Table Of ContentBose-representation for a strongly coupled nonequilibrim fermionic superfluid in a
time-dependent trap
I. V. Tokatly1,2,∗
1Lerhrstuhl fu¨r Theoretische Festk¨orperphysik, Universit¨at Erlangen-Nu¨rnberg, Staudtstrasse 7/B2, 91058 Erlangen, Germany
2Moscow Institute of Electronic Technology, Zelenograd, 124498 Russia
(Dated: February 2, 2008)
4
0 Using the functional integral formulation of a nonequilibrium quantum many-body theory we
0 developaregulardescriptionofaFermisystemwithastrongattractiveinteractioninthepresenceof
2 anexternaltime-dependentpotential. Inthestrongcouplinglimitthisfermionicsystemisequivalent
n toanoequilibriumdiluteBosegasofdiatomicmolecules. Wealsoconsideranonequilibrimstrongly
a coupled Bardeen-Cooper-Schrieffer (BCS) theory and show that it reduces to the full nonlinear
J time-dependent Gross-Pitaevski (GP) equation, which determines an evolution of the condensate
wavefunction.
9
2
PACSnumbers: 03.75.Ss,03.75.Kk,05.30.Jp
]
r
e A remarkable progress in creation of long-lived cold anyorderofnonlinearityprovidedthatexternalfieldsare
h
diatomic molecules in trapped Fermi gases [1, 2, 3, 4, 5] slow functions of time on the scale of the inverse molec-
t
o hasbeenrecentlycrownedwithexperimentaldemonstra- ular binding energy. Despite physical simplicity of this
. tion of the molecular Bose-Einstein condensation(BEC) argumenta formalnonequilibriumtheory ofa superfluid
t
a [6, 7, 8]. One of the main fundamental goals of these Fermi systemin the strong coupling limit is still lacking.
m
studies is to investigate experimentally the problem of The present paper is aimed to fill this gap. We
- a crossover from weakly coupled BCS superfluidity to formulate a regular quantum kinetic description of a
d
the molecular BEC. Possibly the first experimental real- fermionic system with a strong attractive interaction
n
o ization of the crossover regime was reported recently in in the presence of time-dependent trapping potential
c Ref. 9. Inthe lastdecade a theoreticaldescriptionofthe U(x,t). Specifically we consider a two-component sys-
[ crossover problem also attracted a considerable interest tem that is defined by the following Hamiltonian
mainlyinthe contextofhighttemperaturesuperconduc-
1
v tivity [10, 11, 12, 13, 14, 15]. The results of these works 2 ∇2
3 show that if the interaction supports two-particle bound H= dx1 Ψ†j(x1) −2m −µ+U(x,t) Ψj(x1)
0 statesandthe density islowenough,the systembehaves Z nXj=1 (cid:20) (cid:21)
6 as a dilute Bose gas of diatomic molecules. This regime
1 corresponds to a strongly coupled fermionic superfluid − dx2Ψ†1(x1)Ψ†2(x2)V(x1−x2)Ψ2(x2)Ψ1(x1) (1)
0 andamolecularsideoftheabovecrossover. Thedescrip- Z o
4
0 tion of a spatially inhomogeneous condensed Bose gas is where Ψ†j(Ψj) is the creation (annihilation) operator of
/ commonlybasedonGPequationforthecondensatewave a Fermi particle of type j, V(x) is the interparticle in-
t
a function (see Refs. 16, 17 and references therein). Gen- teraction potential and µ is the chemical potential (the
m eralization of the previous theoretical works to spatially
Lagrange multiplier, which is defined by the condition
inhomogeneous Fermi systems demonstrates that in the
- for conservation of the average number of particles N).
d strongcouplinglimittheequilibriumBCStheoryreduces
For simplicity the mass of particles m is assumed to be
n
the common stationary GP equation [18].
independent of j.
o
c One of important features of the experiments with First we briefly summarize some formalities of a
: trapped atomic systems is that measurements are fre- nonequilibrium many-body theory. The complete de-
v
i quentlyperformedundernonequilibriumconditions. The scription of a system with a time-dependent Hamilto-
X populartime-of-flighttechniquerepresentsanexampleof nian H(t) is given by the many-particle density ma-
r experimental methods of this type. Therefore it is desir- trix ρ(t), which satisfies the common equationof motion
a
able to have a consistent kinetic theory of a spatially in- i∂ ρ(t) = [ρ(t),H(t)] with initial condition ρ(t ) = ρ .
t 0 0
homogeneousFermisystemwithstrongpaircorrelations.
