Table Of ContentInfluence of Cooper pairing on the inelastic processes in a gas of Fermi atoms.
V.S.Babichenko and Yu.Kagan
R.S.C. ”Kurchatov Institute”, Kurchatov Square 1, Moscow 123182, Russia.
(Dated:; Received text; Revised text;Accepted text; Published text)
Abstract
CorrelationpropertiesinultracoldFermigaswithnegativescatteringlengthanditsimpactonthethree-bodyrecombination
isanalyzed. WefindthatCooperpairingenhancestherecombinationrateincontrasttothedecreaseofthisrateaccompanying
2 Bose-Einstein condensation in a Bose gas. This trend is characteristic for all interval of temperatures T<Tc.
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The investigations of ultracold atomic gases give unique possibilities for studying quantum correlations and their
role in the kinetics and dynamical properties of macroscopic systems. At early stage of these investigations it has
been predicted that the rate of inelastic processes drastically decreases with the Bose condensation due to principal
reconstruction of quantum correlations [1]. The local three-particle correlator K , responsible for the recombination
3
in a dilute gas, reduces with decreasing the temperature T < T and the ratio K (T)/K (T ) becomes equal to
C 3 3 C
∼1/6asT →0. Asthegasparameterna3 increasestheeffectdecreasesandpracticallyvanishesattheliquiddensities
(a is the scattering length).
For the first time, the effect has been observed in work [2]. The authors studied the decay kinetics of a ultracold
87Rb gas in a trap under the BEC realization. The experimental results turned out in a qualitative and quantitative
agreement with the theoretical predictions. In fact, these experimental results were ones of first evidences for the
quantum correlationformation in the process of the BEC kinetics with a limited lifetime of the system.
Theanalogousphenomenontakesplaceinthe2DcaseatfinitetemperaturesT <T whenthecondensateisabsent
C
[3]. This result reflects a key role of the local correlations which are close to the case of a genuine condensate.
The condensateofCooper pairs occursin a two-componentatomic Fermigaswith the attractionbetweenparticles
at sufficiently low temperatures. It is interesting to reveal how the Cooper pairing and pair condensate affect the
three-particle recombination rate. Analyzing this problem, we suppose that the attraction is a result of the Feshbah
resonancefors-wavescatteringparticles. Herewith,theinteractionissupposedtohavenonzerovaluefortheparticles
of different components only.
The results obtained in the present work lead to a priory not evident conclusion, namely, Cooper pairing results in
enhancing the probability of three-particle recombination. To make the investigation more transparent, we consider
the low density gas, assuming the following conditions
|a|k <<1 and r k <<1 (1)
F 0 F
where r is the characteristic spatial size of the interparticle interaction.
0
In the case of negative scattering length a < 0 the weakly bounded dimers, typical for a > 0, are not created.
The recombination in this case is accompanied by the formation of a molecule at a deep energy level. Completely
three-particlecharacterof the relaxationin the case ofthe negative scatteringlength a<0,including unitary region,
is demonstrated in the work [4]. The formation of the weakly bounded dimers in the case of the positive scattering
length a>0 changes the kinetics of the strongly bound molecule formation essentially [5]. The large released energy
transfers to the kinetic energy of the molecule as a whole and to the third particle involved into the recombination
process. Here the picture is analogous for Bose and Fermi particles and the transition rate is proportional to the
three-particlecorrelatorinthebothcases. However,thiscorrelatorforFermiparticlesdemonstratestheincreasewith
the Cooper pairing in contrast to its decrease in the BEC case.
2. Consideringthree-particlerecombinationinalowdensityFermigaswiththe creationofamoleculeinastrongly
bound state, we present the Hamiltonian of the system in the form
H =H +H′ (2)
0
whereH0 istheHamiltoniancorrespondingtoelastbicprbocessebsandH′ istheHamiltoniancorrespondingtoinelastic
recombination processes, respectively,
b b
1 ψ+(−→r ,→−r )ψ+(−→r )×
H′ = d3r d3r d3r m 1 2 σ 3 (3)
1 2 3 −→ −→ −→
2σ6=σ′Z (×Vψσ(r1)ψσ′(r2)ψσ(r3)+h.c. )
X b b
b
In this expression ψ+ is the creation operator of a moblebcule b b
m
b
ψm+(−→r1,−→r2)= exp −i12−→q (−→r1+−→r2) ϕm(−→r1−−→r2)bm→−q (4)
X→−q (cid:18) (cid:19)
b b
where ϕ is the wave function of a molecule in its center-of-inertia frame. The Fermi operators of particle annihi-
m
lation have the standard form
1
−→ −→−→
ψσ(r)= c→−kσexp ik r (5)
→−
Xk (cid:16) (cid:17)
b b
−→ −→ −→ −→
Conserving the pairwise structure of the interaction, we present V in the form V =U(r − r )+U(r − r ).
