Trapped p-wave superfluids: a local density approach

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by  M. Iskin

Trapped p-wave superfluids: a local density approach
M. Iskin and C. J. Williams
Joint Quantum Institute, National Institute of Standards and Technology,
and University of Maryland, Gaithersburg, Maryland 20899-8423, USA.
(Dated: February 5, 2008)
The local density approximation is used to study the ground state superfluid properties of har-
monically trapped p-wave Fermi gases as a function of fermion-fermion attraction strength. While
8 the density distribution is bimodal on the weakly attracting BCS side, it becomes unimodal with
0 increasing attraction and saturates towards theBEC side. This non-monotonicevolution is related
0 to the topological gapless to gapped phase transition, and may be observed via radio-frequency
2 spectroscopy since quasi-particle transfer current requires a finitethreshold only on theBEC side.

n
PACSnumbers: 03.75.Hh,03.75.Ss,05.30.Fk
a
J

4
Recent experiments measuring momentum distribu- p-wave case may be observed via radio-frequency spec-
2
tion, collective modes, order parameter, quantized vor- troscopy since quasi-particle transfer current requires a

tices, etc. have provided strong evidence for observa- finite threshold only on the BEC side, which is in sharp
]
n tion of a superfluid phase in two-componentcold atomic contrast with the crossover physics found in the s-wave
o
mixtures, interacting with short-range attractive s-wave casewhereafinitethresholdisrequiredthroughoutBCS-
c
interactions [1, 2, 3, 4, 5, 6]. These experiments have BEC evolution.
-
r also shown evidence that the ground state of these s- Local density (LD) approach: To obtain these results,
p
wave mixtures evolves smoothly from a paired Bardeen- we consider a harmonic trapping potential, separate the
u
s Cooper-Schrieffer (BCS) superfluid to a molecular Bose- relativemotion fromthe center-of-massone, anduse LD
. Einstein condensate (BEC) as the attractive interac- approximation to describe the latter. In this approxi-
t
a tion varies from weak to strong values, marking the mation, the system is treated as locally homogenous at
m
first demonstration of theoretically predicted BCS-BEC every position inside the trap, and it is valid as long as
- crossover[7, 8, 9]. the number of fermions is large which is typically satis-
d
On the other hand, there has also been substantial fied in cold atomic systems [22]. In orderto describe the
n
o experimental progress studying p-wave interactions to pairingcorrelationsoccurringintherelativecoordinates,
c observe triplet superfluidity [10, 11, 12, 13, 14]. How- we use the BCS mean field (MF) formalism and neglect
[ ever,controllingthe p-waveinteractionsis provingmuch fluctuations. The MF description is qualitatively valid

1 more difficult due to the narrow nature of p-wave Fes- throughoutBCS-BECevolutiononlyatthelowtempera-
v hbach resonances as well as to the more dramatic two- turesconsideredhere[7,8,9],andithasbeenextensively
5 and also three-body losses [10, 11, 12, 13, 14]. These applied to cold atomic systems describing qualitatively
9 experiments still motivated considerable theoretical in- the experimental observations.
7 terest predicting quantum and topological phase tran- Therefore,westartwiththefollowinglocalMFHamil-
3
sitions [15, 16, 17, 18, 19, 20]. More recently, p-wave tonian (in units of h¯ =kB =1)
.
1 molecules have been produced and their two-body prop-
80 estrutideysinhgavmeabneye-nbostduydpierdop[2er1t],ieospoefnpin-wgatvheespuopsesriflbuiliidtys ionf Hℓ(r) = ξℓ(r,k)a†k,σak,σ+ |∆ℓg(r)|2
0 Xk,σ
the near future.