The average value of any operator O can be calculated
Recently it has been shown that linear response dynam-
as follows
ics of stronglycoupledBCS andbosonic GP systems are
b
also equivalent [19]. However, physically it seems to be Trρ(t)O Trρ U†(t,t )OU(t ,t)
0 0 0
quite likely that the bosonic description of a strongly hO(t)i= = , (2)
Trρ(t) Trρ
0
coupled nonequilibrium Fermi system should be valid to
b b
b
where U(t ,t )=T exp −i t2H(t)dt is the evolution
2 1 t1
operator and T means tnhe uRsual chronoological ordering.
∗Electronicaddress: [email protected] Calculationoftheaveragecanbereformulatedinamore
2
t0 C− t 2
a. S= dt dx ψ∗(x,t)L(x,t)ψ (x,t)+S .(6)
C+ ZC Z j=1 j j int
X
b. t0C C− C+ t Htheereinψit(iaxl,ta)ndistaheGfirnaasslmpaoninntsfioeblfdi,ntte−0graantdiont+0co−ntioβurarCe
T
respectively (see Fig. 1b). One particle operator L(x,t)
t − i/T in Eq. (6) is defined as follows
0
b
FIG.1: Schematicrepresentationoftheintegrationcontours: ∇2
Contour C′ which enters general Eq.(3) (a); and contour C L(x,t)=i∂t−H(x,t)=i∂t+ 2m −UJ(x,t)+µ. (7)
(b),which is used in the generating functional of Eq. (4)
Fobr simplicity webintroduced only one source J(x,t) for
thedensityoperatorandredefinedtheexternalpotential
compact way if we introduce the following generating U = U + J. The second term, S , in the action of
J int
functional Eq. (6) is given by the second term in Eq. (1) with an
additional contour integration and with all Ψ-operators
ZJ =Trρ0TC′exp −i HJ(t)dt , (3) beingreplacedbythecorrespondingGrassmannfieldsψ.
(cid:26) ZC′ (cid:27) Following the standard route we introduce in Eq. (5)
where HJ(t) = H(t)+J(t)O is the Hamiltonian in the Gaussian integral over nonlocal Bose field η(x1,x2,t)
presence of a source field J(t). The time integration in and decouple the four-fermion term S in the action.
int
Eq (3) runs over contour C′b, which is shown in Fig. 1a. It is important that field η(t) should satisfy boundary
ContourC′ consistsoftwobranchesC− andC+ andgoes condition η(t+0 − iβ) = η(t−0), which follows from the
from initial time t0 to infinity (C− branch) and back- boundary condition imposed on Grassmann variables in
wards (C+ branch). Operator TC′ in Eq. (3) orders all Eq.(5). AfterthisHubbard-Stratonovichtransformation
times along the integration contour. It is worth to men- S takes the form
int
tion that generating functional Z , Eq. (3), is nontrivial
J
(differs from Trρ0) only if the source J(t−) at the upper S =− V(x −x )η∗(x ,x ,t)ψ (x ,t)ψ (x ,t)
branchdiffersfromthesourceJ(t+)atthelowerbranch. int 1 2 1 2 2 2 1 1
Z
Using the generating functional of Eq. (3) one can rep- +c.c.(cid:8)+η∗(x ,x ,t)V(x −x )η(x ,x ,t) (8)
1 2 1 2 1 2
resent the average of Eq. (2) in terms of the following
functional derivative with the integration over all internal variables. (cid:9)
δlnZ Obviously, if we now take the integral over field η in
hO(t)i=i .
the stationary phase approximation, we end up with a
δJ(t )
(cid:20) − (cid:21)J=0 nonequilibrium (and nonlocal due to general form of the
Introducing sourcbesfor anyother operatorwe cancalcu- interaction potential) version of BCS theory. We shall,
late the corresponding observables as well as any higher however, postpone this until the very end and proceed
order correlationfunction. The formalism is simplified if further at the formally exact level.
