Zero-Temperature Theory of Collisionless Rapid Adiabatic Passage from a Fermi Degenerate Gas of Atoms to a Bose-Einstein Condensate of Molecules

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by  Matt Mackie

Zero-Temperature Theory of Collisionless Rapid Adiabatic Passage from a
Fermi Degenerate Gas of Atoms to a Bose-Einstein Condensate of Molecules
Matt Mackie1 and Olavi Dannenberg2
1QUANTOP–Danish National Research Foundation Center for Quantum Optics,
Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark
2Helsinki Institute of Physics, PL 64, FIN-00014 Helsingin yliopisto, Finland
(Dated: February 2, 2008)
5
0 Wetheoreticallyexamineazero-temperaturesystemofFermidegenerateatomscoupledtobosonic
0 molecules via collisionless rapid adiabatic passage across a Feshbach resonance, focusing on satu-
2 ration of the molecular conversion efficiency at the slowest magnetic-field sweep rates. Borrowing
n a novel many-fermion Fock-state theory, we find that a proper model of the magnetic-field sweep
a cansystematically removesaturation. Wealsodebunkthecommonmisconceptionthatmany-body
J effects are responsible for molecules existing above thetwo-body threshold.

7
PACSnumbers: 03.75.Ss

]
h
p Introduction.–Magnetoassociation creates a molecule 1.0
- from a pair of colliding atoms when one of the atoms
m
spin flips in the presence of a magnetic field tuned near 0.8
o a Feshbach resonance [1]. Recently, ultracold [2, 3] and
t condensate [4] molecules have been created via magne- 0.6
a 2
. toassociation of a Fermi gas of atoms, in the course of |β
cs efforts to create superfluid Cooper-paired atoms [5, 6] | 0.4
i (seealsoRefs.[7]). Thebackboneoftheseexperimentsis
s
y rapidadiabaticpassage: thegroundstateoftheFeshbach 0.2
h system is all atoms far above the molecular-dissociation
p threshold and all molecules far below it, so that a slow 0.0
2 [ scwoneveperotsftahteommsaginnteotidcifiaetoldmfircommoolneceuelxetsr.emetotheother 0.0 0.2 0.k4T/–hε0.6 0.8 1.0
B F
v Finite-temperature mean-field theory of magnetoasso-
8 ciationofaFermigasofatomsleadstotwotypesofinsta-
04 bilities against molecule formation. One is the thermo- F(|IβG|2.)1o:fPcroeldliisciotendlestsemrappeirdataudreiadbeaptiecndpeansscaegfeorfrtohme eqffiuacnietnucmy
2 dynamic instability of a Fermi sea against the formation degenerate40Katomsto40K2molecules. Asthetemperature
1 ofCooperpairs[8],atraitofsuperconductorswhoseana- inFermiunits(~εF/kB)islowered,atomsbecomemorelikely
4 log is passed on to Feshbach-resonant superfluids [9]. A tobeaffectedbythedynamicalstabilitythatformsmolecules.
0 thermodynamical instability occurs because pairing low- The magnetic field was swept linearly at the (inverse) rate
/ ers the energy, and coupling to a reservoir with a low 1/B˙ =400µs/G. Figure reproduced from Ref. [10].
s
c enough temperature leaves the system prone to pairing.
i The other is a dynamical instability, whereby the larger
s
y state space of the molecules, owing somewhat to Pauli
tures of atoms into Fermi molecules [12]. Unfortunately,
h blocking,leavesthe atomsproneto spontaneousassocia-
p tion[10]. Therolethattemperatureplaysinthisprocess computing power is presently sufficient for calculations
v: is an open question experimentally, as well as a matter with only about 20 atoms total at best, precluding any
brute-force[12]testofsaturation. Hereweapplyanovel
i of theoretical contention.
