Laser probing of atomic Cooper pairs

4 Pages·0.16 MB·

by  P. Torma

Laser probing of atomic Cooper pairs
P. T¨orm¨a1,2,3 and P. Zoller1
1Institute for Theoretical Physics, University of Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria
2Helsinki Institute of Physics, P.O.Box 9, FIN-00014 University of Helsinki, Finland
0 3Laboratory of Computational Engineering, P.O.Box 9400, FIN-02015 Helsinki University of Technology, Finland
0
We consider a gas of attractively interacting cold Fermionic atoms which are manipulated by
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laser light. The laser induces a transition from an internal state with large negative scattering
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length to one with almost no interactions. The process can be viewed as a tunneling of atomic
n
population between thesuperconductingandthenormal statesof thegas. Itcan beused todetect
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the BCS-ground state and tomeasure thesuperconducting orderparameter.
J

4 05.30.Fk, 32.80.-t, 74.25.Gz
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] Successful cooling of trapped Fermionic 40K atoms smallchangeinitis asignificantrelativeeffect. Further-
h
c down to temperatures where the Fermi degeneracy sets more, our work offers a basic starting point for describ-
e in was reported recently [1,2]. This breakthrough opens ing any phenomenon related to the superconducting–
m
upnewopportunitiesforstudyingfundamentalquantum normal interface. For instance, our method can be un-
- statistical and many-body physics. A major advantage derstood as an outcoupler. With small modifications,
t
a of Fermionic atoms compared to electrons in condensed one could also describe outcoupling of pairs instead of
t
s matter is the richness of their internal energy structure single atoms which would create an atomic beam with
t. and the possibility to accurately and efficiently manip- BCS-type statistics.
a
ulate these energy states by laser light and magnetic
m
fields. Furthermore, atomic gases are dilute and weakly a) b)
d- interactingthusofferingtheidealtoolfordevelopingand − ¢G ¹g
n experimentally testing theories of many-body quantum
¢
o physics. A major goal is to observe the predicted [3,4] jei
c BSC-transitionforFermionicatoms–thiswouldcompare jgi jg0i ¹e
[
to the experimental realization of atomic Bose-Einstein
FIG.1. Probingofthegapinagasofattractivelyinteract-
1 condensates [5]. It is still, however, an open question
ing cold Fermionic atoms. a) Laser excitation with the cou-
v
how to observe the BCS-transition, because the value of
pling Ω and the detuning ∆ transfers a Cooper paired atom
1
4 the superconducting order parameter (gap) is expected fromtheinternalstate|gitothestate|ei. b)Theotheratom
3 to be small andelectro-magneticphenomena suchas the in the initial Cooper pair becomes an excitation in the BCS
1 Meissner effect do not take place. state, therefore the laser has to provide also the additional
0 gapenergy∆G. InthispicturetheFermilevelsµg andµe for
0 We propose a method to detect the existence of the theinternalstateshavebeenchosentobedifferentfromeach
0 BCS-ground state and to measure the gap using laser other butthey could also equal.
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t light. The main difference in our method compared to
a
the proposals of using off-resonant light scattering [6,7]
m
is that the light is resonant, that is, population is trans- Weconsideratomswiththreeinternalstatesavailable,
- ferred from one internal state to another. Furthermore, say |ei, |gi, and |g′i. They are chosen so that the inter-
d
n the final state ofthis process is chosento have negligible action between atoms in states |gi and |g′i is relatively
o scattering length compared to the initial one: this effec- strong and attractive, and all other interactions are neg-
c tively createsa superconducting– normalstate interface ligible. The laser frequency is chosen to transfer popu-
:
v across which the atomic population can move. There is lation between |ei and |gi, but is not in resonance with
i aconceptualanalogyto electrontunneling fromasuper- any transition which could move population away from
X
conducting metalto anormalone, althoughthe physical the state |g′i. If there is more or less equal amount of
r
a systems and their describtions differ in an essential way. atoms in states |gi and |g′i they can form the supercon-
Also non-optical phenomena, such as collective and sin- ductingBCSgroundstate;atomsin|eiareinthenormal
gle particle excitations, have been proposed to be used state. For small intensities, the laser-interaction can be
for observing the BCS-transition [8–10]. The specific treated as a perturbation, the unperturbed states being
feature of our method with respect to other suggested the normal and the superconducting state. The transfer
probes, both optical and mechanical (modulating the of atoms from |gi to |ei is then analogous to tunneling
trap frequency), is the utilization of the superconduct- ofelectronsfromanormalmetaltoasuperconductorin-
ing – normal state interface. The advantage is that the duced by an external voltage, which can be used as a
normalstate populationcanbe initally zero,thus evena method to probe the gap and the density of states of
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the superconductor [11]. In our case the tunneling is be- T =0.