:
Xiv moIgnenthoiussmsaynstuesmcrsip[t1,5u,n1li6k,e1t7h,e1p8r,ev1io9u,s2w0]o,rkwseosntuhdoy- − Xk [∆ℓ(r,k)a†k,↑a†−k,↓+H.C.], (1)
the ground state superfluid properties of harmonically
r where ℓ = 0 (ℓ = 1) corresponds to s-wave (p-wave)
a trapped p-wave Fermi gases as a function of fermion-
systems, a† creates a pseudo-spin σ fermion with mo-
fermion attraction strength. Our main results are as k,σ
follows. While we find that the density distribution is mentum k, ξℓ(r,k)=ǫ(k) µℓ(r) is the dispersion with
bimodalontheweaklyattractingBCSsidewherethelo- ǫ(k)=k2/(2m) and µℓ(r)=−µℓ V(r). While the global
−
calchemicalpotentialsarepositiveeverywhereinsidethe chemicalpotential µℓ fixes the total number of fermions,
trap,itbecomesunimodalwithincreasingattractionand the local chemical potential µℓ(r) includes the trapping
saturates towards the BEC side where the local chem- potential V(r) = mω02r2/2. In Eq. (1), ∆ℓ(r,k) =
ical potentials become negative. This non-monotonic ∆ℓ(r)Γℓ(k) is the local MF order parameter where
evolution is related to the topological gapless to gapped ∆ℓ(r) = g kΓℓ(k) a−k,↑ak,↓ describes the spatial de-
h i
phasetransitionoccurringinp-wavesuperfluids,andisin pendence,sPuchthatg >0isthestrenghoftheattractive
sharpcontrastwith the s-wavecase wherethe superfluid fermion-fermion interactions and ... implies a thermal
phase is always gapped leading to a smooth crossover. average. Γℓ(k)determinesthesymhmeitryoftheorderpa-
Lastly,weproposethatthephasetransitionfoundinthe rameter given by Γ (k) = W (k) λ Y (k) where
ℓ ℓ m ℓ,m ℓ,m
P
b
2
Wdeℓp(ekn)d=enkceℓk[01/6(,k129+]. kH02e)r(eℓ+,k1)0/2∼dRes0−c1risbeetsstthheemmoommeennttuumm 11 µµss 1122
scale where R is the interaction range in real space. In 00
0 88
thismanuscript,weassume∆ℓ(r)tobereal,andconsider ((aa)) --11 kk00====1111000000
only the λ = 1 and λ = 0 symmetry. However,
1,0 1,±1 44
weemphasizethatourdiscussionappliesqualitativelyto --22 ∆∆
ss
other p-wave symmetries as well.
--33 00
ThelocalMFHamiltoniancanbesolvedbyusingstan- --22 --11 00 11 22
dard techniques [16, 19], leading to a set of nonlinear 11//((kkFF aass))
equations for ∆ (r) and µ (r). These equations are
ℓ ℓ 11 µµ 4455
pp kk ==110000
00
MV W2(k) 4πΓ2(k) E (r,k) 00 ==1100
= ℓ ℓ tanh ℓ (,2) 3300
4πa k2ℓ (cid:20) 2ǫ(k) − 2E (r,k) 2T (cid:21)
ℓ 0 Xk ℓ ((bb)) --11
1 1 ξℓ(r,k) Eℓ(r,k) --22 ∆∆pp 1155
n (r) = tanh , (3)
ℓ V (cid:20)2 − 2E (r,k) 2T (cid:21)
Xk,σ ℓ --33 00
--11 --00..55 00 00..55 11
wherea istheexperimentallyrelevantscatteringparam- 11//((kk kk22 aa ))
ℓ 00 FF pp
eter which regularizes g, E (r,k)= ξ2(r,k)+∆2(r,k)
ℓ ℓ ℓ
is the local quasi-particle energy, Tpis the temperature FIG. 1: We show (in units of ǫF) the chemical potential µℓ
andV isthevolume. InEq.(3),n (r)isthelocaldensity (left y-axis) and the amplitude of the order parameter ∆ℓ
ℓ
of fermions, and the total number of fermions N is fixed (righty-axis)for(a)s-wavesystemsasafunctionof1/(kFas),
by N = drn (r). Notice that a has units of length and (b) p-wave systems as a function of 1/(k0kF2ap). Here,
(volume)Rin theℓs-wave (p-wave) caℓse, and that our self- solid (dotted) lines corresponds to k0 =100kF (k0=10kF).
consistent solutions also describe the single pseudo-spin
p-wave systems (except for the σ summations here and
on the BEC side. However, both ∆ and µ are non-
throughout), which are presented next. p p
analytic exactly when µ =0 at 1/(k3a ) 0.45, which
BCS-BEC evolution in homogenous systems: To un- p F p ≈
occurs on the BEC side of unitarity. We note that the
derstand the ground state properties of harmonically
non-analyticity of µ is barely seen in Fig. 1(b), and it
trapped p-wave superfluids within the LD approach, it p
is more expilicit in derivatives of µ . Thus, in the p-
is very useful to analyze first the homogenous s- and p
wave case, BCS-BEC evolution is not a crossover, but a
p-wave systems where V(r) = 0. Thus, next we dis-
cuss the s-wave case where ∆ (k) = ∆ W (k)Y (k) quantum phase transition occurs [15, 17, 19].
s s 0 0,0
with Y (k) = 1/√4π, and compare these results with This phase transition can be understood as follows.