the initial condition corresponds to the thermal equilib-
CalculatingGaussianintegraloverGassmannvariables
rium: ρ =exp{−βH(t )}, where β =1/T. In this case
0 0 weobtainthefollowingrepresentationofgeneratingfunc-
thegeneratingfunctionalcanbewrittenasatraceofthe
tional Z
J
only chronologicalexponent
Z =Z(0) Dη∗DηeiSeff[η∗,η]. (9)
Z =TrT exp −i H (t)dt (4) J J
J C (cid:26) ZC J (cid:27) Zη(t+0−iβ)=η(t−0)
where all times are ordered along new three-branchcon- Factor Z(0) in Eq. (9) corresponds to the noninteracting
tour C, which is shown in Fig. 1b (for discussion of the J
Fermi gas, while the effective action S is given by the
three-branch contour in the context of quantum kinet- eff
expression, which we present in the obvious structural
ics see, for example, Refs. 20, 21 andreferences therein).
form
ThemainadvantageofEq.(4)isthepossibilitytorepre-
sentitasacoherentstatefunctionalintegral. Thederiva- S [η∗,η]=−iTrln 1−GVη −η∗Vη
tionofthisrepresentationis,infact,independentofpar- eff
ticulartimecontour. Therefore,itsimplyreproducesthe ∞ 1 (cid:16) 2(cid:17)l
correspondingderivationforthepartitionfunctioninthe =i Tr GVbηb −η∗Vη, (10)
2l
equilibrium statistical mechanics [22]. For our nonequi- Xl=1 (cid:16) (cid:17)
librium system we get the result b b
where we introduced the notations
2
ZJ= Dψj∗DψjeiS[ψj∗,ψ], (5) G(t,t′)= G(t,t′) 0 , η = 0 η(t) . (11)
Zψ(t+0−iβ)=−ψ(t−0)jY=1 (cid:20) 0 −G(t,t′)(cid:21) (cid:20)η∗(t) 0 (cid:21)
b b
3
proximation satisfies Bete-Salpeter equation
t t′ K =K −K VK. (15)
0 0
Usingthe definitionofEq.(15)we transformEq.(14)as
a. b. follows
FIG.2: Diagrammaticrepresentationofthesecond-order(a.) S(2) = dtdt′η∗(t){K−1−K−1K K−1}η(t′). (16)
and the fours-order (b.) terms in effective action. eff 0
ZC
From this point we concentrate on a strongly coupled
One particle propagator G(t,t′) is defined as the inverse superfluid. We assume that the lowesteigenvalue −ε0 of
of one particle operator L, Eq. (7): the two-particle problem
∇2
i∂t−H(x,t)bG(t,t′)=δC(t−t′), (12) − −V(r) χn(r)=−εnχn(r) (17)
m
h i corresponds(cid:20)to a bound(cid:21)state with radius a (ε =
where δ (t − t′)bis the delta-function on contour C. 0 0
C 1/ma2). In the strong coupling limit a defines the
The uniqueness of a solution to Eq. (12) is guaran- 0 0
smallest spatial scale of the problem, which means that
teed by the boundary condition G(t−,t′) = −G(t− −
0 0 a n1/3 ≪ 1 and in addition a /L ≪ 1, where L is the
iβ,t′). Introducingone particle contourevolutionopera- 0 0
characteristic length-scale of external potential U(x,t).
tor UC(t,t′) = TCexp −i tt′H(τ)dτ we represent the Similarly, the binding energy ε0 is the largest energy
solution to Eq. (12) innthe following foorm scale, which should be large than both temperature T
R
b and the inverse characteristic time related to variations
G(t,t′)=−iU (t,t−) (1−n(t ))θ (t−t′) of U(x,t). The above assumptions allow for significant
C 0 0 C
simplification of the effective action, Eq. (10).
−n(t0)θC(t′−(cid:2)t) UC(t−0,t), (13) Intheinitialstateatt=t thesystemisinequilibrium
b 0
where θ (t) is the contour ste(cid:3)p-function and operator and the chemical potential is negative and close to half
C b of the binding energy µ ∼ −ε /2 [10, 11, 12, 13, 14]. In
n(t0)=[eβH(t0)+1]−1 correspondstotheFermi-function fact,thestrongcouplinglimitc0anbeformallyviewedasa
at initial time t .
