From the Bogoliubov-de Gennes equations for superfluid fermions to the Gross-Pitaevskii equation for condensed bosons

4 Pages·0.11 MB·English

by  P. Pieri

From the Bogoliubov-de Gennes equations for superfluid fermions to the
Gross-Pitaevskii equation for condensed bosons
P. Pieri and G.C. Strinati
Dipartimento di Fisica, UdR INFM, Universit`a di Camerino, I-62032 Camerino, Italy
(February 1, 2008)
We derive the time-independent Gross-Pitaevskii equation at zero temperature for condensed
bosons, which form as bound-fermion pairs when the mutual fermionic attractive interaction is
sufficiently strong, from the strong-coupling limit of the Bogoliubov-de Gennes equations that de-
scribe superfluid fermions in the presence of an external potential. Three-body corrections to the
Gross-Pitaevskii equation arealso obtained byourapproach. Ourresultsarerelevant totherecent
advances with ultracold fermionic atoms in a trap.
3
0
PACS numbers: 03.75.Ss, 03.75.Hh, 05.30.Jp
0
2

n Evolutionfromsuperfluid fermions to condensedcom- tion is suitably mapped onto the condensate wave func-
a
posite bosons appears on the verge of being experimen- tionoftheGPequation. Ourderivationresemblesclosely
J
tallyrealizedintermsofdiluteultracoldfermionicatoms Gorkov’smicroscopicderivationofthe Ginzburg-Landau
3
in a trap, although several experimental difficulties still equations from BCS theory near the superconducting

1 remain to be overcome1. Dilute condensed bosons in critical temperature in the weak-coupling limit8. Our
v harmonic traps, in particular, have already been stud- approach further enables us to obtain high-order correc-
3 ied extensively2. From a theoretical point of view, their tions to the GP equation; in particular, the three-body
2
macroscopicproperties (suchas the density profile)have correction is here explicitly derived.
0
beendescribedbythetime-independentGross-Pitaevskii We begin by considering the BdG equations for su-
1
0 (GP) equation for the condensate wave function3 (see perfluid fermions in the presence of a spatially varying
3 also Refs. 4 and 5). Superfluid fermions in the presence external potential V(r):
0 of a spatially varying external potential, on the other
/ (r) ∆(r) u (r) u (r)
at hGaenndn,easr(eBudsGua)lleyqudaestciornibse6d, winhteerremassopfatthiaelBlyovgaorliyuinbgovg-adpe (cid:18)∆H(r)∗ (r) (cid:19)(cid:18) vnn(r) (cid:19) = ǫn (cid:18) vnn(r) (cid:19) . (1)
m −H
functionisobtainedfromasetoftwo-componentfermion
Here
-
d wave functions. Several problems (such as the calcula-
n tion ofthe Josephsoncurrent7) havebeen approachedin 2
(r) = ∇ + V(r) µ (2)
o terms of the BdG equations for superfluid fermions. H −2m −
c
Given the possible experimental connection between
: is the single-particle Hamiltonian reckoned on the
v the properties of superfluid fermions with a strong mu-
(fermionic) chemical potential µ (m being the fermionic
i tual attraction and of condensed bosons, a correspond-
X mass and ¯h=1 throughout), while
ing theoretical connection between these two (GP and
r
a BdG)approachesseems appropriateatthis time. Inthis ∆(r) = v u (r)v (r)∗[1 2f(ǫ )] (3)
0 n n n
way,onecouldevenexploretheinterestingintermediate- − Xn −
coupling (crossover) region, where neither the fermionic
is the gap function that has to be self-consistently
nor the bosonic properties are fully realized. Such
determined6, where f(ǫ ) = [exp(βǫ )+1]−1 is the
a connection might be also relevant for the study of n n
Fermi function with inverse temperature β (the sum in
high-temperature (cuprate) superconductors, for which
Eq. (3) being limited to positive eigenvalues ǫ only).
the coupling strength ofthe superconducting interaction n
The negative constant v in Eq. (3) originates from the
seems to be intermediate between weak coupling (with 0
attractive interaction acting between fermions with op-
largely overlapping Cooper pairs) and strong coupling
posite spins (or fermionic atoms with two different in-
(with nonoverlapping composite bosons).
ternal states) and taken of the contact-potential form
In this paper, we establish this connection by showing
v δ(r r′), which can be conveniently regularized in
that the time-independent GP equation at zero temper- 0 −
terms of the scattering length a of the associated two-
ature can be obtained as the strong-coupling limit (to F
bodyproblem9,10. Correspondingly,thegapfunctionhas
be specified below) of the BdG equations. This result
s-wave component only with spinless structure.
