Polarons in the radio-frequency spectrum of a quasi-two-dimensional Fermi gas

0.57 MB·English

by  Y. Zhang

Supplementary Material: “Polarons in the radio-frequency
spectrum of a quasi-two-dimensional Fermi gas”
Y. Zhang1,2, W. Ong1,2, I. Arakelyan1,2, and J. E. Thomas1
1Department of Physics, North Carolina State University, Raleigh, NC 27695, USA and
2Department of Physics, Duke University, Durham, NC 27708, USA
(Dated: January 18, 2012)
In this supplementary material, we discuss the calculation of the radio-frequency spectra arising
2 fromconfinement-induceddimersandpolaronsinaquasi-two-dimensionalFermigas. Wedetermine
1 the dimer binding energies, including both the tight axial confinement and the nonzero transverse
0
confinement. Weprovidetheprobabilitiesfordimer-to-dimertransitionsandtheshapeofthedimer-
2
to-scatteringstatespectrum. Wealsofindtheenergyandquasi-particleweightsforpolaronsinthe

n two-dimensional gas and thecorresponding spectra for polaron to polaron transitions.
a
J PACSnumbers: 03.75.Ss

7
1
Webeginbyreviewingbrieflyin Itheradio-frequency and dωI(ω)=1.

] spectrum arising from confineme§nt-induced pairs, in- ToRdetermine the spatial wavefunctions and the pair
s cluding final state interactions, but ignoring many- bindingenergies,we notethatthe rangeofthe two-body
a
g body effects, using the method employed for the three- interaction is small compared to the interparticle spac-
- dimensional case by Chin and Julienne [1]. We consider ing as well as to the harmonic oscillator confinement
nt mixtures of the three lowest hyperfine states of 6Li, de- scale lz ¯h/(mωz). In this case, interactions be-
≡
a noted 1 , 2 , 3 . For the conditions of our experi- tweenatomspintwodifferentspinstatesarewelldescribed
u ments |inia|1i2 m| iixture at 720 G, the observed 2 3 by the s-wave pseudopotential in three dimensions [2],
q threshold spectrum is well described by a 12-dime→r-to- V(r) = (4π¯h2a/m)δ(r)∂ (r...), where r is the distance
. r
t 13-scattering-state transition. In contrast, at 834 G, the between the atoms, m is the mass of a single atom and
a
predicteddimerspectrumisinmarkeddisagreementwith a is the magnetically tunable s-wave scattering length.
m
the data. In particular, we find that the difference be- Thespatialwavefunctionsarereadilywrittenintermsof
-
d tweenthe groundand excited state dimer energiesis too theGreen’sfunctionGE(r)fortherelativemotionofthe
n small. In II we determine the energies for noninter- twoatomsintheconfiningpotential,whichwetaketobe
§
o acting confinement-induced polarons. We find that the harmonic, with ground state energy E . The two-atom
0
c locationsoftheobservedresonancesfora12mixturenear scatteringstateswithenergyE =E +E ,whereE 0,
0 s s
[ ≥
834Garewellmodeledbythepredictedenergydifference take the form ψ (r) = ψ(0)(r) aG (r)u′(0), where
1 between isolated state 2 polarons and state 3 polarons, ψ(0) is the inputsstate anEdsψ =−u/rEwsith usregular at
v in a bath of atoms in state 1. Es
r =0. For the bound states, where there is no input, we
0
56 hEabve>ψ0b.(r)U=sin−ga∂Gr[Erbψ(br()ru)]′br(→0)0, w=huer′be(0E) y=ieEld0s−thEebewquitah-
3 I. CONFINEMENT-INDUCED DIMERS tion for the binding energy [2], 1 = a∂ [rG (r)] ,
1. where the right side projects out the−regrularEpbart or→f G0
0 A simple golden rule calculation gives the radio- at r = 0. For a three dimensional harmonic trap, the
2 Green’s function is
frequency-induced transition rate out of the initial state
1
Xiv: tR¯hoΩf←alilFp=oIs(s/2i2bπ.l/eH¯hfi)e|nrH˜ea,fl′iΩs|2taδt(ieEssftRh−ei(RωEarifb−)i f=¯hreωqPrufe)Fn,cRwyfi←ftohir,Hc˜whf′ahinerg=e- Gǫ(r)= lz√i4π Z0∞dηeiǫη Yj eicot(βjη)(cid:16)2xljj(cid:17)2fj(η), (1)
fi fi
h | i
r ing the hyperfine state of a single atom from the cho- where E = E + ǫ¯hω , with ǫ = ǫ for bound
a senpopulatedstate (i)to the initially unpopulatedstate 0 z − b
states and ǫ > 0 for scattering states. Here, f (η) =
j
(f) and F I is the overlap between the initial and fi-
nal waveh-fu|ncitions for the relative motion of the atom- 2βj/(1−e−2iβjη), with βz ≡ 1, βx,y = ωx,y/ωz and
lp= l / β . After subtracting the irregular part of G
pair. Since the center of mass energy does not change j z j
and chanpging to real variables, we determine the dimer
in the rf transition, E E is the total change in the
atomic hyperfine energfy−( i ¯hω ) plus the change in binding energies Eb =ǫb¯hωz >0 from
fi
≡
the energy of the relative motion of the pair E E .