1 2 3 2
−→ −→ −→ −→
From(4) it followsthat | r − r | in (3)has a scale of the molecule size r . The characteristicvalue of| r − r |
1 2 ∗ 3 2
in (3) is close to r∗ since in this case only the third particle shouldbget the momebntum compared in the magnitude
withthe momentumofthe createdmolecule. Analyzing(3)inthe frame whichoriginis atthe center-of-inertiaofthe
three-particle ensemble, we introduce the variables
−→ρ =−→r −−→r , −→ρ′ =−→r −−→r (6)
1 2 3 2
The bare vertex for a product of three Fermi operators on the right hand-side with taking (6), (1) and the trans-
position symmetry into account can be transformed as
−→ −→ −→ −→ −→ −→
exp ik r +ik r +ik r (7)
1 1 2 2 3 3
→ 1hexp i−→k −→ρ +i−→k −→ρ′ 1−iexp −i −→k −−→k (−→ρ −−→ρ′) ≈
1 3 1 3
2
≈ i exp (cid:16)i−→k −→ρ +i−→k −→ρ′(cid:17)h−→k −−→k(cid:16) (−→(cid:16)ρ −−→ρ′) (cid:17) (cid:17)i
1 3 1 3
2
(cid:16) (cid:17)(cid:16) (cid:17)
When the energy of a created molecule E >>ε , the momentum of the third particle and the momentum of the
m F
molecule are practically the same in the absolute magnitude. Taking into account the relation
1(−→r +−→r )−−→r = 1−→ρ +−→ρ′
1 2 3
2 2
for the vertex in (3), we have
−→ −→ −→ →−
Γ q, k , k , k (8)
1 2 3
(cid:16) e(cid:17)xp −i−→q 1−→ρ +−→ρ′ +i −→k −→ρ +−→k −→ρ′ ×
= 4iZ d3ρd3ρ′ ×h(cid:16)−→k1−(cid:0)2−→k3(cid:17)+(−→ρ(−→ρ−(cid:1) ⇄−→ρ′−→(cid:16))ρϕ′m)1(−→ρ)U(3−→ρ′)(cid:17)+i
Considering the Fermi gas with the large magnitude of negative scattering length | a|>> r0, r∗ and conserving
limitation (1), we face with a sharp increase of vertex Γ under conditions of the Feshbah resonance. In fact, for
the singlet pairs a quasi resonance state is realized in the continuous spectrum for energy ε → 0, whereas the real
dimer bound state is absent. The solution of the Schroedinger equation for the two-particle problem demonstrates
−→
that, for | ρ |< r , the wave function obtains an additional large factor | a | /r . The probability amplitude to
0 0
find simultaneously three fermions in the volume ∼ r3 requires a factor (|a|/r )2 as compared with the case of
∗ 0
noninteracting gas (c.p. [6], [7], [8]). Note that r .r . Correspondingly, vertex (8) obtains effectively an additional
∗ 0
factor ∼(|a|/r )2.
0
The integral in (8) can be represented as a sum of two integrals. Each of them is factorized as a product of two
integrals over −→ρ and −→ρ′. Integrating in every case by parts and taking into account that |−→k |<< |−→q |, we find.
i
−→ −→ −→
k − k q
−→ 1 3
Γ=Γ(q) (9)
(cid:16) q2 (cid:17)
e
−→ i |a| 2 dϕm(q/2) dU(q)
Γ(q)=ξ q U(q)+ϕ (q/2)
m
2 r dq dq
(cid:18) 0 (cid:19) (cid:20) (cid:21)
e
2
−→ −→
Hereξ isthe numericalcoefficient. Inallcaseswesupposethe sphericalsymmetryofthe functions U(r), ϕ (r).
m
−→ −→
As a result, Γ(q) depends on the absolute value of vector q alone.