X large-fermion-number theory [13] to demonstrate near-
Physically[10],hightemperaturelessensthe chanceof
r anatomoccupyinganarbitrarylevelintheFermisea,the unit-efficient collisionless rapid adiabatic passage in the
a
limit of zero temperature, thereby ruling out any funda-
dynamical instability becomes less effective and the effi-
mentalceilingtothemolecularconversion,andbolstering
ciency of even the slowestrapid adiabatic passage there-
Ref. [10] (and also Ref. [14]).
fore saturates (c.f., Fig. 1). The mean-field theory be-
hind this understanding agrees semi-quantitatively with This development is outlined as follows. After briefly
experiments [2]; nevertheless, a recent zero-temperature introducing the collisionless model, we focus on rapid
Landau-Zener theory predicts that saturation is funda- adiabatic passage and confirm the reduced-space map-
mental to the collisionless regime [11]. If temperature is ping [13] by comparison with exact few-particle results.
not a limiting factor, then any zero-temperature model Increasing the total particle number to 2 102, we then
×
of collisionless rapidadiabatic passageshould ultimately observe what appears to be saturation at about 50%.
∼
display saturation, e.g., a Fock-state approachsimilar to However,includingfluctuationeffectsintherateatwhich
the theory ofcooperativeassociationofBose-Fermimix- the system is swept across the Feshbach resonance, we
2
find that saturation can be systematically removed, and Hamiltonian (2) yields [13]
near-unitefficiencycanbeachievedforanyparticlenum-
ber. Lastly, from the single pair results we also debunk iC˙ = [N m]δC
m m
−
thecommonlyheldnotionthatmany-bodyeffectsarere-
+κ √N m+1Dm−1C
sponsiblefortheexistenceofmoleculesabovethethresh- h − m m−1
old for molecular dissociation.
+√N mDm+1C . (4)
Collisionless Gas Model.–We model an ideal two- − m m+1i
component gas of fermionic atoms coupled by a Fesh-
Here C (t) C (t) is a column vector of
bach resonance to bosonic molecules. In the language m ≡ N−m,n1,...,nN
all the amplitudes corresponding to the (N) possible
of second-quantization, an atom of mass m and momen- m
tum ~k is described by the annihilation operator ak,1(2), faerrrmaniognemmeondtseso,fanmd DatJomis apnair(sNa)m(oNng) dthimeeNnsioanvaaillambale-
and a molecule of mass 2m and similar momentum is I I × J
trixthatcontainsonlyunitandzeroelementsdetermined
described by the annihilation operator bk. All operators by C and C . The problem with the system (4) is that
obeytheir(anti)commutationrelations. Themicroscopic I J
there are 2N amplitudes, which limits most numerical
Hamiltonian for such a freely-ideal system is written
experiments in rapid adiabatic passage to about N =10
H (see alsoRefs. [12]); however,by multiplying Eqs.(4) by
~ = Xh(ǫk−µ)a†k,σak,σ+(12ǫk+δ−µmol)b†kbki theappropriatecolumnvectorum,N,anyredundantam-
k plitudes can be eliminated [13]. The remaining N +1
+κXhb†k+k′ak,1ak′,2+H.c.i, (1) amplitudes evolve in time according to [13]
k,k′
iα˙ = [N m]δα
m m
−
where repeated greek indices imply a summation (σ = +κ √m(N m+1)αm−1
(cid:2) −
1,2). The free-particle energy is ~ǫk = ~2k2/2m, the +√m+1(N m)α , (5)
atom (molecule) chemical potential is ~µ , and the − m+1(cid:3)
σ(mol)
detuning δ is a measure of the binding energy of the where the sum of all (N) amplitudes with N m
molecule (δ > 0 is taken as above threshold), the mode- moleculesandmfreeatommpairsisdefinedas (N)α−
independent atom-molecule coupling is κ 1/√V with u C = C (with α noprmmalizemd t≡o
V is the quantization volume. ∝ m,N m Pnk N−m,n1,...,nN m
the number of permutations of m atoms in N states).