Since F |hF|Ii|2 = 1, dωrfRi(ωrf) = (π/2)ΩF2fi−. WIe lz = ∞ du 1 2βju 1/2 e−ǫbu. (2)
definePa normalized speRctrum I(ω) where Ri(ωrf) = a Z0√4πu3  −Yj (cid:18)1−e−2βju(cid:19) 
(π/2)Ω2 I(ω) and ω = ω +ω, with ω the frequency
fi rf fi  
relative to the (unshifted) free-atomhyperfine transition The dimer binding energy is significantly increased for
frequency. Then, I(ω)= F I 2¯hδ(E E ¯hω) nonzero transverse confinement. At resonance, where
F |h | i| F − I −
P
2
l /a 0, for ν /ν =0 we obtain E =0.245hν , while
z → ⊥ z b z 1.0
for ν /ν = 1/25 we obtain E = 0.290hν . At 842
⊥ z b z
G in the shallowest trap, the binding energy of the 13
dimerisincreasedfrom0.15kHzwithouttransversecon-
0.8
finement to 0.78 kHz with transverse confinement. We
n
compute the pair binding energy Eb ≡ ǫb¯hωz as a func- ctio
taiomneoafsumraedgnientiRcefife.l[d3]u.sing the s-wavescatteringlengths nd fra 0.6
u
The scattering state is determined using u′(0) = o
(uo′(sn0e)-(t0o)-/o{n1e+coar∂rers[rpGonEdse(rn)c]er)→0b}e.tweUensingthethescarsettlaertiionng und-to-b 0.4
o
length and the bound state energy, we have u′(0) = B
s
u′(0)(0)/ a[G (r) G (r)] ,whereE is the bind- 0.2
s { Es − Eb r→0} b
ing energy corresponding to the scattering length a and
we have used ∂ r[G (r) G (r)] = [G (r)
G (r)] , whric{h isErsegula−r atErb = r0→. 0T}hen, thEes sca−t- 0
Eb r→0 600 700 800
tering state takes the form
Magnetic field (G)
G (r)u′(0)(0) FIG. 1: Bound 12 dimer to bound 13 dimer transition frac-
ψs(r)=ψE(0s)(r)− [GEs(Ers)−GEsb(r)]r→0, (3) (trioend ovnerlisnues):mTarganpetdicepfitehldU0fo=r ν2⊥80/νµzK=, νz1/=258.2.U5pkpHezr;cLuorwveers
where u′(s0)(0) =ψE(0s)(0), since the input state is regular lciunrevsedse(nboltueetohnelisnqeu)arUe0o=f F2ra1nµkK-Caonnddoνnzo=ver2l4a.p5iknHtezg.raSl,oilnid-
at r =0. cludingthefirst50evenaxialstates. Dashedlinesdenotethe
For 2 3 transitions in a 12 mixture at 834 G, the correspondingresultsusingonlythegroundaxialstate,Eq.6.
→
bindingenergiesaresmallcomparedtotheenergydiffer-
encebetweensymmetricaxialstates2¯hω ,whicharecou-
z
pledbythes-wavescatteringinteraction. Inthiscase,the first 50 even axial states. For comparison, we show the
Green’sfunctions,andhencethe1-2and1-3boundstates weak binding approximation given by Eq. 6, which in-
and the 1-3 scattering states, are well approximated by cludes only the groundaxialvibrationalstate. For small
the ground axial state component dzφ (z)G (z,ρ), binding energies, increasing the trap depth significantly
∝ 0 ǫb
yielding the normalized bound state R increases the pairing energy, increasing the overlap inte-
gral for bound-to-bound transitions.
