Post-quench dynamics and pre-thermalization in a resonant Bose gas

2 MB·

by  Xiao Yin

Post-quench dynamics and pre-thermalization in a resonant Bose gas
Xiao Yin∗ and Leo Radzihovsky†
Department of Physics, University of Colorado, Boulder, CO 80309
(Dated: January 8, 2016)
We explore the dynamics of a resonant Bose gas following its quench to a strongly interacting
regimenearaFeshbachresonance. Forsuchdeepquenches,weutilizeaself-consistentdynamicfield
approximationandfindthatafteraninitialregimeofmany-bodyRabi-likeoscillationsbetweenthe
condensateandfinite-momentumquasiparticlepairs,atlongtimes,thegasreachesapre-thermalized
nonequilibriumsteadystate. Weexploretheresultingstatethroughitsbroadstationarymomentum
distribution function, that exhibits a power-law high momentum tail. We study the dynamics and
6
steady-state form of the associated enhanced depletion, quench-rate dependent excitation energy,
1
Tan’scontact,structurefunctionandradiofrequencyspectroscopy. Wefindthesepredictionstobe
0
in a qualitative agreement with recent experiments.
2

n PACSnumbers: 67.85.De,67.85.Jk
a
J
7 I. INTRODUCTION time evolution |ψˆ(t)(cid:105) = eiHˆft|ψˆi(0)(cid:105) of a closed quan-
tum system vis-´a-vis eigenstate thermalization hypoth-

] esis [24, 25], role of conservation laws and obstruction
s A. Background and motivation
a to full equilibration of integrable models argued to in-
g stead be characterized by a generalized Gibb’s ensemble
Degenerate atomic gases have radically expanded the
- (GGE),emergenceofstatisticalmechanicsunderunitary
t scope of quantum many-body physics beyond the tradi-
n time evolution for equilibrated and nonequilibrium sta-
a tional solid-state counter part, offering opportunity to tionary states [26, 27]. These questions of post-quench
u study highly coherent, strongly interacting, and well-
dynamicshavebeenextensivelyexploredinalargenum-
q characterized, defects-free systems. Atomic field-tuned
ber of systems [28–43]
t. Feshbach resonances (FRs) [1–4] have become a pow-
a Early studies of a Feshbach-resonant Fermi gas pre-
erful experimental tool that has been extensively uti-
m dicted persistent coherent post-quench oscillations [30,
lizedtoexplorestrongresonantinteractionsinthesesys-
- tems. Feshbach resonances have thus led to a seminal 44] and, more recently found topological nonequilibrium
d steady states and phase transitions [45, 46].
n realization of paired s-wave fermionic superfluidity, with
o the associated BCS-to-Bose-Einstein condensate (BEC) Resonant Bose gas quenched dynamics studies date
c crossover [3–6] through a universal unitary regime [7–9], back to seminal experiments on 85Rb [47, 48], that
[ andphasetransitionsdrivenbyspeciesimbalance[10,11] demonstrated coherent Rabi-like oscillations between

1 and by Mott-insulating physics in an optical lattice [12– atomic and molecular condensates [49], enabling a mea-
v 16]. Numerous other promising many-body states and surement of the molecular binding energy. More re-
7 phase transitions, such a p-wave fermionic superfluidity cently, oscillations in the dynamic structure function
9 [17–19] and Stoner ferromagnetism [20] have been pro- have also been observed in quasi-2D bosonic 133Cs [38]
3 posed and continue to be explored. andstudiedtheoretically[37,50]forshallowquenchesbe-
1 tweenweakly-repulsiveinteractions(smallgasparameter
Unmatched by their extreme coherence and high tun-
0 na3 (cid:28)1 where a is the s-wave scattering length).