Wehavealreadyimposedtheidealconditionsforatom- Lastly we will need the molecular fraction β 2 =
molecule conversion with µ1 = µ2 = µ. An appropriate 2 b†b /(2N)=(1/N) N (N m)α 2. | |
unitary transformation then shuffles µ into the defini- h i Pm=0 − | m|
Rapid Adiabatic Passage.–Putting fluctuations [17]
tion of µ which, in turn, can be absorbed into the
mol momentarily aside, the relevant frequency scale is Ω =
detuning and written off as an effectively dc bias (see √Nκ √ρ [12, 16], where √ρ is the so-called collec-
alsoRefs.[12]). Since magnetoassociationusuallyoccurs ∝
tiveenhancementfactor. “Adiabatic”isthereforedefined
muchfasterthananytrapped-particlemotion,anexplicit
qualitatively as the detuning changing by an amount Ω
trap can be neglected along with the free-particle ener- inatime1/Ω,or δ˙ .Ω2. Modelingthetime dependent
gies ǫk [15]. For the sake of simplicity, and to compare detuningasδ = |ξ|Ω2t,sweepswithξ 1shouldqualify
withRef.[10],weneglectallmolecularmodesexceptthe − ∼
as adiabatic. Off hand, Fig. 2(a) confirms this intuition
k+k′ =0 mode, b b, so that
0 ≡ for N = 4; also, noting that the full results are shifted
for clarity,the reducedsystem(5) indeed reproducesthe
H
~ = δb†b+κX(cid:2)b†ak,1a−k,2+H.c.(cid:3). (2) full system (4). Making a more full use of the reduced-
k space theory, Fig. 2(b) illustrates that the efficiency of
rapid adiabatic passage in fact decreases for increasing
Absent losses, the total particle number is conserved, particle number, saturating at about 50% for N = 102.
2hb†bi+Pkha†kσakσi = 2n+Pk,σnkσ = 2N, where n Nevertheless,if we accountforfluctuations, then the rel-
is the number of molecules, n = 0,1 is the number of evant frequency scale is Ω/lnN [17]. Now the detuning
k,σ
atoms per mode (k) per species (σ), and 2N is the total should change by Ω/lnN in a time (Ω/lnN)−1, sug-
number of atoms were all the molecules to dissociate. gesting the detuning-sweep model δ(t)= ξ(Ω/lnN)2t.
For a fixed number of particles equal to the number of Indeed, Fig. 2(c) shows that the N =102−and N =1 re-
fermion modes, the Fock-state wavefunction is [13] sultsagreenicely,andareabsentanyevidentsaturation.
We can also make a rough comparison with the zero-
N temperaturelimitinFig.1. Magneticfieldsareconverted
|ψ(t)i= X X CN−m′,n1,...,nN(t).|N −m′,n1,...,nNi into detunings according to δ = ∆µ(B −B0)/~, where
m′=0{nk} the difference in magnetic moments between the atom
(3) pair and a molecule is ∆ , and B is the magnetic-field
µ 0
The time dependence of the system is determined by position of resonance. For N = 105 atoms of 40K in a
the Schr¨odinger equation, i~∂ ψ = H ψ , so that the typical [2] trap the peak density ρ = 2 1013cm−3, so
t
| i | i ×
3
tually for a resonance (atom-molecule coupling) of arbi-
1
1 (a) 0.8 (b) trarystrength. However,the model(5) is broadlyequiv-
0.8 alent to the two-mode model in coherent association of
2 0.6
β|0.6 0.4 condensate [13], and it is well known that strong cou-
|0.4
0.2 0.2 pling can lead to dissociation to modes lying outside the
0 0 two-mode system [16, 19], so-called rogue dissociation.
10 0 -10 10 0 -10 Nevertheless, if the sweep is directed from above to be-
low threshold, then rogue dissociation is negligible and
1 1 the two-mode model is a good approximation. Hence,
0.8 (c) 0.8 (d) the aboveresults areexpectedly reasonableto describe a
2 0.6 0.6 sweep across an arbitrarily strong resonance.