κ
ψǫb(z,ρ)=φ0(z)√π K0(κρ), (4) From Fig. 1, we see that for a trap depth U0 = 21µK
at 720 G, the 2 3 transition in a 12 mixture is dom-
→
inated by bound-to-free transitions. In contrast, at 834
where φ (z) is the axial ground vibrational state for the
0 G, the correspondingbound-to-bound transitionis dom-
relative motion, E = ǫ ¯hω is the pair binding energy
for the given scattberingblengzth, and ¯h2κ2/m = E , i.e., inant and increases with increasing trap depth.
b
In the same approximation, the two dimensional box-
κ=√ǫ /l . The1-2boundstateto1-3boundstatetran-
b z normalized (to area A) 1-3 scattering state takes the
sitionstrengthisthendeterminedbytheoverlapbetween
form ψ (z,ρ) = φ (z)ψ (ρ), with E = ǫ ¯hω =
ofthemodifiedBesselfunctions,K (κ ρ)andK (κ ρ), ǫ⊥ 0 ǫ⊥ s ⊥ z
0 12 0 13 ¯h2k2/m. Assuming that the incident transverse state
where κ is determined by ǫ12(13). The 1-2bound to ⊥
12(13) b is the azimuthally-symmetric part (l = 0) of a box
1-3 bound contribution to the lineshape is then given by
normalized plane-wave state exp(ik x )/√A, i.e.,
⊥ ⊥
·
Ibb(ν)=ǫbb(q)δ[ν−(ǫ1b2−ǫ1b3)νz], (5) J0(k⊥ρ)/√A, Eq. 3 gives
whereν isthe axialharmonicoscillatorfrequencyinHz 1 πi
z ψ (ρ)= J (k ρ) H (k ρ) .
and ν is the rf frequency in Hz, relative to the bare 2- ǫ⊥ √A(cid:26) 0 ⊥ − ln(ǫb/ǫ⊥)+πi 0 ⊥ (cid:27)
3 hyperfine transition frequency of 83 MHz. For the (7)
≃
axial ground state, the frequency integrated bound to Wedetermine theoverlapintegralofthe 1-2boundstate
bound transition strength can be written compactly in withthe 1-3scatteringstate andintegratethe transition
terms of q ln(ǫ13/ǫ12), rate using density of transverse states, A/(2π)2πk dk ,
≡ b b ⊥ ⊥
to obtain the 1-2 bound to 1-3 scattering state contribu-
q2 tiontothelineshape,whichtakestheformofathreshold
dνI (ν) ǫ (q)= . (6)
Z bb ≡ bb 4sinh2(q/2) function,
We plot ǫbb as a function of magnetic field in Fig. 1 for I (ν)= ǫ1b2νz q2θ(ν −ǫ1b2νz) . (8)
thegeneralcase,validforbothweakandtightbindingof bf ν2 2
the1−2or1−3dimers,includingthecontributionofthe hq−ln(cid:16)ǫ1b2ννz −1(cid:17)i +π2
3
Note that dνI (ν)=1 ǫ (q), as it should. We find the physical two-body T-matrix element in 2D, 1/g =
bf bb 0
−
that Eq. 8Rwell fits the radio frequency spectra obtained 1/T2B(k0) − (1/A) k1/(2ǫk − 2ǫk0). This method is
at 720 G, where the dimer binding energy is larger than similar to that emPployed previously [4, 7, 8]. Here,
the transverse Fermi energy. we choose the T-matrix element Tk′k0 = T2B(k0)/A so
that the scattering rate calculated using the generalized
Goldenrulereproducesthescatteringrateobtainedfrom
II. POLARONS IN A TWO-DIMENSIONAL the 2D flux corresponding to Eq. 7. Then T (k ) =
2B 0
FERMI GAS ¯h2f(k )/m, with f(k ) = 4π/[πi + ln(E /ǫ )]. Here,
0 0 b ⊥
ǫ = 2ǫ and E = ǫ ¯hω is the dimer binding energy
⊥ k0 b b z
Forradiofrequency2 3spectraobtainednear834G calculated from Eq. 2. Both the πi term and the k de-
0
→
in a 12 mixture, we find that the difference between the pendenceinT (k )arecanceledbycorrespondingterms
2B 0
calculated dimer binding energies significantly underes- in the sum (1/A) 1/(2ǫ 2ǫ ), so that f−1(E ,q)
k k− k0 pi
timates the observed frequency shifts, as shown in Figs. is independent ofPk0 as it should be. For attractive po-
2, 3, and 4 of the main paper. We consider the possibil- larons with energy E < 0, we then obtain the simple
pi
ity that the spectra may arise from transitions between integral equation ǫ(L )=Σ(L ,ǫ ), where
i i i
polaronic states, as the polaron energy is significantly
1 2du
more negative than the corresponding dimer energy and
Σ − . (12)
is therefore energetically preferred [4]. ≡Z L+ln[ (1 ǫ)2 u+(1 ǫ u)]
0 − − 2 − − 2 − 2
Toestimatethepolaronenergies,weconsidereitheran p
isolatedimpurity atomin state2 orinstate 3,immersed Here, ǫ(Li) ≡ ǫi = Epi/EF and Li = ln(Eb1i/EF) for an
in a bath of atoms in state 1. We employ the method impurity in state i. Eq. 12 yields the polaron energies
describedforpolaronsinthree dimensions inthe supple- Epi =ǫiEF for the initial and final states i=2,3, using
mentary material of Schirotzek at al. [5], which is based thebindingenergiesEb1i =ǫ1bi¯hωz determinedfromEq.2.
