Relaxation of the molecular state in atomic Fermi gases near a Feshbach resonance

0.09 MB·English

by  V. S. Babichenko

cond-mat
Relaxation of the molecular state in atomic Fermi gases near a Feshbach resonance.
V. S. Babichenko
Kurchatov Institute, Moscow.
Abstract
The relaxation processes in an ultra-cold degenerate atomic Fermi gas near a Feshbach resonance are considered. It is
shown that the relaxation rate of the molecules in a resonance with the atomic Fermi system is of the order of the fraction of
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0 the chemical potential determined by the interaction between molecules. In this connection the lower part in the excitation
0 spectrum of thesystem of resonance molecules is not well-defined.
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Thevariousinvestigationsofatomicultra-coldFermigasesclosetoaFeshbachresonance[1]giveauniquepossibility
tostudymany-bodyproblemsbothwiththesmallandthestrongcouplingsbetweenparticles. Inaddition,thestrength
ofthe interactioncanbe regulatedbychangingthe magnitudeofthe externalmagneticfield[2]-[9]. Thereisa lotof
theoreticalworksinwhichthecreationoftheboundstatesofFermiatoms,i.e.,molecules,inthedegenerateFermigas
near a Feshbach resonance at ultra-low temperatures is considered. In majority of these works the internal structure
of molecules and the influence of the many-particle effects on this structure are not taken into account [10]-[16]. In
these works the molecules are considered as point particles and described by the local boson field. However, in the
caseofthestrongscatteringofparticlestheconditionsforcorrectnessofthissuppositionareviolatedandtheinternal
structure of molecules should be taken into account.
We consider the influence of the many-particle correlationson the creation of bound states. In this connection, we
take into account the internal structure of molecules as well as the effect of the many-particle correlations on this
structure. The main attention is given to the case when one of the levels of the bound state lies between zero and
the Fermi energy of the atomic Fermi gas. In this case some fraction of Fermi atoms, namely, those which have the
energy larger than the Feshbach resonance level, creates bound pairs with the radius much smaller than the average
spacing between atoms and these molecules are in the resonance with the remaining Fermi gas. We obtain that,
due to many-particle correlations,the molecular propagatorhas the imaginary part. This results in the relaxationof
resonance molecules to the lower energy state. The inverse relaxation time obtained is lager or about the fraction of
the Bose gas chemical potential connected with the interaction between resonance molecules.
The Hamiltonian of the model we consider is
H = d3r Ψ+ (−→r)ε −→p Ψ (−→r)+H +H (1)
α,σ σ α,σ int tr
Z αX=1,2; (cid:16) (cid:17)
b σ=↑,↓b c b b b
where
H = d3rd3r′ g (−→r −−→r′)Ψ+ (−→r)Ψ+ (−→r′)Ψ (−→r′)Ψ (−→r) (1.a)
int Z σ1,σ2 α1,σ1 α2,σ2 α2,σ2 α1,σ1
σX1;σ2
b α1,α2 b b b b
Htr =tZ d3r Ψ+α1,↑(−→r)Ψ+α2,↓(−→r′)Ψα2,↑(−→r′)Ψα1,↑(−→r)+h.c.
b αX16=α2b b b b 
and
−→ −→p2 −→ −→p2
ε (p)= −µ ; ε (p)= +E −µ ; g <0
↑ F ↓ 0 F ↑,↓
2m 2m
−→ −→
The operator p = −i∇ is the momentum operator. The arrow indexes σ =↑,↓ of Fermi operators Ψ denote
α,σ
the direction of the electron spin of atoms with respect to the external magnetic field. The direction ↑ of the atomic
c b
electronspinmeanstheorientationofthe spinalongthedirectionoftheexternalmagneticfield,indexesαdenotethe
direction of the nuclear spin. The interaction between electron and nuclear spins is treated as a small value. Energy
E isdeterminedbytheinteractionbetweenthe externalmagneticfieldh,whichissupposedtobehomogeneous,and
0
theatomicspin. IthasthemagnitudeE =2µ hwhereµ istheBohrmagneton.ThetermH intheHamiltonianis
0 e e tr
connectedwiththepossibilityofhybridizationbetweenthesingletclosedchannelandthetripletopenchannel. Inthis
b
paper we neglect hybridization H and suppose that conditions for the Feshbach resonance, which are formulated
tr
below, are fulfilled. Note that the interaction between atoms g (−→r −−→r′) is independent of nuclear spins α
b σ1,σ2
and depends on the relative orientations of the atomic electron spins of interacting atoms σ =↑,↓. The value g
↑,↓
is supposed to correspond to a strong attractive potential. At the same time, interaction potentials g , g are
↑,↑ ↓,↓
assumed to have a small value or being repulsive ones.