. abilityofsystemparameters,suchasFRinteractionsand s s
1 single-particle (trap and lattice) potentials, atomic gases Such resonant bosonic gases were also predicted to ex-
0 have also enabled numerous experimental realizations of hibit distinct atomic and molecular superfluid phases,
6
highly nonequilibrium, strongly-interacting many-body separated by a quantum Ising phase transition (rather
1
: states and associated phase transitions [2, 6, 12]. than just a fermionic smooth BCS-BEC crossover) and
v otherrichphenomenology[51–55], therebyprovidingad-
This has motivated extensive theoretical studies [21–
i
X 23], with a particular focus on nonequilibrium dynamics ditional motivation for their studies.
r followingaquenchofHamiltonianparameters,Hˆi →Hˆf. Important recent developments are experiments by
a Makotyn, et al, [56], that explored dynamics of 85Rb fol-
In addition to studies of specific physical systems, ex-
lowingadeepquenchtothevacinityoftheunitarypoint
periments on these closed and highly coherent systems
onthemolecular(positivescatteringlength,a >0)side
have driven theory to address fundamental questions in s
of the Feshbach resonance. It was discovered that even
quantum statistical mechanics. These include the con-
neartheunitarypoint,whereaBosegasisexpectedtobe
ditions for and nature of thermalization under unitary
unstable[57], thethree-bodydecayrateγ (ontheorder
3
of an inverse milli-second) appears to be more than an
order of magnitude slower than the two-body equilibra-
∗[email protected] tion rate γ2 (both measured to be proportional to Fermi
†[email protected] energy, as expected [58, 59]. This thereby opened a win-
2
dow of time scales from a microsecond (a scale of the C. Summary of results
quench)toamilli-secondforobservationofametastable
strongly-interacting nonequilibrium dynamics. Before turning to the derivation and analysis, we
Stimulated by these fascinating experimental develop- brieflysummarizethekeypredictionsofourwork. Work-
mentsandmotivatedbytheaforementionedearlierwork, ing within the upper-branch of a single-channel model of
in a recent brief publication [39] we reported on results a resonantly interacting Bose gas we studied an array
for the upper-branch repulsive dynamics of a resonant ofnonequilibriumobservablesfollowingitsFeshbachres-
Bose gas following a deep-detuning quench close to the onance quench toward the unitary point. One central
unitary point on the molecular side (a > 0) of the FR quantityextensivelystudiedinrecenttimeofflightmea-
s
[56]. Taking the aforementioned slowness of γ (cid:28) γ as surements [38, 56] is the momentum distribution func-
3 2
an empirical fact, consistent with experimental observa- tion,n (t)=(cid:104)gs |a†(t)a (t)|gs (cid:105)attimetafteraquench
k i k k i
tions we predicted a fast evolution to a pre-thermalized from a ground state |gs (cid:105) of an initial Hamiltonian Hˆ to
i i
strongly-interacting stationary state, characterized by a a final Hamiltonian Hˆ . Motivated by experiments we
f
broad, power-law steady-state momentum distribution
take|gs (cid:105)tobeasuperfluidBECgroundstateintheup-
function, nss, with a time scale τ = (cid:126)/E for the pre- i
k k k per branch of the repulsive Bose gas [61]. For a shallow
thermalization of momenta k set by the inverse of the
quench in the scattering length a → a , away from the
i f
excitation spectrum, E . The associated condensate de-
k immediatevicinityoftheunitarypoint,thecalculationis
pletion was found to exhibit a monotonic growth to a
controlled by an expansion in a small interaction param-
nonequilibrium value exceeding that of the correspond- eter, na3 (cid:28)1. Within the lowest, Bogoluibov approxi-
ing ground state. In the current manuscript we present i,f
mation the momentum distribution function is given by
thedetailsoftheanalysesthatledtotheseresultsaswell (choosingunitswhere(cid:126)=1andk =1throughout)[37]
B
as a large number of other predictions.
(cid:113)
n (tˆ)= kˆ2+σ+ 2kˆ(12−+σ2)sin2(tˆ kˆ2(kˆ2+2)) − 1, (1.1)
kˆ 2(cid:113)kˆ2(kˆ2+2σ) 2
B. Outline
whereσ ≡a /a characterizesthe“depth”ofthequench,
i f
and we have rescaled the momentum k and time t
The rest of the paper is organized as follows. We con- (cid:112)
with the coherence length ξ ≡ 1/ 2mng and pre-
f
clude the Introduction with a summary of our key re-
thermalization timescale t = 1/ng , as kˆ = kξ and
sults. In Section II, starting with a one-channel model 0 f
tˆ= t/t , respectively. We start the system in a weakly
of a Feshbach-resonant Bose gas, we develop its approxi- 0
interacting state, characterized by a short positive scat-
mateBogoluibovandself-consistentdynamicfieldforms.
tering length a and quench it to a > a (σ ≤ 1).