tweentwointernalstatesratherthantwospatialregions, Homogeneous case: The assumption of spatial homo-
this resembles the idea of internal Josephson-oscillations geneity is appropriate when the atoms are confined in
in two-component Bose-Einstein condensates [12]. Fig- a trap potential which changes very little compared to
ure 1 illustrates the basic idea. The observable carrying characteristic quantities of the system, such as the co-
essential information about the superconducting state is herence length and the size of the Cooper pairs. In the
the changein the populationofthe state|ei, wecallthis homogeneouscaseweexpandtheFermionfieldsψ into
e/g
the currentI. Below we calculatethe currentfirstin the plane waves. The Hamiltonian becomes
homogeneouscaseandthenassumingthattheatomsare
∆
citoiensfifnoerdeixnpaerhimaremntoanlircetarlaizpa.tiFoinnaolfltyhweiildledai,scaunsdscphoosossibinilg- H =He+Hgg′ + 2 Xk [cek†cek−cgk†cgk]
the states |ei, |gi, and |g′i for 6Li.
+ [T ce†cg+h.c.], (2)
We define a two-component Fermion field ψ(x) = kl k l
(ψ (x) ψ (x))T, where ψ and ψ fulfil standard Xkl
e g e g
Fermioniccommutationrelations. Thefieldsψe/g canbe where Tkl = V1 d3xΩ(x)eik·xe−il·x. We calculate the
expanded using some basis functions (e.g. plane waves current I = −hN˙Ri = −ih[H,N ]i treating H as a per-
e e T
or trap wave functions) and corresponding creation and 2
turbation: terms of higher order than H are neglected.
annihilation operators: ψe/g(x) = jcej/gφej/g(x). The Because we are interestedin the currentTbetweenthe su-
annihilation and creation operatorPs fulfil {ce†,cg} = 0 perconductingandnormalstates,correlationsoftheform
i j
and{cei/g†,cej/g}=δij. The two components ofthe field, hc†ec†ecgcgi (and h.c.) are omitted since they correspond
to tunneling of pairs (Josephson current). The current
corresponding to the internal states |ei and |gi, are cou-
can be written
pled by a laser. This can be either a direct excitation
or a Raman process; we denote the atomic energy level ∞
difference by ω (h¯ ≡1), the laser frequency ω and the I = dtθ(t)(e−i∆˜th[A†(0),A(t)]i−ei∆˜th[A(0),A†(t)]i)
a L Z
−∞
wavevectork –inthecaseofaRamanprocesstheseare
L
effective quantities. In the rotating wave approximation where A(t) = T cg†(t)ce(t), and ce/g(t) =
the Hamiltonian reads kl kl k l l
eiKtce/ge−iKt wherPe K = H −µ N −µ N . The two
l e e g g
H =He+Hgg′ + ∆ d3xψ†(x)σzψ(x) terms in the above equation have the form of retarded
2 Z andadvancedGreen’sfunctions. Theseareevaluatedus-
+Z d3xψ†(x)(cid:18) Ω∗0(x) Ω(0x) (cid:19)ψ(x). (1) itnogIM=atsubkalr|aTkGl|r2ee−n∞’∞sf2duπǫn[cntFio(nǫ)s−tecnhFn(iqǫu+es∆,˜w)]hAicgh(kl,eǫad+s
∆˜)A (l,ǫP), where Rn are the Fermi distribution func-
Here ∆ = ω −ω is the (effective) detuning, and Ω(x) e F
a L tions and A are the spectral functions for the su-
contains the spatial dependence of the laser field mul- g/e
perconducting and normal states. We use the standard
tiplied with the (effective) Rabi frequency. The parts expressions A (l,ǫ) = 2πδ(ǫ − ξ ) and A (k,ǫ + ∆˜) =
He and Hgg′ contain terms which depend only on ψe or 2π[u2δ(ǫ+∆˜−eω )+v2δ(ǫ+∆˜+ω )l][11]. Hegreξ =E −µ
ψg,ψg′,respectively. Possiblespatialinhomogeneity,e.g. and uk , v andkω arke given bykthe Bogoliubovl tranlsfore-
from the trap potential, is also included in He and Hgg′. mationk. Bkecausekweareinterestedinthetunnelingoutof
The observable carrying essential information about