thus shows that the BdG equations, originallyconceived
SolutionoftheBdGequations(1)isequivalenttocon-
for weak coupling, are also appropriate to describe the
sideringtheassociatedGreen’sfunctionequation(inma-
strong-coupling limit, whereby the fermionic gap func-
trix form):
1
(cid:18)iωs∆−(Hr)(∗r) iω−s+∆(r)(r)(cid:19)Gˆ(r,r′;ωs)=1ˆδ(r−r′) (4) ltehnegatshsoscciaalteedfohrotmheogneonne-oiunsteprraocbtilnemg G(wreitehn’Vs(fru)n=cti0o)n,aosf
− H
it can be seen from the expression
whereω =(2s+1)π/β(sinteger)isafermionicMatsub-
s
ara frequency, 1ˆ is the unit dyadic, and ˆ is the single- dk eik·(r−r′)
G (r r′;ω µ)=
particle Green’s function in Nambu’s notation11. Solu- G0 − s| Z (2π)3iω k2/(2m)+µ
s
tions of Eqs. (1) and (4) are, in fact, related by the ex- m −
= exp isgn(ω ) 2m(µ+iω )r r′ (11)
pression: −2π|r−r′| n s p s | − |o
ˆ(r,r′;ω )= un(r) 1 (u (r′)∗,v (r′)∗) that holds for any coupling. With the conditions (10), it
G s Xn (cid:18) vn(r) (cid:19) iωs − ǫn n n can be readily verified that the position
+Xn (cid:18)−uvnn((rr))∗∗ (cid:19) iωs 1+ ǫn (−vn(r′),un(r′)) . (5) wG˜h0e(rre,bry′;ωths)e≃chGe0m(ric−alrp′;oωtesn|µti−al(µV(irn)e+xpVr(ers′s)i)o/n2)(,11()12is)
We adopt at this point Gorkov’s procedure8 for ex- replacedby the localformµ (V(r)+V(r′))/2, satisfies
−
pressing the solution of Eq. (4) in terms of the non- Eq. (6) in the presence of the external potential. [The
interacting Green’s function ˜ that satisfies the equa- midpointrule,albeitsuperfluousformostofthefollowing
0
tion G argumentsowingtotheslownessofthepotentialoverthe
distance a , has been adopted for later convenience in
F
[iω (r)] ˜(r,r′;ω ) = δ(r r′) (6) the derivation of the current density.] The approximate
s 0 s
− H G −
expression(12)playsinthepresentcontextananalogous
andsubjectto thesame externalpotentialV(r). We are role to the eikonal approximation for Gorkov’s problem
thus led to consider the two coupled integral equations: in the presence of an external magnetic field8.
From a physical point of view, the first of the con-
11(r,r′;ωs)= ˜0(r,r′;ωs) + dr′′ ˜0(r,r′′;ωs)∆(r′′) ditions (10) implies that the external potential is much
G G Z G
smaller than the binding energy ε , so that the compos-
(r′′,r′;ω ) (7) 0
×G21 s ite boson does not break apart by scattering against the
(r,r′;ω )= dr′′ ˜(r′′,r; ω )∆(r′′)∗ potential V(r). The two additional conditions (10), on
21 s 0 s
G −Z G − the other hand, imply that the external potential is suf-
(r′′,r′;ω ) (8) ficiently slowly varying over the characteristic size a of
11 s F
×G
the composite boson, so that the composite boson itself
for the normal ( 11) and anomalous( 21) single-particle probes the potential as if it were a point-like particle.