In Section III, as a warmup we analyze the equilibrium i f i
Following coherent oscillations, the gas then exhibits
self-consistentmodelforthestronglyinteractingcaseand
pre-thermalization dynamics, where after a dephasing
compare its predictions to that of the Bogoluibov ap-
time τ , set by the inverse of the excitation spectrum
proximation. In Section IV we utilize the Bogoluibov k
(cid:113)
model to study the nonequilibrium dynamics following a 1/E =1/ kˆ2(kˆ2+2)consistentwithexperiments[56],
k
shallow-quench, computing the momentum distribution the initial narrow Bogoluibov momentum distribution
function nk(t) probed in the time-of-flight, the radio- evolves to a stationary state, characterized by a broad-
frequency (RF) spectroscopy signal, I(ω,t), and the ened distribution function
structure function S (t) probed via Bragg spectroscopy.
k  
Then in Section V we generalize the quench to a more 1 (kˆ2+σ)(kˆ2+2)+1−σ
estxupdeyrimtheenteafflelyctroefalriastmicpcraastee.ofInaSfiecntiitoen-raVtIewraememppalnody nskˆs = 2 (kˆ2+2)(cid:113)kˆ2(kˆ2+2σ) −1, (1.2a)
the self-consistent dynamic field theory to study these 
Css/k4, for kξ (cid:29)1,
and a number of other observables for deep quenches in 
∼ 1/k2, for σ (cid:28)kξ (cid:28)1, (1.2b)
a strongly interacting regime relevant to JILA experi-
1/k, for kξ (cid:28)σ,
ments[56]. InSectionVIIwestudyexcitationenergy,an
important measure of long time nonequilibrium station- where we defined Css as the nonequilibrium analog of
ary state, for both sudden quench and finite ramp-rate
Tan’s contact for the nonequilibrium steady state, given
cases, and discuss its dependence on quench depth and
by
ramp rate. We generalize Tan’s Contact to nonequilib-
rium process and study its long time behavior in Section Css =(4πa n)2[1+(1−σ)2]. (1.3)
f
VIII.FinallyinSectionIXweconcludewithadiscussion
of our predictions for experiments and of the future di- Within above approximation the quasi-particles do not
rections for this work. We relegate the details of most scatter, precluding full thermalization, and the above fi-
calculations to Appendices. nalstateremainsnonequilibrium,completelydetermined
3
bythedepth-quenchparameterσ,withtheassociateddi- 1.0
agonal density matrix ensemble.
t=0.1
The associated condensate depletion nd(t) = 0.8 t=0.5
1 (cid:80) n (t) is then straightforwardly computed and t=1.0
N k(cid:54)=0 k t=6.0
monotonically pre-thermalizes to 0.6 t=10
t=∞
nss(σ)= √8 (cid:0)na3(cid:1)1/2(cid:20)σ3/2+ 3√1−σarccos(√σ)(cid:21),
d 3 π f 2 0.4
(1.4)
0.2
a value exceeding that for the ground state of the final
scattering length a and greater than the initial ground
f
state depletion ni =nss(σ =1)= √8 (cid:0)na3(cid:1)1/2 at scat- 0.2 0.4 0.6 0.8 1.0 1.2 1.4
d d 3 π i
tering length ai. FIG.1: (Coloronline)Timeevolutionofthe(column-density)
With the goal of understanding deep quenches of a momentum distribution function, n˜ (t) ≡ (cid:82) dk n (t) fol-
k⊥ z k
strongly interacting Bose gas [39, 56, 62] near a Fesh- lowing a deep scattering length quench k a = 0.01 →
n i
bach resonance, we developed a self-consistent dynamic knaf = 1 in a resonant Bose gas (where kn ≡ n1/3), com-
field theory of coupled Gross-Petaevskii equation for the puted within a self-consistent dynamic field approximation.