The propagatorof bound states of atoms, i.e. molecules, is determined by a sum of the ladder diagramswhich can
be found by solving the Bethe-Salpeter equation. This equation can be written in the form
T =T(0)+T(0)g T (2)
↑,↓
1
Here T(0) is one link of the ladder diagramin which the integrationover the internal four-momentum is performed
over zero component of this momentum, i.e. over the internal frequency alone. It has the form
dε P P
T(0)(P,p,p′)=−iZ 2πG(↑0)(cid:18)2 +p(cid:19)G(↓0)(cid:18)2 −p(cid:19)δpp′ (3)
The momentums P and p are the total four-momentum and the four-momentum of the relative motion of two
particles,respectively,andεiszerocomponentorthefrequencyoftherelativemotion. Notethatthe integrationover
theinternalfrequencyεcanbeperformedineachlinkoftheladderdiagramssincetheinteractionistime-independent.
The calculation of T(0) gives
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T(0)(P,p,p′)= Ω− →−P2 −−→p2−E +2µ +iδδpp′ (4)
4 0 F
Here and henceforth we choose the system of units so as the magnitude of the atomic mass is equal to unity m=1.
Formally, the solution of Eq.(2) can be rewritten in the form
1
T = (5)
−1
T(0) −g
↑,↓
(cid:0) (cid:1)
It can easily be seen that Eq.(5) is analogous to the expression for the Green function of one particle with the
−→
reduced mass in the external field g (r). The solution of this equation can be written in the form
↑,↓
T (P,p,p′)= ψn(−→p)ψn(−→p′) + d3k ψ→−k (−→p)ψ→−k (−→p′) (6)
Xn Ω+2µF − →−P42 −E0−En+iδ Z (2π)3Ω+2µF − →−P42 −E0−−→k2+iδ
−→2 −→
Wave functions ψ , ψ are the eigen-functions of Hamiltonian H = p + g (r). The functions ψ , ψ
n →− m ↑,↓ n →−
k k
correspond to the discrete and continuous spectra, respectively, and obey the equations
b c
−→
Hmψn =Enψn; Hmψ→−k = k2ψ→−k (7)
In the case of the Feshbach resonancbe there is a discrete lbevel E ≡ E ≈ −E . Usually, the conditions of
n0 Fesh 0
experimentaresuchthatthisdiscretelevelisnotthelowestone. TheexistenceoftheFeshbachresonancemeansthat
the energy ∆E = E −E obeys inequalities | ∆E |<< E and | ∆E |. ε . Different states ψ of the discrete
0 Fesh 0 F n
spectrum E describe the differentstates of moleculesand the propagatorsofmolecules in these states have the form
n
ofthetermsinthediscretesumofEq.(6). Atthesametime,thesecondsummandinEq.(6)determinesthemolecular
states of the continuous part of the molecular spectrum.
Considering the ladder diagrams alone, we take the two-particle processes into account and neglect the many-
particle correlations completely. To involve these correlations, we dress the fermion Green functions in the ladder
−→ −→
diagramsby the self-energy part. In the approximationof the point-like interactionbetweenatoms g(r)=g δ(r)
aa
this self-energy part can be written in the form
Σ (x ,x )=ig2 G(x ,x )G(x ,x )G(x ,x ) (8)
F 1 2 aa 1 2 1 2 2 1
where g is the effective interaction between atoms. The self-energy part Σ has an imaginary part. Its retarded
aa F
−→
component for the large external momentum p, compared with the Fermi momentum p of the atomic gas, can be
F
calculated as
ImΣ(R)(p)=γ =−g2 |−→p |ρ (9)
F↑ aa
where ρ is the density of the Fermi gas. Note that, if we consider a portion in a sum of the ladder diagrams
−→ −→
corresponding to the molecule propagator, the average value | p | of the momentum | p | in this case is of the
av
2
order of the inverse radius of the bound state. In the case of the small density of Fermi gas it obeys inequality
−→
| p | >> p . Due to finite imaginary part in the self-energy part of Fermi propagator, the finite imaginary part
av F
appears also in the propagatorof a molecule, i.e., in the terms of a discrete sum in Eq. (6). The calculationof a sum
oftheladderdiagramswiththedressedfermionpropagatorsunderconditionsofthe thermalequilibriumoftheFermi
gas in the Keldysh - Schwinger technique [17], [18] gives the following expressions for the retarded D(R), advanced
D(A), and kinetic D(K) propagatorsof molecules
1
D(R,A)(Ω)= (10)
→−
Ω+2µ − P2 −E −E ±iΓ
F 4 0 Fesh 1
Ω
D(K)(Ω)=coth D(R)(Ω)−D(A)(Ω) (11)
(cid:18)2T(cid:19)h i
The value E < 0 is the binding energy of the molecule. It is necessary to emphasize that Eq.(11) is not an
Fesh
assumptionwhichcanbe madeonthe naturalbasiswhenthe Fermigasstateissupposedtobe equilibriumone. This
canbe obtainedin the Keldysh- Schwingertechnique by directsummation of the ladder diagramswith the bare link
(3) consisting of the equilibrium Fermi propagators. Note that, in the case of ∆E =E −|E |<0 near the pole
0 Fesh
−→
of D(R,A), the value of D(K) Ω,P becomes negative in the case Γ >0. However,this fact is in contradictionwith
1
(cid:16) −→(cid:17)
the general properties of D(K) Ω,P [17]. The change of Green functions D(R) →D(A), and vice versa, in Eq.(11)
(cid:16) (cid:17)
with the negative value of coth Ω restores the correct sign of D(K). This change in fact means a change of the
2T
sign of Γ . Thus, in the case of(cid:0)∆E(cid:1)< 0 near the pole of D(R,A) the value Γ becomes negative Γ < 0. This, in its
1 1 1
turn, results in an instability of the system and exponential growth of the number of molecules. At the same time,
the Fermi distribution becomes unstable too.