on the zero momentum polaron wavefunction proposed Fig. 2 shows the attractive polaron energies obtained
by Chevy [6], which for a polaron in state i = 2,3 takes fromEq.12, which agreewith those obtainedin Ref.[4].
the form We see that the polaron energy Ep is a large fraction of
the local Fermi energy, which can be much larger than
the corresponding dimer binding energy E .
b
E =ϕ 0 FS + ϕ q k c† c FS . (9)
| ii 0i| ii| i1 kq| − ii k1 q1| i1
q<XkF<k 0
Here, the first term describes an impurity i of zero mo-
mentum in a Fermi sea of atoms in state 1 for which the
-1
net momentum is zero. Collisions between the impurity
and the background atoms couple the zero momentum
impuritystatetothatwithmomentumq k,producing -2
a particle-hole pair from the Fermi sea of−atoms in state EF
1 with net momentum k q, conserving the total zero E/p
− -3
momentum.
Forthe2Dcalculations,wereplacetheboxnormaliza-
tion volume in the supplementary materialof Schirotzek -4
atal.[5]bythecorrespondingareaA,sothatthepolaron
energy in 2D takes the form
-5
1
E = f(E ,q), (10)
pi A pi -5 -4 -3 -2 -1 0 1 2
qX<kF
ln[Eb/EF]
where
FIG. 2: Polaron energy Ep/EF versus ln[Eb/EF], where Eb
f−1(E ,q)= 1 + 1 1 . (11) isthedimerbindingenergyandEF isthelocalFermienergy.
pi
g A ǫ ǫ +ǫ E
0 kX>kF k− q q−k− pi Forradiofrequencytransitionsbetweenimpuritystates
Here,ǫk =¯h2k2/(2m)andkF isthelocalFermiwavevec- 2 → 3 in a bath of atoms in state 1, the momentum of
torwithE =¯h2k2/(2m)thecorrespondinglocaltrans- the impurity does not change. We therefore assume the
F F coherent part of the spectrum is given by
verse Fermi energy.
Following Zo¨llner et al., Ref. [4], we assume that the
I(h¯ω)=Z Z δ[¯hω E +E ], (13)
2 3 p3 p2
effective bare interaction U arises from a short range −
2D potential, so that the matrix elements Ukk′ = g0/A where |ϕ∗03ϕ02|2 = Z2Z3 is the square of the overlap
are momentum independent. g can be rewritten using integral between the part of the initial and final polaron
0
4
dimerbindingenergiesina12mixturenear834G,Table
1.0
1 of the main paper, we find that Z 0.85 and Z
2 3
≃ ≃
0.94 for the shallowest trap depth, both close to unity.
0.9
Hence,weexpectthattheoverlapbetweentheinitialand
0.8 finalpolaronstatesisstrongandthattransitionsbetween
polaronstatesshouldmakeanimportantcontributionto
0.7 the spectrum.
Z In the limit that the dimer binding energy is small
0.6 compared to the local Fermi energy, i.e., E << E ,
b F
one verifies from Eq. 12 that the corresponding polaron
0.5 energy yields the limiting form [4]
0.4
2E
F
E − , (14)
p
0.3 ≃ ln(2EF/Eb)
-5 -4 -3 -2 -1 0 1 2
ln[Eb/EF] which can be interpreted as an effective mean field shift,
sinceE isproportionaltothe2Ddensityn (ρ). At842
FIG. 3: Polaron quasiparticle weight Z versus ln[Eb/EF], F ⊥
where Eb is the dimer binding energy and EF is the local G, where the dimer binding energy is reasonably small
Fermi energy. compared to the transverse Fermi energy in our experi-
ments, this formula overestimates the magnitude of the
polaronenergydifferenceE E byabout10%forthe
p3 p2
−
states that yields the coherentpart of the spectrum. We mostshallowtrap. At811G,itoverestimatestheenergy
determine Z = ϕ 2 using Z−1 = 1 ∂Σ(L ,ǫ)/∂ǫ, differencebyabout40%. AsE isnotsmallcomparedto
2 | 02| 2 − 2 b
with ǫ E /E , as described in Ref. [5] for the 3D E formostofthedata,itisnotsurprisingthatE can-
p2 F F F
problem→, and similarly for Z = ϕ 2. Fig. 3 shows notbeadjustedinEq.14togivethemeasuredfrequency
3 03
| |
thequasiparticleweightZ obtainedfromEq.12. Forthe differences.
[1] C.ChinandP.Julienne,Phys.Rev.A71,012713(2005). 83, 021603(R) (2011).
[2] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. [5] A. Schirotzek, C.-H. Wu, A. Sommer, and M. W. Zwier-
80, 885 (2008). lein, Phys. Rev.Lett. 102, 230402 (2009).