Here momentum is rescaled by the coherence length ξ as
condensate n (t) and a Heisenberg equation for atoms √
c kˆ = kξ ≡ k/ 2mng . Lowest curve corresponds to earlier
aˆ (t) excited out of the condensate. It accounts for f
k(cid:54)=0 timeattˆ≡t/t =0.1inunitsofpre-thermalizationtimescale
strong time-dependent depletion of the condensate, with 0
t = 1/ng = m/(4πa n) while the dashed-thick black one
feedbackondynamicsofexcitations. Withinthisnonper- 0 f f
represents the asymptotic steady-state distribution. The fig-
tubative (but uncontrolled) approximation this amounts
ureillustratestheinitialnarrowmomentumdistribution(low-
to solving for a Heisenberg evolution of aˆk(t) with a estcurve)evolvingtoamuchbroadermomentumdistribution
time-dependent Bogoluibov-like Hamiltonian, parame- (highest curve), corresponding to a pre-thermalized steady
terized by a condensate density n (t). The latter is self- state. The grey region indicates a range of momenta not re-
c
consistently determined by the atom-number constraint solvedinJILAexperiments,duetoinitialinhomogeneousreal
equation, n (t) = n − (cid:80) n (t,[n (t)]) [30, 39]. Our space density profile and finite trap size.
c k k c
treatment here is closely related to the analysis of post-
quench quantum coarsenning dynamics of the O(N) [36] n /n
c
and Ising [35] models. The resulting momentum distri-
1.0
bution function, n˜ (t) (projected column density mea-
k⊥
sured in experiments [56]) and the corresponding deple-
0.8
tion n (t) are illustrated in Figs. 1,3.
d
We also studied the excitation energy after a constant
0.6
ramp rate γ between a and a scattering lengths. As
i f √
illustrated in Fig. 4, wefound thatit displays a γ form self-consistent
0.4
E (γ) 4(σ−1)2n2a 0.2 Bogoluibov
exc = fa Λf(γ/E ), (1.5a)
V m f Λ
k a
∝ (1−σ)3/2√γ, for γ (cid:28)E , 0.00.0 0.5 1.0 1.5 2.0 2.5 3.0 n s
Λ
∝ (1−σ)2a Λ , for γ (cid:29)E ,
f Λ FIG. 2: (Color online) Ground state condensate fraction as
(1.5b) a function of a dimensionless measure of atom density and
interaction, k a (with k ≡ n1/3), computed within a self-
n s n
foraramp-ratebelowthemicroscopicenergycutoffE =
Λ consistent dynamic field approximation (solid red curve), as
Λ2/2m. compared to Bogoluibov approximation result (dashed blue
To further characterize the post-quench evolution curve).
and the resulting pre-thermalized steady-state we have
also computed a time dependent structure function
S(q,t)=(cid:104)gs |n(−q,t)n(q,t)|gs (cid:105), a Fourier transform of first computed and measured in Ref. [38], and after pre-
i i
the density-density connected correlation function. For thermalization reduces to a time-independent form [39],
the weakly interacting, shallow-quench regime, at tem-
(cid:32) (cid:33)
perature 1/β it is given by n (cid:15) E2
Sss = 0 q coth(βE /2) 1+ qi . (1.7)
(cid:34) (cid:35) q 2E2 qi E2
n (cid:15) E2 −E2 qf qf
S(q,t)= 0 q coth(βE /2) 1+ qi qf sin2(E t) ,
Eq2f qi Eq2f qf Utilizing our self-consistent dynamic field theory we
(1.6) extendedabovecalculationofS(q,t)todeepquenchesof
4
n /n S (t)
d q
1.0
100
t=0
1.0 t=1
0.8 80 t=50
0.7 t=∞
0.6 60
0.5
0.4 0.3 40
0.2 20
knaf=0.1
0.00.0 0.5 1.0 1.5 2.0t/t0 0.5 1.0 1.5 2.0 2.5 qξ
FIG. 3: (Color online) Time evolution of the condensate de- FIG. 5: (Color online) Time evolution of the structure func-
pletion fraction nd(t)/n (treated within a self-consistent dy- tion Sq(t) defined in the text following a scattering length
namic field analysis), following a scattering length quench quench from 0.1af →af with knaf =0.7 (where kn ≡n1/3),
fromk a =0.01tovariousk a inaresonantBosegas. Here referringtoEq.(6.17)usingquasi-adiabaticself-consistentap-
n i n f
we normalize the time with the pre-thermalization timescale proximation (see Sec. VIA). It illustrates the initial ground
t = 1/ng = m/(4πa n) associated with k a = 1 (where state structure function (blue curve), that following the
0 f f n f
k ≡n1/3). quench develops oscillations and after a pre-thermalization
n
time approaches a steady-state distribution (dashed black
curve), which within-quasi-adiabatic approximation almost
ε /ε
exc 0 collapseswiththeinitialgroundstatecurve. Heremomentum
√
● andtimearerescaledwithξ≡1/ 2mngf andt0 ≡1/(ngf),
respectively.