0,0 b The quasi-particle excitation spectrum E (k) is gapless
the p-wavbe case where ∆p(k) = ∆pW1(k)Y1,0(k) with when the conditions ∆ℓ(k) = 0 and ξℓ(k)ℓ= 0 are both
Y1,0(k) = 3/(4π)cos(θk). In the numerical bcalcula- satisfiedforsomek-spaceregions. Whilethesecondcon-
tions, whilepwe mainly consider k0 = 100kF to describe dition is satisfied for both s- and p-wave symmetries on
b
realistically the short-ranged atomic interactions, some the BCS side where µ > 0, the first condition is only
ℓ
of the k0 =10kF results are also shown for comparison. satisfied by the p-wave order parameter. Therefore, un-
In Fig. 1(a), we show ∆s and µs at zero tempera- like the s-wave case, Ep(k) is gapless on the BCS side
ture (T = 0) for the s-wave case where the BCS-BEC (µ >0)butitisgappedontheBECside(µ <0),lead-
p p
evolution range in 1/(kFas) is of order 1. Notice that, ingtothephasetransitiondiscussedabove[7,15,19,23].
∆ grows continuously without saturation with increas-
s Havingdiscussedthe groundstate ofhomogenoussys-
ing attraction, while µ decreases continuously from the
s tems, next we analyze the trapped case.
Fermi energy ǫ =k2/(2m) on the BCS side to the half
F F BCS-BEC evolution in trapped systems: For this
of the binding energy ǫ /2 = 1/(2ma2) on the BEC
b,s − s purpose, similar to the analysis of homogenous sys-
side [9]. Here, k is the Fermi momentum which fixes
F tems, first we discuss the s-wave case where ∆ (r,k) =
the total density n = k3/(6π2) of fermions. Thus, s
σ F ∆ (r)W (k)Y (k), and compare these results with the
we conclude that the ePvolution of ∆s and µs as a func- s 0 0,0
tion of 1/(kFas) is analytic throughout, and BCS-BEC p-wave case whebre ∆p(r,k) = ∆p(r)W1(k)Y1,0(k). In
evolution is a smooth crossover[7, 8, 9]. the numerical calculations, we again choose k0 = 100kF
b
In Fig. 1(b), we show ∆p and µp at T = 0 for the where kF = mω0rF is the global Fermi momentum.
p-wave case where the BCS-BEC evolution range in Here, rF is the Thomas-Fermi radius determined by
1/(k0kF2ap)isoforder1. Noticethat,∆p isexponentially V(rF) = ǫF = kF2/(2m), and fixes the total number of
smallbutstillfiniteintheBCSlimitwhenµ ǫ andit fermions to N = k3r3/48.
p F σ F F
≈
grows rapidly with increasing attraction but almost sat- WithintheLDPapproximation,thedensitydistribution
urates for large 1/(k k2a ), while µ decreases continu- of trapped non-interacting (g = 0 or a 0−) gas is
ously from ǫ on the0BFCSp side to ǫp /2 = 1/(mk a ) n (r) = k3(r)/(6π2), where k (r) isℓth→e local Fermi
F b,p − 0 p ℓ σ F F
P
3
1188 best on the BEC side where strongly attracting fermion
11//((kkFF aass))==--∞∞ pairsformweaklyrepulsivelocalmoleculeswhichcanbe
==--00..55
== 00 well described by the Bogoliubov theory [9, 19]. On this
1122
== 00..55
((aa))n(r)n(r)ss stridaer,illtyhesmsiazlel ovfaltuheessa-swaξve molaecu>les0dwechreenaseksato arb1i-,
66 B,s ∼ s 0 s ≫
leading to arbitrarilly weak molecule-molecule repulsion
U = 2πa /m where a = 2a within the
00 BB,s BB,s BB,s s
00 00..55 rr//rrFF 11 Born approximation [9]. However, the size of the p-
wave molecules saturates to small but finite values as
112200
11//((kk00 kk2F2F aapp))====----∞∞00.. 44 ξbBu,tpfin∼ite1/mko0lewcuhleanrrke03pauplsi≫onU1, lead=in2gπaalso t/omawwheearke
8800 == 00 BB,p BB,p
== 00..44 a = 9/k [19]. Since our LD approximation recov-
((bb))n(r)n(r)pp erBsBt,phe Thom0as-Fermi approximation for the resultant
4400 molecules in the BEC limit, n (r) saturates rapidly for
p
1/(k k2a ) > 0 due to the presence of weak but finite
0 F p
00 molecular repulsion.