β|0.4 0.4
| Beforeclosing,weturntoarelatedmatterofprinciple:
0.2 0.2
the nature of above-threshold molecules. Below thresh-
0 0
old (δ < 0), Fourier analysis delivers the binding en-
10 δ/0Ω -10 10 δ/0Ω -10 ergy, ~ωB <0, of the Bose-condensedmolecules [10, 19]:
ω δ Σ′(ω )+ iη = 0, where Σ′(ω ) is the finite
B B B
− −
self-energy of the Bose molecules and η = 0+. Tun-
FIG. 2: Molecular condensate fraction as a function of the
ing the system above the two-body threshold (δ > 0)
detuning in rapid adiabatic passage across a Feshbach res-
onance, beginning above threshold (δ > 0). (a) For N = 4, gives an imaginary ωB, and the bound state ceases to
comparisonofsolutiontothefullequationsofmotion[Eq.(4), exist; nevertheless,Fig.3 showsa largeN =1molecular
dashedline]with thesolution tothereduced-spaceequations fraction. This result is not really a surprise, since the
of motion [Eq. (5), solid line]. The full results are shifted fraction of molecules must vary continuously from zero
for clarity; the two calculations are otherwise indistinguish- to unity across threshold. We conclude that any theory
able. Thedetuningsweepmodelisδ(t)=−ξΩ2t,withξ =1. in which molecules abruptly cease to exist at threshold,
(b)Usingthesamesweepmodel,wefindapparentsaturation
while useful in their own right (e.g., for modeling bind-
for increasing particle number: N = 4 (solid line), N = 10
(dashed line), and N = 102 (dotted line). (c) A fluctuation- ing energies [10]), are not a good rule of thumb for pre-
adjusted sweep model, δ(t) = −ξ(Ω/lnN)2t, leads to near- dicting the existence of above-threshold molecules. Our
unitefficiency for N =102 (solid line), aswell as solid agree- interpretation is that, as usual in cooperative behavior,
ment with the N = 1 results. Here the dimensionless sweep a macroscopic number of particles respond as a unit to
rate is again unity, ξ = 1. (d) Results for N = 102 and a given external drive, thereby mimicking one- or two-
ξ=7.6,anestimateofazero-temperaturesweepfor40K(see body physics. At the least, this implies that many-body
text). effects are sufficient but not necessary for the existence
of above-thresholdmolecules. Moreover,we see in Fig. 3
that the above-threshold molecular fraction for δ/Ω . 2
that the coupling strength is Ω = 0.3 2π MHz [10];
×
the difference in magnetic moments is ∆ 0.19µ [10],
µ 0
≈
where µ is the Bohr magneton. The results of Fig. 1
0
are for 1/B˙ = 400µs/G [10], which corresponds to 0.5
ξ = (lnN)2∆ B˙/(~Ω2) 7.9 for N = 105 atoms per
µ
≈
species. Of course, even the reduced-space model [13] 0.4
cannot handle N = 105 atoms, but for ξ = 7.9 then
N =102 willactuallyunderestimatetheN =105 results. 0.3
HencethealreadygoodagreementbetweenFig.2(d)and 2
β|
Fig. 1 would actually improve if resourceswere available |0.2
to manage the correct number of particles.
We pause briefly to justify the ideal gas model. The
0.1
collisional interaction strength is roughly Λ=2π~ρa/m,
where a is the off-resonant atomic s-wave scattering
length. The40Kscatteringlengthisa=176a [20],with 0
0
a the Bohr radius. For a typical density ρ 1013cm−3,
0 ∼ 10 8 6 4 2 0
the collisional coupling strength, in units of the atom- δ/Ω
molecule coupling, is Λ/Ω 10−3. Collisions should
| | ≈
therefore be broadly negligible. In particular, a system FIG.3: Fractionofabove-thresholdmolecularcondensatefor
of Fermi atoms coupled to Bose molecules is formally N = 100 (from Fig. 2(c), dashed) compared to ground-state
identical to a system of only bosons [13], and collisions N =1(solidline). Evidently,above-thresholdmoleculesexist
are negligible for bosons under such conditions [18]. in theabsence of many-bodyeffectsand, oddly enough, such
Also, it should be noted that, because we have cho- effects can act to suppress the molecular fraction for positive
detuningsδ/Ω&2.
sen Ω as the frequency scale, the above results are ac-
4
is actually suppressed for the many-body case N =102. ouspathwayscouplingthestateshavingN mmolecules
−
Of course, the idea of many-body stabilization of andm dissociatedatom pairswith the states having one
above-thresholdmoleculesgenerallyarisesinthe context more (less) molecules and one less (more) dissociated
ofequilibriumthermodynamics,whereasthecollisionless pair, and the timescale for N-particle interference turns
model describes non-equilibrium processes. The many- outtobe lnN/Ω[13]. Itthenmakesperfectsensethat
∼
body suppression of the above-threshold molecular frac- cooperative (near-unit-efficient and macroscopic) rapid
tion may or may not carry over to the collisional regime adiabatic passage will only occur over a timescale that
(although we find elsewhere that, to a certain degree, it is commensurate with constructive interference. Next
may [21]). However, we expect that the two-body equi- we saw that our zero-temperature model agrees semi-
libration time is sufficiently long compared to the atom- quantitativelywithour mean-fieldmodel [10],indicating
molecule conversiontimescale that, even in the presence thattemperature[10]andpaircorrelations[14]are–asof
ofcollisions,atwo-bodysystemcanalwaysbeconsidered yet–the main obstacles to collisionless cooperative con-
out of equilibrium; hence, the two-body ground state of version to molecules with near-unit efficiency. Finally,
Fig. 3 should apply to the collisional regime as well. whereasstudiesofFeshbachresonancesforbothfermions
Conclusions.–We have investigated saturation in col- andbosonshaveimplicatedmany-bodyeffects inthe ex-
lisionless rapid adiabatic passage from a two-component istenceofmoleculesabovethetwo-bodythresholdfordis-
degenerate Fermi gas to a Bose-Einstein condensate of sociation,wefindthatitisnotnecessarytoinvokemany-
molecules. Saturation indeed arises, but can be system- body effects to explain the existence of above-threshold
aticallyeliminatedbyintroducingthetimescaleappropri- molecules.