●
3
●
tional to Tan’s contact, that in the simplest Bogoluibov
2 approximation is given by C =(4πnaf)2.
● We next turn to a single-channel Feshbach resonant
● model, followedbyitsdetailedanalysisthatledtoabove
1 ● and other results.
●
●
●
● γ
50 100 150 200 II. MODEL OF A RESONANT SUPERFLUID
FIG. 4: (Color online) Excitation energy (scaled by LHY A resonant gas of bosonic atoms can be modeled by
correction to the ground state energy) following a scatter-
a single-channel grand-canonical Hamiltonian, (defining
inglengthrampasafunctionoframprateγ (asa“zoom-in” (cid:82) ≡(cid:82) d3r)
for Fig. 22, see Sec. VII). The red data points are obtained r
fmoromeaecnhtucmhocseuntoγffaΛˆt=ai/Λaξf==110/02,(ξwi≡th1s/c√al2emdndgimeisnstihoenlceoss- Hˆ =(cid:90) [ψˆ†((cid:15)ˆ−µ)ψˆ+ gψˆ†ψˆ†ψˆψˆ], (2.1)
f 2
herencelength);thebluecurverepresentsthefittingfunction r
√
y=0.26 x. whereψˆ(r)isabosonicatomfieldoperator,(cid:15)ˆ=−∇2 isa
2m
single-particle Hamiltonian, µ is the chemical potential,
andthepseudo-potentialg characterizestheatomictwo-
strongly interacting resonant condensates. The resulting
bodyinteractiononthescalelongerthanitsmicroscopic
time-dependent structure function and its steady-state
range r =1/Λ, typically on the order of ten angstroms.
form are illustrated in Fig. 5. 0
For simplicy, we have set (cid:126)=1.
We also computed the RF spectroscopy signal I(ω )
RF AsdiscussedindetailinRef.[4]andreferencestherein,
[63, 64], that measures the transition rate of atoms from
near a Feshbach resonance the magnetic field-dependent
two resonantly interacting hyperfine states into a third
coupling g(B) controls the s-wave scattering length a
weakly interacting hyperfine state, for the quench pro- s
through the renormalized coupling (T-matrix) g˜−1 =
cess. WithintheBogoluibovapproximationtheresponse g−1+(cid:82) 1 =g−1+mΛ/(2π2),
is given by k 2(cid:15)k
√ g
2τVI2(4πna )2 g˜= , (2.2)
I(ωRF)= 4√πm0 ω3/2f , (1.8) 1+g/gc
RF related to the scattering length via g˜ = 4πa /m. As
s
as measured experimentally, with the amplitude propor- illustrated in Fig. 6, for a sufficiently strong attractive
5
g (cid:90)
interaction, in a vacuum, the two-atom scattering length Hˆ = aˆ†aˆ†aˆaˆ. (2.6d)
diverges at g =2π2/(mΛ)=2π2r /m, as the two-body 4 2 r
c 0
bound state forms for g < −g and a turns positive on are the operator components organized by respective or-
c s
the so-called “BEC” side of the Feshbach resonance. r ders in the excitation aˆ.
0
is the range of the potential and Λ is the corresponding
momentum cutoff. It is this scattering-length tunabil-
ity that enables studies of phase transitions in resonant A. Bogoluibov approximation for weakly
Bose [51–55] (and BCS-BEC crossover in Fermi [1–3, 3– interacting bosons
6]) gases and quenched dynamics [38, 39, 56, 62] that is
our focus here. We set the stage for the study of dynamics following a
shallowquench[38]andofaself-consistentdynamicfield
a treatment [39] of a deep quench [56] by first briefly sum-
s
marizing the results for the ground state and excitations
in the Bogoluibov approximation [65, 66].
