get the values of ρ for
≫ ≪ which the slope of µ exhibits discontinuities, and calcu-
In the collisional regime by adopting the equations of
latethecorrespondingdiscontinuitiesinc . Forexample,
hydrodynamics [23, 25, 27] the planar first sound veloc- s
at the first discontinuity, which appears when l goes
ity c of the Fermi gas is given by the zero-temperature max
s
from 0 to 1 the value of the density is
formula
1 1 1 1
cs = ρ ∂µ . (37) ρc = πa2z 1+gu0,0 = πa2z 1+F0 . (44)
sm∂ρ
The corresponding discontinuity exhibited by the first
By using this formula and Eq. (18) we determine the sound velocity is
behavior of c versus ρ and of c /v as a function of µ,
as shown in sFig. 4 and in Figs. 5F,,0respectively. These ∆cs =c−s −c+s , (45)
figures, obtained both for g = 0 and for g > 0 display
where
shell effects, namely jumps of the first sound velocity c
s
when the atomic fermions occupy a new axial harmonic
1
mode. c− =ω a , (46)
If the interaction is negligible in the equation of state, s z zs(1+gu0,0)[1+g(u0,0−u1,1)]
the values ρ of the density at which such jumps occur
c and
may be analytically obtained from Eq. (18). In fact, we
obtain the behavior of the planar density as a function
of the chemical potential by setting g = 0 in the Eq. c+ = ω a
(18) and performing the sum over the quantum number s z z
j. Then, we get 1+g[u0,0+u1,1+g(u0,0u1,1 u20,1)]
− ,
s[1+g(u0,0 u1,1)][2+g(u0,0 2u0,1+u1,1)]
I[h¯ωµz] 1 µ 1 µ − − (47)
ρ= j = S( ), (38)
πa2 ¯hω − πa2 ¯hω
Xj=0 z (cid:18) z (cid:19) z z wherethe u’saregiveninTab. 1. We observethatthese
shell effects are similar to the ones occurring in the 1D-
where
3D crossover [26], where the jumps are sharper because
1 the sound velocity goes to zero in correspondence of the
S(x)=(1+I[x])(x I[x]) (39)
fillingofplanarharmonicmodes,showingthecrucialrole
− 2
playedbythedimensionality. Inthenextsectionweshall
with I[x] the integer part of x. Eq. (38) with Eq. (39) discuss the experimentalconditions necessaryto observe
represents a simple analytical formula which gives the shell effects in the velocity of first sound.
uniform planar density ρ of the Fermi gas as a function Now we comment on the 2D-3D crossover in our
of the chemical potential µ. The values ρ signing the fermionic gas, starting from the strictly 2D case, cor-
c
jumpscanbeobtainedfromEq. (38)withwithEq. (39). responding to l = 0 in Eq. (16). In this case the
max
The jumps of the first sound velocity are given by chemical potential µ is given by Eq. (35) and therefore
from Eqs. (31) and (37) one finds
k(k+1)
ρc = 2πa2z with k =1,2,3,.... (40) cs = vF,0 (1+gu0,0), (48)
√2
q
Whenthecriticalvalueofthedensityisapproachedfrom
or, in terms of the Landau parameter F
the left, from Eq. (37) we find that the sound velocity is 0
2(µ π¯hω a2ρF )
c− =ω a k+1 , (41) c = m − z z 0 1+F (49)
s z z 2 s q √2 0
r
p
while when n is approached form the right, we find recovering the usual result for a 2D Fermi system. Such
c
a behavior is illustrated by the first plateau in Fig. 5.