0 limitµ→−∞,whilethe quantityε +2µremainsfinite.
b Letus cobnsiderquadratic(Gaussian)part,Se(ff2),ofthe In this limit n(t0) → 0, and the one0particle propagator
effective action, Eq. (10),
of Eq. (13)reduces to the operator of retardedevolution
along contour C: G(t,t′)=−iU (t,t′)θ (t−t′). There-
C C
Se(ff2) =− dtdt′η∗(t){VδC(t−t′)+VK0(t,t′)V}η(t′) forethebaretwoparticleGreen’sfunctionK0(t,t′)takes
ZC a form of the two particle retarded propagator
(14)
where t
K01,2(t,t′)=−iTCe−i t′[H1(τ)+H2(τ)]dτθC(t−t′), (18)
K0(x1,x2,t;x′1,x′2,t′)=iG(x1,t;x′1,t′)G(x2,t;x′2,t′) R
where indeces 1 and 2 labbel thebpropagating particles.
isthebaretwo-particlepropagator. Diagrammaticrepre- Finally, the full two particle contour propagator K(t,t′)
sentationofthesecondterminEq.14isshowninFig.2a. of Eq. (15) can be found as a solution to the following
Full two-particle propagator K(t,t′) in the ladder ap- equation
∇2 r r ∇2
i∂ + x −U x+ ,t −U x+ ,t + r +V(r)+2µ+i0 K =δ(x−x′)δ(r−r′)δ (t−t′). (19)
t J J C
4m 2 2 m
(cid:20) (cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
In Eq.(19) we introducedthe notationsx=(x +x )/2 operator K−1, which enters Gaussian part, Eq. (16), of
1 2
and r = x −x for the center-of-mass and the relative the effective action.
1 2
coordinates respectively. The infinitesimally small term,
i0,inEq.(19)meansthatthis equationshouldbesolved
with retarded boundary conditions. By definition, the To derive a low energy form of the Gaussian action,
expression in square brackets in Eq. (19) is the inverse Eq. (16), we expand fluctuating Bose field η(x1,x2,t)
in terms of eigen functions χ of the two-body problem
n
4
Eq. (17) circles are boson-fermions “vertices” Λ , which depend
0
onlyonrelativecoordinater=x −x (see Eqs.(8)and
1 2
η(x ,x ,t)= ϕ (x,t)χ (r). (20) (20))
1 2 n n
n
X ∇2
Λ (r)=V(r)χ (r)= − +ε χ (r), (24)
Substitution of this expansion into Eq. (16) leads to the 0 0 m 0 0
(cid:20) (cid:21)
result
whereas solid lines stand for the one particle fermionic
S(2) = ϕ∗ hχ |K−1−K−1K K−1|χ iϕ . (21) Green’s functions G. Physically interaction of Fig. 2b
eff m m 0 n n
corresponds to a scattering via virtual decay of two
ZCm,n
X composite bosons with subsequent exchange by the con-
Letuscalculatethelowenergyformofmatrixelements stituentfermions. Themaincontributiontointernalinte-
in Eq. (21). The leading contribution to the low energy gralsinthe diagramofFig.2bcomesfroma highenergy
action is given by the term ∼ ϕ∗0ϕ0. Using the explicit region. In fact, the excitation energy ε0 sets an effective
form of K−1 (operator in the brackets in Eq. (19)) and lowercutofffortheseintegrals. Thereforealleffectsofin-
Eq.(17)weobtainthefollowingexpressionforthematrix homogeneityand deviations fromthe equilibrium, which
element hχ |K−1|χ i enter the diagram via the one particle Green’s functions
0 0
G, are irrelevant for the calculation of this term. The
∇2 above arguments are also applicable to all terms with
hχ |K−1|χ i≈i∂ + x −2U (x,t)+2µ+ε . (22)
0 0 t 4m J 0 higherl. Totheleadingorderinthestrongcouplinglimit
all coefficients in terms with l > l simply coincide with
In the derivation of Eq. (22) we have used the fact that thoseobtainedinthe equilibriumtheory. Thisshouldbe
function χ0(r) is localized on the scale a0, which by ba- contrasted to the Gaussian part (l = 1) where the high
sic assumption satisfies the condition a0/L ≪ 1. To the energycontributionto the diagramofFig.2ais canceled
same level of accuracy all off-diagonal matrix elements out by the term η∗Vη in the effective action of Eq. (10).
hχn6=0|K−1|χ0i vanish due to orthogonality of functions What is left after this cancellation gives exactly the ac-
χn with different quantum numbers. The contributions tion,Eq.(23),ofnonequilibriumBosegasinthepresence
of the type hχn|K−1K0K−1|χ0i describe corrections to of the external time dependent inhomogeneity.