[3] M. Bartenstein, A. Altmeyer, S. Riedl, R. Geursen, [6] F. Chevy,Phys. Rev.A 74, 063628 (2006).
S. Jochim, C. Chin, J. H. Denschlag, R. Grimm, A. Si- [7] M. Randeria, j. M. Duan, and L.-Y. Shieh, Phys. Rev.
moni, E. Tiesinga, et al., Phys. Rev. Lett. 94, 103201 Lett. 62, 981 (1989).
(2005). [8] M. M. Parish, Phys. Rev.A 83, 051603 (R) (2011).
[4] S.Zo¨llner, G.M. Bruun,and C.J. Pethick,Phys.Rev.A
Polarons in the radio-frequency
spectrum of a quasi-two-dimensional Fermi gas
Y. Zhang1,2, W. Ong1,2, I. Arakelyan1,2, and J. E. Thomas1
1Department of Physics, North Carolina State University, Raleigh, NC 27695, USA and
2Department of Physics, Duke University, Durham, NC 27708, USA
(Dated: January 18, 2012)
We measure radio-frequency spectra for a two-component mixture of a 6Li atomic Fermi gas in
2
thequasi-two-dimensionalregime. NeartheFeshbachresonance,wherethetransverseFermienergy
1
islargecomparedtotheconfinement-induceddimerbindingenergiesfortheinitialandfinalstates,
0
wefindthattheobservedresonancesdonotcorrespondtotransitionsbetweenconfinement-induced
2
dimers. The spectrum appears to be well-described by transitions between noninteracting polaron

n states in two dimensions.
a
J PACSnumbers: 03.75.Ss

7
1
Quantum degenerate atomic Fermi gases, with mag- be energetically more favorable than the corresponding

] netically controlled interactions, are ideally suited for dimers [12–14].
s
a exploring pairing interactions in reduced dimensions [1– Prior experiments have been performed with a dilute
g 6]. Near a broad collisional (Feshbach) resonance, a gas, where the local transverse Fermi energy µ is small
- two-component gas in three dimensions can be continu- comparedto the binding energy E of a single dimer. In
t b
n ouslytunedfromaBardeen-Cooper-Schrieffer(BCS)su- this case,the dimersaresmallcomparedto the interpar-
a perfluid, which exhibits weakly bound Cooper pairs, to ticle spacing and one expects the measured rf spectra to
u
a resonant strongly interacting superfluid and finally to be consistent with predictions based on molecular bind-
q
. a Bose-Einstein condensate (BEC) of molecular dimers. ing in the BEC regime and confinement-induced dimers
t
a Since 2002, degenerate strongly interacting Fermi gases in the BCS regime, as observedpreviously by measuring
m have been studied in three-dimensional geometries, pro- thethresholdforfree-to-boundtransitions[5]andbound-
- vidingaparadigmforstronglyinteractingsystemsinna- to-free transitions [6].
d
ture, from high temperature superconductors to nuclear
n Our experiments are performed instead in the many-
matter [7–10]. In contrast to free space, where bound
o body limit, where the the transverse chemical potential
c dimers exist only in the BEC regime, two-dimensional µ > E over most of the trap, but µ 1.5hν is small
b z
[ (2D) confinement also stabilizes bound dimers in the ≃
enough to be in the quasi-two-dimensional regime [3].
BCS regime [1, 11]. The interplay between confinement-
1 For a simple BCS approximation in two dimensions, the
v induced pairing and many-body physics in 2D confined trap-averaged rf transition rate to excite an atom from
0 mesoscopic systems of several hundred atoms has not
one populated state in a 50-50 mixture to an unpopu-
6 been previously explored and offers new challenges for lated noninteracting final state is ∞d2x ∆2θ[¯hω+
5 predictions [1, 3, 4]. ∝ 0 ⊥| |
3 µ µ2+ ∆2]/ω2, where ω is theRradio frequency rel-
− | |
1. We study a quasi-two dimensional Fermi gas of 6Li in ativepto the bare atomic transition frequency and ∆ is
0 the many-body regime. Radio-frequency (rf) spectra are the pairing gap for the initial mixture. Here, we assume
2 obtained for a 50-50 mixture of the two lowest hyper- for simplicity that the temperature T << µ over most
1 fine states denoted 1 and 2, by driving transitions to an of the trap, so that the number of excitations is neg-
:
v initially empty hyperfine state 3 and measuring the de- ligible. For µ >> Eb, predictions for a 2D gas [1] give
Xi pletionofstate2. At720G,well-belowtheFeshbachres- ∆=√2µEb inameanfield(BCS)approximation,yield-
onance, the molecular 12 dimer binding energy is larger ingathresholdforthe bound(Cooperpair)-to-freespec-
ar than the local Fermi energy, and the observed spectra trum at h¯ω ∆2/(2µ) = Eb, which is just the dimer
≃
exhibit the expected threshold form, arising from disso- binding energy, as noted previously [6]. Hence, the pre-
ciation of 12 dimers into 13 scattering (“free”) states. dicted spectra are identical with the dilute gas limit, in
However,near the Feshbachresonance at 834 G the pre- contrast to our data taken in the same regime.