The imaginary part Γ of the self-energy part of the molecular propagator can be estimated just as value (9) with
1
the momentum p of about inverse radius of a molecule, i.e., p∼ |E |,
Fesh
p
1
Γ ∼g2 |E |ρ∼ ρ (12)
1 aa Fesh
|E |
p Fesh
p
−→
The relaxationΓ near the pole of propagatorD(R,A) is practicallyindependent of frequency Ω and momentum P
1 −→
for smallvalues ofΩ and P comparedwith the bounding energyof a molecule and its inverseradius,simultaneously.
This fact distinguishes essentially the Bose gas of resonance molecules coexisting with the degenerate Fermi gas of
atomsfromtheBosegasofmoleculesinthelackofdegenerateFermigasofatoms.Thephysicalreasonforanexistence
of this imaginary part is due to creation of Fermi gas excitations as a result of the interaction of the Fermi gas with
the molecule. Note that the existence of the Cooper gap at the Fermi surface has no influence on the magnitude Γ
1
because the main contribution to the link of ladder diagramsis given by the large momentums much larger than the
Fermi momentum p .
F
The diagrams more complicated than considered before, for the self-energy part of the molecular propagator,
namely, the diagrams involving intersections of the interaction lines with the ladder links can be taken into account.
These diagrams take into account the internal structure of molecules and describe the process in which, due to the
atom-molecule interaction, the molecule transfers to the lower energy state, strongly exciting an atom and obtaining
the momentum almost opposite to the momentum of the excited atom. This process results in the relaxation of the
resonance molecular system to the lower energy state. The self-energy part corresponding to this diagram has the
form
Σ(n)(x ,x )=ig2 D(n−1)(x ,x )G (x ,x )G (x ,x ) (13)
M 1 2 aM M 1 2 F 1 2 F 2 1
Here gaM is the effective interactionbetweenatomandresonancemolecule, Dm(n−1)(x1,x2) denotes the propagator
of the molecule which has the energy E lower than the energy E corresponding to the molecular self-energy
n−1 n
part Σ(n)(x ,x ). Note that the atom-resonance molecule effective interaction g is significantly larger than the
M 1 2 aM
atom-atom interaction g due to large size of resonance molecule compared with the size of atoms. The imaginary
aa
part of this self-energy part can be written as
3
−→ −→ 2
−→ d3p p2 −p + P p2
ImΣ(n) Ω,P =ρg2 δ Ω− −E −(cid:16) (cid:17) − F
M (cid:16) (cid:17) amZ (2π)3 (cid:18) 4 n−1(cid:19) 2 2
  
  
Thus, the relaxation rate of the molecular state Γ2 =ImΣm(n) can be calculated as
ρ
Γ ∼ρg2 |E |∼ρg2 |E(bound) |∼
2 aMp n−1 aMq aM |E(bound) |
aM
q
In Eq. (12) Σ(n) denotes the self-energy part of the molecule which corresponds to such molecular binding energy
M
that the energy of the molecule E in the centre-of-mass frame of the molecule is in the resonance with the atomic
n
Fermi gas, i.e., Σ(n) is the self-energy part of resonance molecules. At the same time, the molecular propagator
M
D(n−1) corresponds to the lower bound state of the molecule. The energy of such molecule in the case of zero total
M
momentumhasvalueE <0. The energyE(bound) isthe binding energyofaresonancemoleculeandanatom,and
n−1 aM
E =E −|E(bound) |.
n−1 n aM
If the bare value of the molecular level is less than the Fermi level of the atomic system and positive, a fraction of
fermions bound to the molecules, such that the molecular level lies at the Fermi level of the residuary Fermi system.
During the times of about 1/Γ the created resonance molecules transfer to the lower energy state and escape from
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the resonance. Note that the value of this relaxation rate is of the order of the chemical potential in the system of
resonance molecules determined by the interaction g ∼ g between molecules. As the result, the lower part of
MM aM
the spectrum in the system of resonance molecules has a large imaginary part and is not well-defined.
This work was supported by the RFFI grant.
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