00 00..55 rr//rr 11
FF
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
FIG. 2: (Color online) We show [in units of kF3/(2π)3] the (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)
Fully (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)F(cid:0)(cid:1)ul(cid:0)(cid:1)ly(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
density distribution nℓ(r) for (a) s-, and (b) p-wave systems Gapless (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)G(cid:0)(cid:0)(cid:1)(cid:1)a(cid:0)(cid:0)(cid:1)(cid:1)pp(cid:0)(cid:0)(cid:1)(cid:1)e(cid:0)(cid:0)(cid:1)(cid:1)d(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
as a function of the trap radius r (in units of rF). Here, we (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
set k0 =100kF. (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(a) BCS (b) Unitarity (c) BEC
momentumdetermined byµ =k2(r)/(2m)+V(r) with FIG. 3: Schematic diagrams showing (a) a fully gapless su-
ℓ F
µℓ =ǫF. Therefore, both kF(r) and nℓ(r) are highest at perfluid on the BCS side when µp(0 ≤ r <∼ rF) > 0, (b)
the center of the trap as can be also seen in Figs. 2(a) a partially gapped superfluid around unitarity when µp(0 ≤
and 2(b) when 1/(k a ) = and 1/(k k2a ) = , r ≤ r∗) > 0 but µp(r > r∗) < 0, and (c) a fully gapped
respectively. F s −∞ 0 F p −∞ superfluid on the BEC side when µp(r≥0)<0.
In the presence of weak attraction, while µ deviates
ℓ
This non-monotonic evolution of n (r) is also related
from ǫ , the density distribution is still well described p
F
to the topological phase transition discussed above for
by the non-interacting expression. For fixed N, n (r)
ℓ
the homogenous systems. In the trapped case, the lo-
is expected to squeeze and increase towards the center
cal quasi-particle excitation spectrum E (r,k) at posi-
of the trap since µ decreases with increasing attrac- ℓ
ℓ tion r is gapless when the conditions ∆ (r,k) = 0 and
tion. The squeezing effect of the weak attractive inter- ℓ
ξ (r,k) = 0 are both satisfied for some k-space regions.
actions can be seen in Fig. 2(a) for the s-wave and in ℓ
While these conditions are both satisfied everywhere in-
Fig. 2(b) for the p-wave systems when 1/(k a ) = 0.5
F s
and 1/(k k2a )= 0.4, respectively. − side the trap leading to a fully gapless superfluid on the
0 F p − BCS side, they are only satisfied around the center of
However,weaklyattractingp-wavesuperfluidshavebi-
the trap close to unitarity leading to a partially gapped
modal density distribution as shown in Fig. 2(b), which
superfluid. Furtherincreasingtheattractiontowardsthe
is in sharp contrast with the unimodal s-wave distribu-
BEC limit, the second condition is not satisfied, and the
tionofFig.2(a). Thisdifferencecanbe understoodfrom
entiresuperfluidbecomesfullygapped. Thesephasesare
the homogenous results shown in Fig. 1(a) and 1(b) as
schematically shown in Fig. 3(a), 3(b) and 3(c), respec-
follows. First,sincek (r)decreasesawayfromthecenter
F tively, and next we discuss their experimental detection.
of the trap, the local s- and p-wave scattering parame-
Radio-frequency (RF) spectroscopy: The gapless to
ters1/[k (r)a ]and1/[k k2(r)a ],respectively,increase
F s 0 F p gappedphasetransitiondiscussedabovemaybeobserved
as a function of r if a and a are fixed. Second, notice
s p for the first time in cold atomic systems via for instance
inFig.1(b)that∆ increasesrapidlyfromexponentially
p RF spectroscopy, where atoms are transferred from one
small to larger values as a function of 1/(k k2a ), un-
0 F p hyperfinestatetoanothergeneratingaquasi-particlecur-
like ∆ which increases smoothly as shown in Fig. 2(a).
s rent [22, 24, 25, 26]. This is analogous to electrons tun-
These two observations combined shows that the almost
neling from a superconducting to normal metal, and it
non-interactingn (r) distributiontowardsthe tailisdue
p has been used in atomic systems to observe pairing cor-
to finite but exponentially small ∆ (r).
p relationsinunpolarized[25]aswellaspolarized[26]mix-
Withincreasingattractiontowardsunitarity,whilethe
tures.