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the superconductor, only the term proportional to v is
the superconducting state is the rate of change in pop- k
considered. Thelaserfieldischosentobearunningwave,
ulation of the |ei state. We may call it, after the
electron tunneling analogy, the current I(t) = −hN˙ i, that is Ω(x) =ΩeikL·x. The term |Tkl|2 now produces a
where N = d3xψ†(x)ψ (x). The current I(t) is calceu- delta-function enforcing momentum conservation. Note
e e e that this is very different from the assumption of a con-
lated consideRring the tunneling part of the Hamiltonian, stant transfer matrix ( |T |2 −→ |T|2 ) made in
HT = H −(He+Hgg′ +∆/2(Ne−Ng)) as a perturba- the standard calculatioPnkflor tkulnneling of Pelekcltrons over
tion; the current I becomes the first order response to
a superconductor – normal metal surface [11]. The final
the external perturbation caused by the laser. We cal-
result becomes (assuming T =0)
culate it both in the homogeneous case and in the case
othfehgarramnodncicancoonnificnalemenesnetm.bTleh,ethcearlecfuolraetitohnescahreemdicoanlepoin- I =−2πΩ2ρθ(−∆˜ −ω˜ −∆µ) ωk˜−kL −ξk˜−kL
tentials µg and µe are introduced. Also the detuning ∆ k−kL ωk˜−kL +ξk˜−kL 1− kk˜L
h i
acts like a difference in chemical potentials, thus it be-
comes useful to define an effective quantity of the form where k˜ is given by the following energy conservation
∆˜ =µe−µg+∆≡∆µ+∆. Inthe derivationweassume condition: −∆˜ +ωk˜−kL +ξk˜−kL = 0, ωk = ξk2+∆2G,
finitetemperature,buthereweonlyquotetheresultsfor and ρ is the density of states which appearps when the
2
summation over momenta is changed into an integration Here u (x), v (x) and ω are given by the Bogoliubov–
n n n
over energies. DeGennes equations [13]. We also use the fact that the
The laser momentum k can be very small compared trapwavefunctionsφ (x) arereal. The derivationgives
L m
to the momentum of the atoms, especially in the case of
a Raman process. By setting k =0 the result becomes
L
(choosing ∆˜ <0 i.e. current from |gi to |ei)
2
I =−2πΩ2ρθ(∆˜2−∆2G+2∆µ∆˜)∆∆˜2G2. (3) I =−2πXn,m(cid:12)(cid:12)(cid:12)Z d3xΩ(x)un(x)φm(x)(cid:12)(cid:12)(cid:12) [nF(ωn)−nF(ξm)]
(cid:12) (cid:12) 2
To understand the results in terms of physics, let δ(ξ +∆˜ −ω )+ d3xΩ(x)v∗(x)φ (x)
us first consider the case of equal chemical potentials m n (cid:12)Z n m (cid:12)
(cid:12) (cid:12)
∆µ=0: [nF(−ωn)−nF(ξm(cid:12)(cid:12))]δ(ξm+∆˜ +ωn). (cid:12)(cid:12) (8)
∆2
I =−2πΩ2ρθ(−∆−∆ ) G. (4)
G ∆2
In order to transfer one atom from the state |gi to |ei
the laser has to break a Cooper pair. The minimum en- Again, we consider only the term proportionalto v , as-
n
ergyrequiredforthisisthegapenergy∆ ,thereforethe sume zero temperature and use the known form for the
G
current does not flow before the laser detuning provides trap energy ξ =mΩ −µ to obtain the final result
m t e
this energy – this is expressed by the step function in
(4). As |∆| increases further, the current will decrease
quadratically. This is because the case |∆| = ∆ corre-
G 2
2π
lsaprognedrs|∆to|tmheeatnrsanlasfregrerofmpoamrteicnlteas,wanitdhtph=erepFar,ewshimerpealys I = Ωt Xn θ(−∆˜ +ωn)(cid:12)(cid:12)Z d3xΩ(x)vn∗(x)φM(x)(cid:12)(cid:12) ,
fewer Cooper-pairs away from the Fermi surface. This (cid:12) (cid:12)
(cid:12) (cid:12)
behaviour is very different from the electron tunneling
where the current grows as (eV)2−∆2 [11] (the volt-
G
age eV corresponds to the pdetuning ∆ in our case) be-
where the quantum number M = (µ − ∆˜ + ω )/Ω .