ate to cooperative interference effects. Physically, coop- Acknowledgements.–One of us (O.D.) kindly thanks
erativeinterferenceeffects arisefromaddingup the vari- the Magnus Ehrnrooth Foundation for support.
[1] W. C. Stwalley, Phys. Rev. Lett. 37, 1628 (1976); E. Phys. Rev. Lett. 87, 120406 (2001); Y. Ohashi and A.
Tiesinga, B. J. Verhaar, and H. T. C. Stoof, Phys. Rev. Griffin, Phys.Rev.Lett. 89, 130402 (2002).
A 47, 4114 (1993). [10] J.Javanainenet al.,Phys.Rev.Lett.92,200402 (2004).
[2] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Na- [11] J.Chweden´czuk,K.G´oral,T.K¨ohler,andP.S.Julienne,
ture(London) 424, 47 (2003). e-print cond-mat/0409192.
[3] K. E. Strecker, G. B. Partridge, and R. G. Hulet, Phys. [12] O. Dannenberg, M. Mackie and K.-A. Suominen, Phys.
Rev. Lett. 91, 080406 (2003); J. Cubizolles et al., Phys. Rev. Lett. 91, 210404 (2003); M. Mackie et al., Phys.
Rev.Lett. 91, 240401 (2003). Rev. A 69 053614 (2004).
[4] M. Greiner, C. A. Regal, and D. S. Jin, Nature (Lon- [13] I.TikhonenkovandA.Vardi,e-printcond-mat/0407424.
don) 426, 537 (2003); S. Jochim et al., Science 302, [14] E. Pazy, A. Vardi, Y. B. Band, Phys. Rev. Lett. 93,
2101 (2003); M. W. Zwierlein et al., Phys. Rev. Lett. 120409 (2004).
91, 250401 (2003). [15] Thismeanswewillbeabletofullyaccountfortheatoms
[5] C.A.Regal, M.Greiner,andD.S.Jin,Phys.Rev.Lett. being fermions without, per say, stacking them in trap
92, 040403 (2004). levels, and that densities are understood to beuniform.
[6] M. W. Zwierlein et al., Phys. Rev. Lett. 92, 120403 [16] J. Javanainen and M. Mackie, Phys. Rev. A 59, R3186
(2004). (1999).
[7] C.Chinet al.,Science305,1128(2004);J.Kinastet al., [17] A. Vardi, V. A. Yurovsky, and J. R. Anglin, Phys. Rev.
Phys. Rev. Lett. 92, 150402 (2004); T. Bourdel et al., A 64, 063611 (2001).
Phys.Rev.Lett. 93, 050401 (2004). [18] A. Ishkhanyanet al.,Phys. Rev.A 69, 043612 (2004).
[8] M.Tinkham, Intro. to Superconductivity, (McGraw-Hill, [19] M. Mackie, K.-A. Suominen, and J. Javanainen, Phys.
NewYork, 1975). Rev. Lett.89, 180403 (2002).
[9] E. Timmermans, K. Furuya, P. W. Milonni and A. K. [20] J. L. Bohn, Phys.Rev.A 61, 053409 (2000).
Kerman, Phys. Lett. A 285, 228 (2001); M. Holland, S. [21] M. Mackie and J. Piilo (unpublished).
J. J. M. F. Kokkelmans, M. L. Chiofalo, and R. Walser,