Eq. (22) (for n = 0) and the coupling of the lowest and Forcollidingcompositeparticlesoflowenergy(smaller
excitedtwo-particlestates (forn6=0). Inthe lowenergy than ε ) the diagram Fig. 2b is independent of four-
0
cornerthesetermsarealsosmallsincetheycontainafac- momenta. The corresponding constant defines the
tor K0 ∼1/µ∼1/ε0 which is the smallest parameter of strength ,g(0), of an effective short range two-body in-
thetheory. Matrixelementshχ |K−1−K−1K K−1|χ i 2
n 0 m teraction. Using Green’s functions of the homogeneous
with n,m 6= 0 define contributions of excited states to
system as the solid lines in Fig. 2b and performing fre-
the effective action. They are at least of the order of the
quencyintegrationwearriveatthe result(see,forexam-
excitation energy ε . Since the excited states are decou-
0 ple, similar calculations in Refs. 11, 14)
pled from the lowest bound state, they do not change
low energy physics. Actually they do contribute only to Λ4(p) p2
rloewno).rmItaliiszwatoiortnhonfotthinegbtohsaotni-nbotshoenstirnotnergaccotiuopnli(nsgeelimbeit- g2(0) =2Xp pm2 0+ε0 3 =2Xp (cid:18)m +ε0(cid:19)χ40(p).
the ideal gas contribution Z(0) to the generating func- (cid:16) (cid:17) (25)
J
tionalofEq.(9)isalsoirrelevantasitcorrespondstothe In the second equality in Eq. (25) we make an explicit
ideal Fermi gas with a large negative chemicalpotential. use of Eq. (24), which relates the boson-fermion vertex
Therefore the low energy contribution to the Gaussian Λ to the bound state wave function χ . Similarly one
0 0
part of the effective action takes a clear physical form can calculate all higher order bare coupling constants
Se(ff2) = dtdxϕ∗ i∂t+ 2∇M2x −2UJ(x,t)+λ ϕ, (23) gl(0) =(−1)l[((2ll−−11)!)]!2 pm2 +ε0 χ20l(p). (26)
ZC (cid:20) (cid:21) Xp (cid:18) (cid:19)
which corresponds to the Keldysh action of an ideal It is, however, well established that in the strong cou-
nonequilibrium gasof Bose particles with the mass M = pling limit the pairwise interactionalwaysdominates for
2mandthechemicalpotentialλ=2µ+ε0inthepresence any dimension d of space [11, 15]. Consequently the un-
of the effective external potential UB(x,t) = 2U(x,t). renormalizedactionforlowenergyfluctuatingBosefields
J
Equation (23) is, in fact, one of the main formal results of fermionic pairs take the form
of the present paper.
Terms with l > 1 in the series in Eq. (10) describe l- S = dtdx ϕ∗ i∂ + ∇2x −2U (x,t)+λ ϕ
bosoninteractions. Diagrammaticrepresentationfor the eff t 2M J
two-boson term is shown in Fig. 2b. Wiggled lines in ZC n h 1 i
this figure correspond to fluctuating Bose fields ϕ(x,t), −2g2(0)(ϕ∗ϕ)2 . (27)
o
5
Let us consider 3d system with a short range interac- has first integrate the high energy degrees of freedom
tionpotentialV(r). Ifthecharacteristicrangeofinterac- out. After the integration we obtain the final effective
tionRismuchsmallerthanzeroenergyscatteringlength kinetic theory. This theory is formally defined by the
a , all relevant quantities can be expressed in terms of generating functional of Eq. (29) and the effective ac-
F
aF. The bound state energy takes a form ε0 = 1/ma2F, tion of Eq. (27) with g2(0) being replaced by the physical
whichmeansthata0 =aF. Substitutinganexplicitform interaction constant g2. As usual, the calculation of ob-
of the normalized bound state wave function servables with such an effective field theory requires a
proper regularization procedure (see, for example, [23]).