cause all momentum states are coupled to each other. e n t
Although the result is not as transparent as in the ho-
When the chemical potentials are not equal the situa-
mogeneous case, the characteristic treshold behaviour is
tion is more complicated, but the basic features are the
expressed by the step function. Since the functions in-
same: i) treshold for the onset of the current given ba-
side the absolute square are typically oscillating ones,
sically by the gap energy, and ii) further decay of the
this term would restrict the values of n in the summa-
current because the density of the states that can fulfil
tion. We have checked that with strong approximations
energy and momentum conservation decreases.
one obtains again the homogeneous case result.
Harmonic confinement: The trap wave functions pro-
vide the natural basis of expansion when the atoms are
Experimental realization: Typical experimental pa-
confined in a harmonic trap. We will not, however, do rameters in the case of 6Li are Ω /2π ∼ 150Hz for
thisexpansionfromthebeginningofthecalculationsbut t
the trap frequency and µ ∼ 100kHz for the chemical
ratherderivethecurrentandthecorresponding(position g
potential [14]. The order of magnitude of the gap is
dependent) Green’s functions directly for the Fermion
∆ /ǫ ∼ 0.1 (see e.g. [6]). In Figure 2 we show the
fields ψ . The current becomes I =2Im[X (−∆˜)], G F
e/g ret dependence of the current on the detuning in the homo-
∞ dǫ geneous case, for ∆ = 0.1µ . The sharp peak given by
X (−∆˜)= d3x d3x′Ω∗(x)Ω(x′) G g
ret Z 2π Z Z theresultofEq.(3)willbebroadenedinanexperimental
−∞
situation. One of the main causes of broadening is that
[A˜ (x,x′,ǫ)Gg (x′,x,ǫ+∆˜)
e adv it is difficult to fix the particle number with high accu-
+Geret(x,x′,ǫ−∆˜)A˜g(x′,x,ǫ)], (5) racy. We have simulated this by assuming that the par-
ticle number (µ ) varies from experiment to experiment
and A˜ are defined A˜ (x,x′,ǫ) = i(Ge/g(x,x′,ǫ)− g
e/g e/g ret according to a Gaussian distribution (the width of the
Gea/dgv(x,x′,ǫ)). In order to take the imaginary part of Gaussian is used as a measure of the fluctuations), and
the expression(5)we usethe followingGreen’sfunctions averaged over the different results. In these calculations
u (x′)u∗(x) v∗(x′)v (x) we have introduced corresponding Gaussian fluctuations
Gg (x′,x,ǫ)= n n + n n (6)
adv ǫ−ω −iδ ǫ+ω −iδ in the gap energy, because it is determined by the parti-
Xn n n cle number. For as large as 10% fluctuations in the gap
φ (x)φ∗(x′)
Ge (x,x′,ǫ)= n n . (7) the peak is still clearly visible although broadened and
ret Xn ǫ−ξn+iδ slightly shifted.
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trap should be optical in order to confine also the high
1600 field seekers. Alternatively, in a magnetic trap atoms in
1400 these states (correspondsto the non-interactingstate |ei
Exact
1200 inourcalculation)wouldsimplyflyout—thiscouldeven
Current/arb.units1046800000000 5% 1% moodffiarotekhucetertrbahetaesosodminceastonebiccnseteitorflhvnueaoobrsfleteastchtIeee,nc|cceoueiu,r,lrodterhnbatwethedaiesson,inetaerh.RefoTamrmhieenaasndstueaprtneercomcetceieobnsnyst
200 10% isused,indirectlyfromtheintensitydifferenceofthetwo
0 Raman beams.
96.5 97 97.5 98 98.5 99 99.5 100
In conclusion, we have proposed a method to observe
Detuning/kHz
the superconducting order parameter (gap) in a gas of
FIG.2. The absolute value of the current I =−hN˙ei as a cold Fermionic atoms. The idea is based on creating a
functionofthelaserdetuning∆. Thechemicalpotentialsare
normal phase – superconductor interface by coupling in-
µg =100kHzandµe =0. Thecurrentiscalculatedassuming ternal states of different scattering lengths by a laser.