unimodaln (r)distributionsmoothlysqueezesfurtheras
s Thelocalquasi-particletransfercurrent,withintheLD
shownin Fig. 2(a) for 1/(k a )=0 and 1/(k a )=0.5,
F s F s approximation, is given by [22, 24, 27]
the bimodal n (r) distribution becomes unimodal and
p
s1a/t(ukr0aktF2easpa)s=sh0o.w4.nTinhiFsidgi.ff2e(rbe)ncfeorca1n/(bk0ekuF2nadpe)rs=to0odanadt Iℓ(r,ω)=t2F Xk Aℓ[k,ξℓ(r,k)−ω]F[ξℓ(r,k)−ω], (4)
4
2200 C(r,ω) = [ω2 ∆2(r)/(4π)]/(2ω) +µs(r), and θ(x) is
11//((kkFF aass))==--∞∞ the theta funct−ion. Therefore, I (r,ω) flows when the
==--00..55 s
== 00 threshold ω (r) µ (r)+ µ2(r)+∆2(r)/(4π) 0
((aa))ωωI()I()ss 1100 == 00..55 iins rtehaechBedC,Swthah,snicdhωr≥ed−u(cres)s to 2ωptµh,s((srr)) ≥in∆th2sse(rB)/E[8Cπµlism≥(ri)t].
th,s s
≥ | |
Therefore, w (r) = 0 everywhere inside the trap
th,s
6
throughout BCS-BEC evolution [22, 24, 25, 26]. These
00
00 22 ωω 44 finite detuning thresholds can be also seen in Fig. 4(a),
where we show I (ω) for homogenous systems.
00..22 s
11//((kk kk22 aa))==--∞∞
00 FF pp
==--00..44 Thep-wavecurrentI (r,ω)isdifficulttoevaluateana-
== 00 p
== 00..44 lytically. However,unlikethes-wavecase,weexpectthat
((bb))ωωI()I()pp 00..11 swidthe,pb(rey)o6=nd0uenveitrayrwithyerwehienrseidEet(hre,kt)raips goanplypeodn,tahnedBaElsCo
p
that w (r)=0 everywhereinside the traponthe BCS
th,p
00 side where Ep(r,k) is gapless. The absence (presence)
00 22 ωω 44 of finite thresholds in gapless (gapped) superfluids can
be seen in our homogenous results shown in Fig. 4(b).
FpaIGrt.ic4l:e(tCraonlosfreorncluinrere)nWteIℓs(hωo)wfo(rinhuonmitosgoefnρoFust2F(/a2))st-,heanqdua(sbi)- Nanodti0ce,athfianti,tewthhilreesωhtohl,dp =isr0eqfourir1e/d(kfo0rkF12/a(pk)=k2−a∞),=−00..44.
p-wave systems as a function of the effective detuning ω (in 0 F p
Based on these results, we hope that spatially resolved
unitsof ǫF). Here, we set k0 =100kF. RF spectroscopy measurements (similar to [26]) may be
used to identify all three phases proposed in Fig. 3.
where tF is the transfer amplitude, Aℓ(k,x) is the spec- Conclusions: To summarize,we showedthatwhile the
tral function corresponding to the superfluid state, and density distribution of p-wave systems is bimodal on the
F(x)=1/[exp(x/T)+1]istheFermifunction. Here,ω = weakly attracting BCS side, it saturates and becomes
ωL ωH is the effective detuning where ωL and ωH are unimodal with increasing attraction towards the BEC
−
RF laser frequency and hyperfine splitting, respectively. side. We discussed that this non-monotonic evolution is
WeevaluateEq.(4)withthestandardBCSspectralfunc- related to the topological gapless to gapped phase tran-
vtEiℓ2oℓ((nrrs,,kkA))ℓ]=(}k,0,wǫ.5)h[e=1r−e2uπξℓ2ℓ{((urr,2ℓ,(kkr)),/k=E)δℓ0([.ǫr5,−[k1)E+]ℓaξ(rℓre(,rkc,o)k]h)+e/rEvenℓ2ℓ((crre,,fkka))c]δt[aoǫnr+sd, scisiotnaiotlwnraasoytcscwguiarthrpipnteghdeinlse-awpda-iwvneagvcteaosseauwpshemrefloreuotitdhhse,crsaounspsdeorviflseuri.indLpsahhsatalrsype,
and δ(x) is the delta function. we proposed that this phase transition may be observed
At T = 0, the s-wave current Is(r,ω) can via RF spectroscopysince quasi-particletransfer current
be evaluated analytically leading to Is(r,ω) = requires a finite threshold only on the BEC side, which
πρ t2[∆ (r)/(4πω)]2 C(r,ω)θ(ω)θ[C(r,ω)], where is in sharp contrast with the s-wave case where a finite
F F s
ρF = mVkF/(2πp2) is the density of states, threshold is required throughout BCS-BEC evolution.
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