8π 1
χ0(p)=raF p2+a−F2 Tcohueprlienggulcaornizsatatinotngshoinultdhetaakcetiionntoisacacloreuandtytheequfaaclttothiatst
2
physicalzero energy value. Numerical value of the phys-
into Eq. (25) we arrive at the well known result
ical (renormalized) interaction g is obtained as a zero
2
4πa 4πa(0) energy limit of the scattering matrix T, which satisfies
g2(0) = mF = MB , (28) the following equation
where a(0) = 2a is the bare (unrenormalized) bosonic T =T(0)+T(0)GGT, (32)
B F
scattering length.
where G = (K−1 − K−1K K−1)−1 is the full bosonic
Nonequilibrium BCS theory corresponds to the sta- 0
propagator (see Eq. (16)), and T(0) is the full momen-
tionary phase approximation for generating functional
tum dependent bare scattering matrix of Fig. 2b. Since
Z of Eq. (9). In the strong coupling limit it is suffi-
J the relevant energy scale in Eq. (32) is defined by the
cient to take the stationary point of the integral
bindingenergyε ,slowexternalpotentialU (x,t),which
0 J
Z = Dϕ∗DϕeiSeff[ϕ∗,ϕ] (29) entersEq.(32)viaT(0) andG,isagainirrelevant. Inthe
J homogeneous3dFermisystemwithashortrangeattrac-
Zϕ(t+0−iβ)=ϕ(t−0) tion a numerical solution to Eq. (32) has been found by
with S [ϕ∗,ϕ] defined by Eq. (27). Stationary point Pieri and Strinati [14]. The result reported in Ref. 14
eff
value Φ(x,t) of Bose field ϕ(x,t) satisfies the time- (g =4πa /M with a ≈0.75a ) is somewhat different
2 B B F
dependent GP equation fromthe resultofanexplicittreatmentofthe fourparti-
cle problem(a ≈0.6a ) [24]. This discrepancy,though
B F
∇2 4πa(0) not strong, still requires a clarification, since Eq. (32)
i∂ Φ= − x +2U (x,t)+ B |Φ|2−λ Φ (30)
t 2M J M should be formally equivalent to the four particle scat-
" #
tering problem of Ref. 24.
with initial condition Φ(x,t ) = Φ (x), where Φ (x) is In contrast to the 3d case where a ∼ a and g ∼
0 0 0 B F 2
the solution to the stationary GP equation g(0), in strongly anisotropic traps the renormalization
2
effects should be more pronounced. For example, in a
−∇2x +2U (x,t )+ 4πa(B0)|Φ |2−λ Φ =0. (31) quasi-2d trap the physical coupling constant, which en-
J 0 0 0
" 2M M # ters an effective 2d GP equation, should take the form
[25, 26]
FunctionΦ(x,t)playsaroleofthecondensatewavefunc-
tion. The relation of Φ(x,t) to the nonlocal BCS order 4π
g = .
parameter follow Eqs. (8) and (20) 2 M|lnna2|
B
∆(r,x,t)=V(r)χ (r)Φ(x,t). In this case a can be quite different from a since the
0 B F
bosonicscatteringlengtha isproportionaltothebound
B
In the case of short range interatomic interaction BCS
stateradiusa ,whichstronglydependsonthetransverse
0
theory becomes local with the local order parameter of
size of a quasi-2d trap.
the form
In conclusion we developed a regular quantum kinetic
1 8π theoryofadiluteFermigaswithstrongpaircorrelations,
∆(x,t)= ∆(r,x,t)dr= Φ(x,t). whichistrappedbyatime-dependentexternalpotential.
m a
Z r F Thedescriptionofthisstronglycoupledfermionicsuper-
Equations (30), (31) prove the equivalence of strongly fluid reduces the theory of a dilute nonequilibrium Bose
couplednonequilibriumBCStheoryandtime-dependent gas of diatomic molecules. In particular, we proved the
GPtheory. Asonecouldexpectonthephysicalgrounds, equivalence of nonequilibrium BCS theory in the strong
the equivalence goes far beyond the linear response coupling limit andthe full nonlineartime-dependent GP
regime. equation. Despite these results may seem to be quite
Stationary phase approximation on the level of gener- obvious physically, we believe that our general approach
ating functional Eq. (9) (BCS theory) neglects high en- canbeusefulinstudyingmorecomplicatedsituationsre-
ergy (∼ ε ) fluctuations of Bose fields ϕ(x,t). To con- lated to new nonequilibrium phenomena in the problem
0
struct a regular description of low energy physics one of BCS-BEC crossover.
6
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