G G
Green’s functions. Equations (7) and (8), together with In particular, for the corresponding one-dimensional
the self-consistency equation problem with a generic (albeit non pathological) poten-
tial V(x) that satisfies the first two conditions (10) (a
1 F
∆∗(r)=v0 21(r,r;ωs)e−iωsη (9) therein being replaced by a = (2mµ)−1/2), one can
β G | |
Xs readily verify that our position (12), namely,
(tηion→s (01+)),anadre(f3u)l,lyanedquhivoaldlefnotrtaontyhecoourpigliinnga.l BdG equa- ˜(x,x′;ω ) = me−√2m(|µ|+(V(x)+V(x′))/2−iωs)|x−x′|
0 s
G − 2m(µ +(V(x)+V(x′))/2 iω )
We pass now to specifically consider the strong- s
| | −
coupling limit of Eqs. (7)-(9), and show under what cir- p (13)
cumstances they reduce to the GP equation for spinless
composite bosons with mass 2m and subject to the po- isfullyequivalenttothesolutionofthe(one-dimensional
tential 2V(r). In this limit, the fermionic chemical po- version of the) Green’s function equation (6) as ob-
tential approaches ε /2, where ε = (ma2)−1 is the tained in terms of the asymptotic (a 0+) WKB
binding energy of th−e c0omposite bo0son whichFrepresents approximation12. This identification hold→s in the rele-
the largest energy scale of the problem. vantrange x x′ <aandprovided ωs µ (thatcan
In the present context, achieving the strong-coupling be always s|at−isfied|∼for sufficiently lar|ge|µ≪).| |
| |
limit implies that the following conditions on the poten- Derivation of the GP equation from the BdG equa-
tial tions in the strong-coupling limit exploits the position
(12) and proceeds by using Eq. (7) to expand pertur-
|V(r)|≪|µ|, aF|∇V(r)|≪|V(r)|, a2F|∇2V(r)|≪|V(r)| batively Eq. (8) up to third order in ∆(r) (the ratio
(10) ∆(r)/µ provides the small parameter for this expan-
| |
sion, by taking advantage of the “diluteness” condition
hold for all relevant values of r. In the strong-coupling of the system). Equation (9) yields eventually the fol-
limit, a (2mµ)−1/2 represents the characteristic lowing integral equation for the gap function:
F
∼ | |
2
1
∆(r)∗ = dr Q(r,r ) ∆(r )∗ Entering the approximate expressions (17)-(21) into
1 1 1
−v0 Z the integral equation (14) for the gap function and in-
troducing the condensate wave function
+ dr dr dr R(r,r ,r .r ) ∆(r )∗∆(r )∆(r )∗ (14)
1 2 3 1 2 3 1 2 3
Z
m2a
Φ(r) = F ∆(r) , (22)
with the notation r 8π
1
Q(r,r ) = ˜(r ,r; ω ) ˜(r ,r;ω ) (15) we obtain eventually:
1 0 1 s 0 1 s
β G − G
Xs 1 8πa
2Φ(r)+2V(r)Φ(r)+ F Φ(r)2Φ(r)=µ Φ(r).
B
and −4m∇ 2m | |
(23)
1
R(r,r ,r ,r )= ˜(r ,r; ω ) ˜(r ,r ;ω )
1 2 3 0 1 s 0 1 2 s
−β Xs G − G This is just the time-independent Gross-Pitaevskii equa-
˜(r ,r ; ω ) ˜(r ,r;ω ) . (16) tion for composite bosons of mass mB = 2m, chemical
0 3 2 s 0 3 s
×G − G potential µ , subject to the external potential 2V(r),
B
The position (12) implies that, in strong coupling, and mutually interacting via the short-range repulsive
Q(r,r1) vanishes for r r1 > aF. Since (as a con- potential4πaB/mB (whereaB =2aF isthebosonicscat-
sequence of the condit|io−ns (1|0)∼) ∆(r) is slowly varying tering length within the Bornapproximation13,10). Note
over the length scale aF, we can set ∆(r1)∗ ∆(r)∗ on thatthetwo-bodybindingenergyε0 hasbeeneliminated
the left-hand side of Eq. (14) and write ≃ fromexplicitconsiderationviathebound-stateequation.
Note also that the nontrivial rescaling in Eq. (22) be-
1 tweenthegapfunctionandthecondensatewavefunction
dr Q(r,r )∆(r )∗ a (r)∆(r)∗+ b (r) 2∆(r)∗
Z 1 1 1 ≃ 0 2 0 ∇ has been fixed by the nonlinear term in Eq. (23).
Equation (23) has been formally obtained from the
(17)
original BdG equations in the limit βµ . In this
→ −∞
where respect,itwouldseemthatthisequationholdsevennear
thecondensationtemperatureT ,wheretheGPequation
c
dk 1
a (r)= dr Q(r,r )= is instead known not to be valid. On physical grounds,
0 Z 1 1 Z (2π)3 k2 +2µ(r) however, at finite temperature excitations of bosons out
m | |
1 m2a of the condensate (that are not included in the present
+ F (µB 2V(r)) (18) treatment) are expected to be important especially in
≃−v 8π −
0
the strong-coupling regime of the BdG equations, thus
and restrictingtherangeofvalidityofEq.(23)nearzerotem-
perature.