dicted spectra for transitions between dimer states are Table I lists the experimental parameters for sev-
in marked disagreement with the measurements, where eral different magnetic fields and trap depths. The
the transverse Fermi energy is larger than the 2D dimer CO laser standing-wave trapping potential is charac-
2
binding energy. Here, we find that the resonance lo- terized by using parametric resonance in the weakly in-
cations appear to be described by transitions between teracting regime near 300 G to determine the oscilla-
noninteracting polaron states, which describe an impu- tion frequencies of the atoms in the transverse direc-
rity atom in state 2 or in state 3, immersed in a bath of tions (ν ,ν ) and in the tightly confined axial direc-
x y
atomsinstate1. Inthisregime,polaronsareexpectedto tion (ν ). The axial trapping potential is taken to be
z
2
B(G) U0(µK) νz(kHz) Nsite EF⊥(µK) Eb12 Eb13 ǫbb tion fraction ǫbb = 0.27, we expect bound-to-free tran-
719 27.5 26.0 1298 1.87 145 2.9 0.27 sitions to dominate the resonance spectrum [15]. The
809 23.5 24.0 1951 2.12 12.6 0.88 0.53
810 294 85 1134 5.73 33.1 6.69 0.79 30
832 24.4 24.5 1620 1.97 7.25 0.81 0.66 3 0)
1
832 274 82.0 1500 6.36 23.9 5.79 0.83 2 (
e
832 742 135 1800 11.47 39.1 12.45 0.89 at
st 25
831 1304 179 1250 12.65 51.9 18.9 0.91 n
r i
842 24.4 24.5 1420 1.85 5.91 0.78 0.70 be
m
842 277 82.5 1617 6.64 21.4 5.66 0.85 u
m n 20
TABLE I: Parameters for the radiofrequency spectra: Trap Ato
depthU0;Axialfrequencyνz;Totalnumberofatomsperpan-
caketrapNsite;TransverseFermienergyEF⊥ =hν⊥√Nsite;
E12 and E13 are the dimer binding energies in kHz, for -50 0 50 100 150 200 250 300
b b
ν⊥/νz = 1/25. The dimer binding energies are obtained us- Frequency (kHz)
ingtheGreen’sfunctionmethoddescribedinthesupplemen-
FIG. 1: RF spectra in a 1-2 mixture for a 2 3 transitions
tary material [15]. ǫbb is the corresponding bound dimer to inaquasi-two-dimensional6LiFermigasnear→720G.Theleft
bound dimer transition fraction (area under the bound to
resonanceoccursatthebareatomictransition frequencyand
bound spectrum) determined from the binding energies ob-
thethresholdresonancespectrumontherightisinverygood
tained with thescattering lengths measured in Ref. [16].
agreement with predictions for molecular dimers.
U = U sin2(2πz/λ), where the trap depth is read- resonance near 150 kHz is well fit by a threshold func-
axial 0
tionforaquasi-two-dimensionalgasusingthe calculated
ily determined from ν in the harmonic approximation,
z
12 and 13 dimer binding energies [15],
U0 = m(νzλ)2/2. We find ν⊥ √νxνy = νz/25. The
≡
number of atoms per site is estimated using the lattice ǫ12ν q2θ(ν ǫ12ν )
spacingof5.3µmandthenumberofatomsinthecentral Ibf(ν)= bν2z − b z2 , (2)
part (along z) of the cloud, as measured by absorption hq−ln(cid:16)ǫ1b2ννz −1(cid:17)i +π2
imaging.