Considering the recombinationtransitions and, correspondingly, H′ as a small perturbation, the number of transi-
tions per uniet time takes the form (~=1)
b
W =2π ρ |H′ |2 δ(E −E )= dt<H′(0)H′(t)> (10)
i fi f i
i,f Z
X
b b b b
H′(t)=exp iH t H′exp −iH t
0 0
(cid:16) (cid:17) (cid:16) (cid:17)
Here the operator ρ is the equilibriumb density matrbix defibned by thebHamiltonian H . Summing over index f in
i 0
(10), we use the inequality ε <<E . Then the block
F m
b b
−→k1−−→k3 −→q −→k′1−−→k′3 −→q ∞ 3q2
|Γ(q)|2 dtexp −i E − t
(cid:16)(cid:16) q2 (cid:17) (cid:17)(cid:16)(cid:16) q2 (cid:17) (cid:17) m 4m
X→−q −Z∞ (cid:18) (cid:18) (cid:19) (cid:19)
e
can be singled out in (10). Here the value 3 q2/m is the overall kinetic energy of the created molecule and fast
4
atom with the momenta equal in magnitude and opposite in the direction. The direct calculation of this expression
(cid:0) (cid:1)
gives
−→ −→ −→ −→
k − k k′ − k′
82 1 3 1 3
q |Γ(q )|2
9π ∗ ∗ (cid:16)(cid:16) (cid:17)q2(cid:16) (cid:17)(cid:17)
∗
As a result, for the probability (10) we fined
−→ −→ −→ −→
k − k k′ − k′
1 3 1 3 −→ −→
W =B K(3) k , k′;σ,σ′ (11)
(cid:16)(cid:16) (cid:17)q2(cid:16) (cid:17)(cid:17) i i
→−kXi,→−k′i ∗ (cid:16) (cid:17)
i=1,2,3
where K(3) is the three-particle correlator
−→ −→
K(3) k , k′ =<c+ (0)c+ (0)c+ (0)c (0)c (0)c (0)> (12)
i i k1′↑ k2′↓ k3′↑ k3↑ k2↓ k1↑
(cid:16) (cid:17)
The value of the constant B is determined by the expression
b b b b b b
4
B = mq |Γ(q )|2 (13)
∗ ∗
9π
The characteristic transition time 1/E is small comparedewith the characteristic correlation time in (11), and it
m
determines effectively the correlator (12) for the coincident time moments.
3. We begin with the calculation of the probability W (11) at zero temperature T =0. In the lack of interparticle
interaction the correlator K(3) reads
−→ −→
K(3) ki, k′i =n→−k1↑n→−k2↓n→−k3↑ δ→−k1;→−k′1δ→−k3;→−k′3 −δ→−k1;→−k′3δ→−k3;→−k′1 δ→−k2;→−k′2
(cid:16) (cid:17) (cid:16) (cid:17)
Substituting this expression into (11), we find
12k2
W =B Fn3, (14)
0 5 q2
∗
3
where n is the particle density with the fixed spin projection. If the interaction between particles is taken into
account, correlator (12) should be averagedover the changed ground state. We start from the case when the Cooper
pairing is absent and then find the correlator K(3) in first order in the parameter |a|k <<1.
F
The interaction Hamiltonian for low energy particles can be written as
Hbint =−→−→−ppX1′1,→−→−pp′22V (−→p)bc→+−k′1↑bc→+−k′2↓bc→−k2↓bc→−k1↑,
where −→p =−→p′ −−→p and an additional condition for summing in this expression is −→p +−→p =−→p′ +−→p′. In first
1 1 1 2 1 2
order in the interaction the wave function of the ground state can be represented as
<S |H |0>
ψ =ψ + int ψ(S)
0 E −E 0
0 S
S6=0
X b
The direct calculationof the correlatorK(3) with taking this reconstructioninto accountresults in the appearance
−→ −→ −→ −→
offour terms. Ifthe factor k − k k′ − k′ is involved,thesetermsgivethe identicalsummandsinEq. (11),
1 3 1 3
and the correlator can be w(cid:16)ritten as (cid:17)(cid:16) (cid:17)
K1(3)(cid:16)−→ki,−→k′i(cid:17)=−V (cid:16)−→k(cid:17)nε→−k→−k′1↑′1n+→−kε′2→−↓k(cid:16)′2n−→−kε2→−↓k1+−n→ε−k→−k1↑2(cid:17) + ε→−k′1 +εn→−k→−k′2′1↑−n→−εk→−k′2↓1 −ε→−k2n→−k3↑δ→−k3;→−k′3 (15)
−→ −→ −→
where k = k′2− k2. Inthis expressionthe termwith aproductoffour functions n→−ki in the numeratorgiveszero
contribution to (11) due to the symmetry and can be omitted.