1 ma
b (r) = dr Q(r,r ) r r2 F . (19) The physical interpretation of the condensate wave
0 1 1 1
3 Z | − | ≃ 16π
function(22)canbeobtainedfromthegeneralexpression
for the density n(r) = (2/β) exp(iω η) (r,r;ω ),
Here,µ(r)=µ V(r)andwehaveintroducedthebosonic s s G11 s
chemicalpoten−tialµ =2µ+ε suchthatµ ε . Note where 11 is obtained by cPombining Eqs. (7) and
B 0 B 0 G
≪ (8) and expanding perturbatively up to second order
that(V(r)+V(r ))/2inthelocalchemicalpotentialhas
1
in ∆(r). Recalling that the fermionic contribution
been replaced by V(r) at the relevant order, since the
(2/β) exp(iω η) (r,r;ω ) vanishes in the strong-
externalpotentialis slowlyvaryingoverthe distanceaF. s s G0 s
Disposal of the ultravioletdivergence in Eq. (18) via the coupliPng limit, one is left with the expression
regularizationof the contact potential9,10 in terms of aF dk 1
hanavdeulseedotfotthheesatpropnrogx-cimouaptleinegxparsessusmiopntsio(n18β)µan→d (−19∞). n(r)≃−2|∆(r)|2 Z (2π)3 β Xs G0(k,ωs)2G0(k,−ωs)
By a similar token, we obtain 2 ∆(r)2 m2aF =2 Φ(r)2 (24)
≃− | | (cid:18)− 8π (cid:19) | |
dr dr dr R(r,r ,r ,r ) ∆(r )∗∆(r )∆(r )∗
Z 1 2 3 1 2 3 1 2 3 evaluated in terms of the wave-vector representation of
c0 ∆(r)2∆(r)∗ (20) G0. Since the bosonic density nB(r) is just half the
≃ | | fermionic density n(r), from Eq. (24) we obtain that
with nB(r) = Φ(r)2, as usual with the GP equation. The
| |
normalization condition N = drn(r) (or, equivalently,
m2aF 2 8πaF its bosonic counterpart) fixes,Rin turn, the overall chem-
c . (21)
0 ≃ − (cid:18) 8π (cid:19) 2m ical potential µ (or directly µB in the strong-coupling
limit).
3
Byasimilartoken,thecurrentdensitycanbeobtained (bosonic) Thomas-Fermi approximation in the strong-
from the general expression coupling limit, thus missing the contribution of the ki-
netic energy. The present derivation of the GP equation
1 1
j(r)= ( ′) eiωsη 11(r,r′;ωs)r=r′ (25) fromtheBdGequationsshows,inthisrespect,thatsolu-
im ∇−∇ β G |
Xs tion of the BdG equations for fermionic atoms in a trap
represents a refined approach that correctly treats the
by combining Eqs. (7) and (8) as before. The
kinetic energy for all couplings.
independent-particle contribution to the current now
The connection we have established between the BdG
vanishes exactly with the midpoint rule (12). After long
and the GP equations might be also useful for deter-
butstraightforwardmanipulationsoneobtainsforthere-
mining the limiting strong-coupling behavior of super-
maining contributions in the limit βµ :
→−∞ fluid fermions in the presence of an external potential,
1 as,forinstance,whenstudying strong-couplingeffects in
j(r) [Φ(r)∗ Φ(r) Φ(r) Φ(r)∗] (26) the Josephson and related problems.
≃ 2im ∇ − ∇
In conclusion, we have shown that the time-
which is, as expected, twice the value of the quantum- independent GP equation at zero temperature for com-
mechanicalexpressionof the currentfor a composite bo- posite bosons (formed as bound-fermion pairs) can be
son with mass mB =2m and wave function Φ(r). obtainedongeneralgroundsfromtheBdGequationsde-
It is clear from the above derivation that higher-order scribingsuperfluidfermionssubjecttoanexternalpoten-
correctionstothe GPequation(23)canalsobe obtained tial, in the strong-couplinglimit of the mutual fermionic
by expanding Eqs. (7) and (8) to higher than the third attraction. Corrections to the GP equation have also
orderin∆(r). Inparticular,tofifthorderin∆(r)thefol- been derived in powers of the condensate wave function.
lowing term adds to the right-hand side of Eq. (8) (with We are indebted to D. Neilson for discussions.
r=r′) once summed over ω :
s
dk 1
∆(r)4∆(r)∗ (k,ω )3 (k, ω )3.
−| | Z (2π)3β G0 s G0 − s
Xs
(27)
1G.Modugno,G.Roati,F.Riboli,F.Ferlaino,R.J.Brecha,
EvaluatingtheintegralinEq.(27)inthestrong-coupling
and M. Inguscio, Science 297, 2240 (2002).
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| |
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the mass of 85Rb and the typical value a 250a.u., JETP 13, 451 (1961).
B
one gets g /¯h 10−27cm6s−1 with the corre∼ct order of 4F. Dalfovo, S. Giorgini, L.P. Pitaevskii, and S. Stringari,
3
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in Dilute Gases (Cambridge University Press, Cambridge,
in this way with not much additional burden. Alterna-
2002).
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6P.G. De Gennes, Superconductivity of Metals and Alloys
posite bosons are needed, one could directly proceed
(Benjamin, NewYork,1966).
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density approximation. Suchanapproachreducesto the
4