1%,5%,and10%fluctuationsinthegapenergy. Correspond-
The advantage of our scheme is the sensitivity to even
ing fluctuations in the particle number (µg) are taken into
small currents of atomic population caused by the laser
accountusingtheexponentialdependencebetweentheFermi
interaction. Furthermore, our results extend beyond
energy and the gap energy. The unit of the current is arbi-
trary. In the case of a normal state, there would be a sharp measuring the order parameter: they can be used as a
peakat ∆=100=µg−µe. Duetothesuperconductinggap starting point in describing various processes related to
this peak is shifted and its shape modified. thetransferofatomsbetweentwogascomponents(insu-
perconducting or normal states). If both of the internal
states used were strongly interacting, one could perhaps
It is possible to prepare the desired initial number of
observe Josepson tunneling of pairs from one supercon-
atomsinstates|ei,|giand|g′iveryaccuratelybyapply-
ductor to another. With small modifications, our results
ing suitable incoherent laser pulses — this is one of the
alsodescribe the functioning of anoutcoupler froma su-
advantagesofatomicFermigasescomparedtocondensed
perconducting gas of Fermionic atoms.
matter systems. The number of atoms in the states |gi
Acknowledgements We thank H.T.C. Stoof for useful
and|g′ishouldbe nearlyequalto producethe supercon-
discussions. PT acknowledges the support by the TMR
ducting state, whereas the number of atoms in state |ei
programme of the European Commission (ERBFM-
is not critical. In the simplest case the state |ei would
BICT983061) and the Academy of Finland (projects
be empty in the beginning. The use of our method then
42588 and 47140).
requirestoknowµ ratheraccurately,sinceitappearsin
g
thetunableargument∆˜ =∆µ+∆=−µ +∆ofthecur-
g
rentI(∆˜). Ifthis isdifficult, itmightbe betterto usean
incoherent pulse to prepare initially ∆µ = µ −µ = 0.
e g
Inthiscasethechemicalpotentialdependencedisappears
from the expressions in the homogeneous case.
[1] B. DeMarco and D.S.Jin, Science 285, 1703 (1999).
Finally,wediscusshowtochoosethestates|ei,|giand
|g′iforarealatom(forinstance6Li). Itwillprobablybe [2] For theoretical work on cold atomic Fermi-gases see e.g.
J. Ruostekoski and J. Javanainen, Phys. Rev. Lett. 82,
difficultto findstatesinwhichthe atomsdonotinteract
4741 (1999) and references therein.
at all. A more likely solution is to choose the states so [3] H.T.C. Stoof et al., Phys.Rev.Lett. 76, 10 (1996).
that the interaction between all of them is weak and re- [4] L.YouandM.Marinescu,Phys.Rev.A60,2324(1999).
pulsive,exceptbetween|giand|g′istrongandattractive. [5] For recent reviews, see E.A. Cornell et al., cond-
Even a weak and attractive interaction instead of weak mat/9903109; W. Ketterle et al.,cond-mat/9904034.
andrepulsivewoulddo,becausethegap,criticaltemper- [6] F. Weig and W. Zwerger, cond-mat/9908176.
ature, and other essential BCS quantities have an expo- [7] W. Zhang et al.,Phys. Rev.A 60, 504 (1999).
nentialdependenceonthescatteringlength,sotheeffect [8] M.A. Baranov, cond-mat/9801142.
becomes easily negligible. Particularly6Liseems to offer [9] M.A. Baranov and D.S. Petrov, cond-mat/9901108.
[10] G.M. Bruun and C.W. Clark, cond-mat/990392.
avarietyofinteractionstrengthsbetweendifferentstates.
[11] G.D.Mahan,Many-ParticlePhysics(PlenumPress,New
As shownin[15],inamagneticfieldofaboutB ≃0.02T
York, 1990).
thescatteringlenghtofseverallowfieldseekinghyperfine [12] J. Williams et al.,Phys.Rev. A 59, R31 (1999).
states is as strong as aS ∼−2000a0 whereas the scatter- [13] P.G. deGennes, Superconductivity of Metals and Alloys,
inglengthbetweentwohighfieldseekingstatesisaround (Addison-Wesley Publishing Company, 1989).
zero. This is an example of the potential of modifying [14] M. Houbierset al.,Phys. Rev.A 56, 4864 (1997).
thescatteringlengthsbymagneticfields. Inthiscasethe [15] M. Houbierset al.,Phys. Rev.A 57, R1497 (1998).
4