To compare the measured spectra to predictions, we where ν is the rf frequency in Hz, relative to the bare
begin by determining the 2D dimer binding energies, atomic transition frequency, and Eb12 = ǫ1b2hνz is the 12
Einbijth≡e sǫuibjpphlνezmfeonrtaartyommsatinerisatlat[1e5s]iaannddgijv,enasindeTsacbrilbeeId. 1dimeǫrbb(bqi)n,dainsgitesnheoruglyd.in Hz. Note that R dνIbf(ν) =
−
Thecalculationincludesthefinitetransverseconfinement Next, we examine spectra for the 12 mixture near the
ofthetrappingpotential,whereν /ν =1/25,whichsig- Feshbach resonance, at 832 G, Fig. 2, at 810 G, Fig. 3,
⊥ z
nificantly increases the dimer binding energy, especially and at 842 G, Fig. 4. For several of the spectra, the
for weakly bound dimers. bare-atomresonancepeak on the left side exhibits a fast
The contribution to the spectrum from a dimer-to- riseandatailtowardhigherfrequency,whichweassume
dimer transition is determined by computing the corre- arises from a density-dependent mean field shift [17]. At
sponding fraction ǫ . In the weak binding approxima- 832 G, the sharp threshold remains at nearly the same
bb
tion, including only the axial ground state part of the frequency (which we have set to 0 in Fig. 2), even when
dimer wavefunction, we obtain, the trapdepthis increasedto maximum,wherethe ideal
gas Fermi energy at the trap center is E = 12.7µK,
F⊥
q2 i.e., 260 kHz. We assume that the threshold loca-
Z dνIbb(ν)≡ǫbb(q)= 4sinh2(q/2), (1) tion ≃is determined by the lowest density region at the
cloudedges,wherethe meanfieldshiftis negligible. The
where q ln(ǫ13/ǫ12) for a 2 3 transition in a 12 observedbare-atom resonance frequency is in agreement
≡ b b →
mixture. We plot ǫ as a function of magnetic field in with that obtained at high temperature as well as for
bb
the supplementary material [15]. There we show that 2 3 transitions with all the atoms initially in the 2
→
similar results for ǫ are obtained including the first 50 state.
bb
evenaxialstates,whichdeterminesǫ forbothweakand Near834G,for the conditions ofour experiments,the
bb
tight binding of the 1 2 or 1 3 dimers. bound-to-bound transition fractions in Table I are close
− −
Fig.1showsthemeasuredspectraat720G,wellbelow to unity [15] so that bound-to-bound transitions should
theFeshbachresonance,wherethemoleculardimerbind- dominate the spectrum abovethe bare atomic resonance
ing energy is larger than the local Fermi energy. At this frequency. Todeterminethefrequencyshift∆ν between
field and trap depth, where the bound-to-bound transi- the bare atom resonance and the second resonance peak
3
40
30
30
20 (a) 3 0)20
1
2 (
3 (10) 2350 n state 10 (a)
n state 2 1250 (b) number i20
er i om 15
mb 50 At
u
m n 40
10
Ato 30 (b)
(c)
20 -10 0 10 20 30 40 50 60
Frequency (kHz)
30
FIG. 3: RF spectra for 2 3 transitions in a quasi-two-
25 dimensional 6Li Fermi gas a→t 810 G: a) U0 = 23.5µK, νz =
20 24.0kHz;b)U0 =294µK,νz =85.0kHz. Verticallinesshow
15 (d) the measured bare atomic resonance position (solid) and the
predictedfrequenciesforconfinement-induceddimers: bound-
-20 -10 0 10 20 30 40 50 60 70 80 to-boundtransitionresonancehν =Eb12 Eb13 (dashed)and
−
Frequency (kHz) thethresholdforthebound-to-freetransitionhν =Eb12 (dot-
ted).
FIG. 2: RF spectra for a 1-2 mixture (12 13 transi-
→
tion) near the Feshbach resonance at 834 G, versus trap
depth U0. The atom number remaining in state 2 is shown
versus rf frequency: a) U0 = 21µK, νz = 24.5 kHz; b) determinedbytheoverlapoftheinitialandfinalpolaron
U0 = 280µK, νz = 82.5 kHz; c) U0 = 742µK, νz = 135 momentum space wavefunctions [15]. For experiments
kHz;d)U0 =1304µK,νz =179 kHz. Verticallinesshow the at 832 G, we find that the individual polaron quasipar-
measuredbareatomicresonanceposition(solid)andthepre- ticle weights are Z > 0.8 and Z > 0.9 [15], assuming
2 3
dictedfrequenciesforconfinement-induceddimers: bound-to-
that the local density does not change in the transition.
boundtransitionresonancehν =E12 E13 (dashed)andthe
thresholdforthebound-to-freetranbsit−ionbhν =E12 (dotted). Therefore, we expect strong overlap between the initial
b and final polaron states, so that transitions between po-
According to Table I, the bound-to-free transition (dotted
line) should makea negligible contribution to thespectrum. laron states should make an important contribution to
the spectrum.