c+ =ω a k . (42) Incidentally, note that when F0 ≃ 0 the zero sound ve-
s z z 2 locity (36) is larger than the first sound velocity (49) by
r
7
2 0.8
0.75
1.5 0.7
0.65
0
F,
s 1 v 0.6
c /
s
c 0.55
0.5
0.5
0.45
0.4
0 0 1 2 3 4
0 2 4 6 8 µ
ρ π
FIG. 5: (Color online). The ratio cs/vF,0 versus chemical
FIG. 4: (Color online). First sound velocity cs as a function potentialµ. Chemicalpotentialisinunitsof¯hωz. Wereport
ofthetransversedensityρ. Threevaluesofthescaledinterac-
behavior for g = 0 (dotted curve), g = 0.35 (dashed curve),
tion strength: g=0(dottedcurve),g=0.35 (dashed curve),
g =0.6 (solid curve). The velocities cs and vF,0 are in units
g = 0.6 (solid curve). The first sound velocity is in units of
azωz, density in unitsof 1/a2z, and lengths in unitsof az. oufniatzsωozf,¯hleωnzg.ths in units of az, and the chemical potential in
a factor of √2. In our case, the quantity F is smaller We observe that starting from the familiar Thomas-
0
than one. Then the two velocities are different between Fermi formula for the 3D local density ρ(z) of the Fermi
each other also when F remains finite. gas under axial harmonic confinement, given by
0
We observe that the first sound velocity depends ex-
plictly on the chemical potential. In fact, remembering ρ(z)= (2m)3/2 µ 1mω2z2 3/2 , (54)
the relation between µ and ρ given by Eq. (18), and us- 3π2¯h3 − 2 z
(cid:18) (cid:19)
ing Eqs. (38) and (39), one may show that the velocity
and integrating over z, one obtains exactly Eq. (52).
of first sound takes the form:
2µ 1 1I[µ/¯hωz] t(g) V. EXPERIMENTAL FEASIBILITY
c = 1 + , (50)
s
rms2 − 2 µ/¯hωz µ/¯hωz
h i
We now discuss the experimental conditions to detect
where t(g) is a complicated but analytical function of g
the zero and first sound in a quasi-2D two-component
and of the matrix elements u’s, which vanishes if g =
Fermigas. Wesupposethatinthe(x,y)planethesystem
0. We stress that the chemical potential µ depends on
isinasquareboxoflengthL=20µm,whileintheaxial
the planar density ρ and on the interaction strength g.
directionzthereisaharmonicconfinementcharacterized
Fig. 5 shows how c /v explictly depends on µ when
s F,0 by a frequency of 160 kHz, which corresponds to a char-
axial harmonic states other than the ground states are acteristic length a 0.1 µm. We consider 40K atoms
z
occupied. FromEq. (50)onefinds thatforanyg atvery ≃
trapped in the two hyperfine states f =9/2,m =7/2
f
largeρ(i.e. alsoverylargeµ)thespeedofsoundisgiven | i
and f = 9/2,m = 7/2 with f the total atomic spin
f
by: | − i
andm its axialprojection. Inthis casethes-wavescat-
f
tering length reads a =174 a , where a =0.53 10−10
F 0 0
1 2µ m is the Bohr radius [35]. For this value of a , w· e can
c = . (51) F
s
2 m determine the value of the collective mode frequency ω
r c
whichdiscriminatesbetweenthe collisionlessandthehy-
In absence of two-body interactions (g = 0) for ρ > drodynamic regimes. In fact, the collision time τ can
1th/e(πaax2zia),lih.ea.rmfoornµic>we¯hlωlaz,reseovcecruapliseidngalned-ptahretigclaessetxahteibsiotsf bcreiteicsatilmfraetqeudenacsyτis∼ω1/=(ω2zπρ/aτ2F) aωnd(2tπhρeac2o)r.reIsfptohnedfirneg-
the 2D-3D crossover. Finally, for ρ 1/(πa2), i.e. for c ∼ z F
≫ z quency ω of the collective mode is much greater than ωc
µEq≫s. (¯h3ω7z),,t(h38e)Faenrmdi(3g9a)sobneecofimndess 3D. In this case from (i.e. forω ≫ωc)thesystemisinthecollisionlessregime;
otherwise (i.e. for ω ω ) the collective excitation is
c
≪
hydrodynamic-like.