−→ −→
We use the usual method of substituting genuine potential V (r) for an effective potential V (r) with the same
magnitude of the scattering length a for small energies ε →0 and conserve a possibility to employ the perturbation
−→
theory. The potential V (r) can be taken as a simple spherical rectangular well with the depth V >> ε and the
0 F
radiusR . ApplicabilityoftheperturbationtheoryimpliesthevalidityoftheinequalityκR <<1whereκ = mV ,
0 0 0
herewith, the bound state in the well is absent. It can easily be revealed that, for these requirements, the relation
p
R =η |a| with η >>1 takes place as well as combined inequalities k |a|<<k R <<1. As a consequence, the
0 F F 0
−→ −→
Fourier component obeys the equality V k −→0 ≈ (4π/m)a ≡ g and the function V k begins drastically to
decrease for kR0 ∼ 1 (k >>kF). The la(cid:16)st point i(cid:17)s important since with the substitution(cid:16)(15(cid:17)) into (11) and under
−→
the integration over momentum k the divergent of the second term in the brackets in (15) is eliminated when the
−→
dependence V k is taken into account. The magnitude of the integral proves to be ∼ 1/R and the ratio g/R
0 0
becomes indep(cid:16)ende(cid:17)nt of the scattering length. In fact, the contribution of the corresponding term in the correlator
K(3) determines some renormalizationof the probability W (14).
0
Inthe firsttermin(15)allthe momentaobeyinequalityk .k andthe effectiveinteractioncanbe approximated
i F
−→
as V k ≃ g. Substituting K(3) into (11), we find the probability of the transition W and, at the same time, the
1 1
gener(cid:16)al e(cid:17)xpression for W in the normal phase in the linear approximation in interaction
6
W =W +W =W 1− (11−2ln2)k |a| (16)
n 0 1 0 F
35π
(cid:18) (cid:19)
4. Consider now the contribution of the Cooper pairing to the formation of the three-particle correlator (12). For
this purpose, averaging over the ground state, we should include the anomalous averages
KS(3) =−<c→+−k′1↑c+−→−k′1↓ ><c→−k1↑c−→−k1↓ >n→−k3δ→−k3;→−k′3δ→−k′2;−→−k′1δ→−k2;−→−k1 +... (17)
Inthis expressionthere areaddbitionbalthree terbms rebsultingfromthe commutationsofthe momenta andgivingthe
identical contribution to (11).
Using the standardmethod ofthe u-v transformationfor the calculationofthe anomalousaverages,see e.g. [9], we
have at zero temperature
4
∆
<c→− c →− >=
k1↑ −k1↓ Ek1
Here E = |∆|2 + k12 −µ 2 and ∆ is thebCoobper gap in the one-particle excitation spectrum. Within the
k1 2m
r
framework of the method(cid:16)consider(cid:17)ed the integral in the expression obtained is well known
d3k ∆ 2∆
1
=
Z (2π)3Ek1 g
One cansee that the resultholds for T 6=0 if the gap∆is the temperature-dependentquantity ∆(T). As a result,
after substituting K(3) (17) into (11), we find directly
S
9π2 1 ∆(T) 2
W (T)=W (18)
S 0 4 (kFa)2 (cid:18) εF (cid:19)
FromEq. (18)itfollowsthemainqualitativeresult: theCooperpairingorcondensateofCooperpairsintensifiesthe
three-particlerecombinationandthustherelaxationrateoftheattractiveFermigas. Inthisaspecttheresultdiffersin
kindfromtheBECcaseinadiluteatomicBosegasinwhich,onthecontrary,therateofthree-particlerecombination
decreases. Due tothe behaviourof∆asthe functionofthe parameterk athe three-particlerecombinationraterises
F
exponentially when k | a | increases, at least up to k | a |. 1. The exterpolation of the results obtained from the
F F
regionk |a|<<1 to the regionk |a|.1 demonstrates the realpossibility of the experimentalstudy of the effect.
F F
The leading temperature correction for W is connected with W and equal to π2/3 (T/ε )2. Since T ≃
n 0 F c
∆(T =0), it is readily seen that, for the interval T < T , the temperature behavior of W is governed by the de-
c
(cid:0) (cid:1)
pendence W (T) (18).
S
[1] Yu.Kagan, B. V.Svistunov,G. V. Shlyapnikov,JETP Lett., 42, 169 (1985).
[2] E. A. Burt, R.W. Ghrist, C. J. Myatt, E. A. Cornell, and C. E. Wieman, Phy.Rev.Lett.,79, 337 (1997).
[3] Yu.Kagan, B. V.Svistunov,G. V. Shlyapnikov,JETP, 93, 552 (1987).
[4] X. Du.Y. Zhang, and J. E. Thomas, Phys.Rev.Lett. 102, 250402 (2009).
[5] D. S.Petrov, C. Salomon, and G. V. Shlyapnikov,Phys.Rev.Lett. 93, 090404 (2009).
[6] D. S.Petrov, Phys. Rev.A 67, 010703 (2003).
[7] E. Braaten, H.-W. Hammer, Phys.Rept.,428, 259 (2006). .
[8] K. Helfrich, H.-W. Hammer, and D. S.Petrov, Phys. Rev.A81, 042715 (2010).
[9] L. D. Landau,E. M. Lifshitz, ”Statistical Physics”, Part 2, Pergamon, 1980.
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