We determine the 2D polaron energies for an isolated
to the right of it, we fit a threshold line shape to the impurity atom in state 3 or state 2 in a bath of atoms
atomic peak, which is convolved with a narrow Lorenz- in state 1, using the method described in the supple-
tian. Inthiscase,thepeakpositionisshiftedtotheright mentary material [15]. The method is based on that de-
ofthe bareatomthresholdlocation,whichisdetermined scribed for a 3D gas in the supplementary material of
fromthefit. AsecondLorentzian,withalargerwidth,is Schirotzek at al. [12], which utilizes the zero momentum
fit to the shifted resonance peak. The results for ∆ν are polaron wavefunction proposed by Chevy [18]. We ex-
given in Table II and do not agree with the predictions tend that method to the 2D problem, by renormalizing
for dimer-to-dimer transitions, h∆ν = E12 E13, the interactionstrengthas in Refs. [1, 13, 14]. Using the
dimer b − b
shown as dashed lines in Figs. 2-4. calculated dimer binding energies to determine the po-
We consider the possibility that polarons and not laronenergies,we find thatfor anatomin state 3 at 832
dimers determine the primary spectral features, so that G, the energy for an isolated polaron is attractive and
the difference between the initial and final state polaron morethanhalfoftheFermienergy,muchlargerthanthe
energies determines the observed frequency shifts. We corresponding dimer binding energy. For an impurity in
assume that the coherent part of the spectrum is ap- state 2, for which the 12 scattering length diverges, the
proximatelygivenbyZδ[¯hω E (3,1)+E (2,1)],where polaron energy is attractive and somewhat larger than
p p
−
E (i,1) is the polaron energy for an impurity atom in theFermienergy. Inthiscase,the polaronislocalizedto
p
statei=2,3immersedinabathofstate1. Sincethemo- approximatelythe interparticle spacing, but is not small
mentumdoesnotchangeintherftransitionZ Z Z is compared to the interparticle spacing, as it is at 720 G.
2 3
≃
4
B(G) νz(kHz) ∆νmeas(kHz) ∆νpolaron(kHz)
809 24.0 18.7 18.3
25
810 85 37.1 37.0
3 0)20 842 24.5 10.1 9.7
1
2 (15 842 82.5 27.2 26.7
n state 10 (a) 883322 2842..50 1228..33 1219..61
r i 832 135 38.8 42.8
e
b
m40 831 179 44.5 48.3
u
n
m
Ato30 aTnAdBtLhEesIeIc:oFnrdeqrueseonncaynscheifpte∆akνfobretaw1e2enmtihxetubrearneeaatrotmhepFeeask-
hbach resonance at different trap depths. The corresponding
20 axial trap frequency is νz. The measured values of ∆ν are
comparedtothevaluescalculatedassumingatransitionfrom
(b) apolaroninstate2toapolaroninstate3,inabathofatoms
10
in state 1. For determination of the polaron frequency shift,
-10 0 10 20 30 40 50 60
the dimer binding energies are taken from Table I and the
Frequency (kHz) localFermienergyistakentobeEF =0.67EF⊥,whereEF⊥
is theideal gas global Fermienergy given in Table I.
FIG. 4: RF spectra for 2 3 transitions in a quasi-two-
dimensional6LiFermigasne→ar842G:a)U0 =24.4µK,νz =
25.0kHz;b)U0 =277µK,νz =82.5kHz. Verticallinesshow
the measured bare atomic resonance position (solid) and the
In summary, we have observed rf spectra in a dense
predictedfrequenciesforconfinement-induceddimers: bound-
quasi-2D Fermi gas regime near a Feshbach resonance,
to-boundtransitionresonancehν =Eb12 Eb13 (dashed)and
− wherethedimerbinding energiesforboththe initialand
thethresholdforthebound-to-freetransitionhν =Eb12(dot-
ted). final states are smaller than the transverse Fermi en-
ergy. The spectra appear to be explained by transitions
between noninteracting polaronic states, despite the 50-
In calculating the polaron frequency shift, we assume 50 mixture employed in the experiments. These results
support the conjecture [12] that at low temperature, a
the peak position is determined by a local Fermi energy
strongly interacting Fermi gas in a balanced mixture of
E = λ E , where E is the ideal gas global Fermi
F 1 F⊥ F⊥
two spin states may be described as a gas of polaron
energy given in Table I. Fixing λ =0.67, we obtain the
1
pairs.
polaronfrequencyshiftsh∆ν E (3,1) E (2,1),
polaron p p
≡ −
which are compared to the measurements in Table II. This research is supported by the Physics divisions of
For the data just below and just above the resonance, the Army Research Office, the Air Force Office Office of
at 810 and 842 G, Table II, we find excellent agreement. Sponsored Research, and the National Science Founda-
Further,at832G,thecalculatedpolaronfrequencyshifts tion,andtheDivisionofMaterialsScienceandEngineer-
agree very well with the measured frequency shifts at ing,theOfficeofBasicEnergySciences,OfficeofScience,
all four trap depths, Table II, demonstrating nontrivial U.S.DepartmentofEnergy. Theauthorsareindebtedto
scaling with the trap depth and axial frequency. ThomasScha¨ferandLubosMitas forstimulating discus-
We have also obtained spectra for a 50-50 mixture of sions.
states 1 and 3 near the Feshbach resonance at 690 G,
where we drive either the 13 12 transition or the
→
13 23 transition. In both cases, the binding energy of
→
the final state 12 or 23 dimers is large compared to the
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