µ=(2π)1/2¯hωz(a2zρ)1/2 , (52) One can experimentally change ωc in the Fermi gas
of 40K atoms by varying the planar density ρ. For in-
stance, from Fig. 4 and Eq. (39) one finds that in corre-
cs =√2 (2π)1/4azωz(a2zρ)1/4 . (53) spondence to (ρa2z)π = 1/2 the system is strictly 2D. In
8
this case, the planar density is ρ 1.6 1013 atoms/m2, radial plane instead of the axial direction (see [26]). In
the total number of atoms is N =≃ρL2· 6.4 103, and this case, the treatment of the problem becomes much
≃ ·
the critical frequency reads ω 1.4 kHz. In addition, more complicated. In fact, the cigar-shaped configura-
c
the zero sound velocity is c0 =≃v 0.02 m/s, while tion is characterized by two quantun mumbers related
s F,0 ≃
the first sound velocity is c = v /√2 0.014 m/s. to the harmonic energetic levels in the planar directions.
s F,0
Moreover, one finds that the scaled interac≃tion strength However,we expect for the zerosound velocity the same
is g = 2a /a 0.18. To conclude, we observe that one behavior as in the case of the only axial harmonic trap-
F z
can change the≃critical frequency ω also by varying the ping.
c
scattering length a via magnetic Feshbach resonances.
F Inthesecondpartofthepaper,fromtheformulawhich
relatesthe chemicalpotentialofa disk-shapedFermigas
toits uniformplanardensity,wehavecalculatedthe col-
VI. CONCLUSIONS
lisional sound velocity of the system. We have found
that this sound velocity gives a clear signature of the
In the present paper we have considered a dilute and dimensional crossover of the two-component Fermi gas.
ultracold interacting disk-shaped Fermi gas confined in Our calculations suggest that the dimensional crossover
the axial direction by a strong harmonic trap and uni- induces shell effects, which can be experimentally de-
formin the two planardirections. In particular,we have tected. In fact, as in 1D-3D crossover [26], also in the
analyzed the behavior of the velocity of propagation of 2D-3Dcrossoverthesoundvelocityexhibitsjumpsincor-
the zero and first sound modes in this gas. respondence of the filling of axial harmonic modes. We
Inthe firstpartofthe paper,we havestudiedthe zero have investigated the behavior of the ratio between the
sound mode from the density-density response function first sound velocity c and the planar Fermi velocity as
s
within the linear response theory and under the random a function of the chemical potential. We stress that our
phase approximation. We have obtained the zero-sound study shows that when only the groundstate of the har-
dispersion law by calculating the poles of the density- monicwellispopulatedandwhenthe chemicalpotential
density response function. Then we have analyzed the is sufficiently large so many excited axial modes are oc-
behaviorofthevelocityofthezerosoundasafunctionof cupied, ourresults reproducethe behaviorofa 2Dand a
theplanardensityfordifferentvaluesoftheFermi-Fermi 3D Fermi system, respectively.
scatteringlength. Wehaveverifiedthatasinthe strictly
Finally, we have discussed the possibility of observing
two dimensionalcase, in correspondenceofa givenvalue
inexperiments the zeroandfirstsoundby using gasesof
of the planar density, the zero sound velocity increases
40Katomstrappedinthe twolowesthyperfinestates. In
as a consequence of the increasing of the strength of the
particular,we haveestimated the sound-mode frequency
fermion-fermioninteraction. We havestudied,moreover,
which discriminates between the collisionless and hydro-
thebehavioroftheratiobetweenthezerosoundvelocity
dynamic regime. We have found that these regimes re-
c0 and the planar Fermi velocity v at varying of the
s F,0 quireseveregeometricandthermodynamicalconstraints,
chemical potential; when, in presence of a non vanish-
but they can be reached with the available experimental
ing fermion-fermion interaction, the chemical potential
setups.
isgreaterthanacriticalvalue,excitedaxialmodesbegin
to be occupied by the fermions and c0/v grows with ThisworkhasbeenpartiallysupportedbyFondazione
s F,0
µ. Inprinciple, the same kindofanalysismaybe carried CARIPARO. The authors thank Giovanni Modugno for
outin presence ofa harmonic potentialin the transverse useful suggestions.
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