In the weakly interacting limit the atomic gas is char-
acterized by a small gas parameter na3 (cid:28) 1, well-
s
approximatedbytheBogoluibovquadraticHamiltonian,
neglecting the nonlinear Hˆ components of Hˆ. Focus-
3,4
πr0/2 ingontheuniform(bulk)condensateandeliminatingthe
chemical potential in favor the condensate density by re-
g quiring the vanishing of the Hˆ1 component (equivalent
-gc 0 to a minimization of Hˆ over Ψ ), µ = g|Ψ |2 ≈ gn,
0 0 0
neglectingthedifferencebetweenthecondensatedensity,
|Ψ |2 ≡n andtotalatomdensity,n,thegrand-canonical
0 c
Hamiltonian reduces to Hˆ ≈−1Vgn2+Hˆ ,
2 B
FIG. 6: (Color online) A plot of the s-wave scattering length (cid:18) (cid:19)(cid:18) (cid:19)
ainsa(reFneoshrmbaaclhizerdescoonuapnlcine.gg˜H)earesagcfu=nc2tπio2nr0o/fmbairsetchoeucprliitnigcagl HˆB = −21k(cid:88)(cid:54)=0εk+ 21k(cid:88)(cid:54)=0(cid:0)aˆ†k aˆ−k(cid:1) gεnkc gεnkc aˆaˆ†−kk ,
coupling strength at which a diverges.
s = −1 (cid:88)ε + 1 (cid:88)Φˆ† h Φˆ ,
2 k 2 k,i k,ij k,j
To allow for dynamics within a Bose-condensed state k(cid:54)=0 k(cid:54)=0
explored experimentally [38, 56], we decompose the = −1 (cid:88)ε + 1 (cid:88)E Ψˆ† Ψˆ ,
atomic field operator ψˆ(r) = √1V (cid:80)kaˆkeik·r, into a c- 2k(cid:54)=0 k 2k(cid:54)=0 k k,s k,s
field condensate Ψ and a fluctuation field aˆ(r),
0 1 (cid:88) (cid:88)
= − (ε −E )+ E αˆ†αˆ , (2.7)
ψˆ=Ψ0+aˆ. (2.3) 2k(cid:54)=0 k k k(cid:54)=0 k k k
where the quadratic Hamiltonian was straightforwardly
Expressing the Hamiltonian, (2.1) in terms of the opera-
diagonalized in terms of the Bogoluibov quasi-particles
tor aˆ, it decomposes into
Ψˆ = (αˆ ,αˆ† ), related to the atomic Nambu spinor
k k −k
Hˆ =Hˆ0+Hˆ1+Hˆ2+Hˆ3+Hˆ4, (2.4) Φˆk =(aˆk,aˆ†−k) by a pseudo-unitary transformation, Uk
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
where aˆk = uk vk αˆk, (2.8a)
aˆ† v∗ u∗ αˆ† ,
(cid:90) g −k k k −k
Hˆ0 = [Ψ∗0((cid:15)ˆ−µ)Ψ0+ 2|Ψ0|4], (2.5) Φˆk = UkΨˆk. (2.8b)
r
U satisfies a pseudo eigenvalue equation h U =
is the lowest order mean-field ground-state energy, and k k k
E σzU and preserves the canonical commutation re-
k k
(cid:90) lation, [a ,a† ] = δ , corresponding to [Φˆ ,Φˆ† ] =
Hˆ1 = [aˆ†((cid:15)ˆ+g|Ψ0|2−µ)Ψ0]+h.c., (2.6a) σzδ , dkefinke(cid:48)d by k,k(cid:48) ik jk(cid:48)
ij k,k(cid:48)
r
Uσ U† =σ , (2.9)
z z
Hˆ2 =(cid:90)r(cid:104)aˆ†(cid:0)(cid:15)ˆ+2g|Ψ0|2−µ(cid:1)aˆ+ g2(cid:16)Ψ∗02aˆaˆ+Ψ20aˆ†aˆ†(cid:17)(cid:105), wiWthi|tuhkε|2k−=|vkk2|/22=m1+agnndiσnz(2th.7e),ththiredBPoaguolilumibaotvrixsp.ec-
(2.6b) trum is given by a well-known gapless form,
(cid:113) (cid:113) (cid:112)
(cid:90) Ek = ε2k−g2n2 = (cid:15)2k+2gn(cid:15)k =ck 1+ξ2k2/2,
Hˆ =g [Ψ aˆ†aˆ†aˆ+Ψ∗aˆ†aˆaˆ], (2.6c)
3 0 0 (2.10)
r
6
that interpolates between the low-momentum zeroth- Wick’s theorem, we have
(cid:112)
sound with velocity c = gn/m (a Goldstone mode of
the U(1) symmetry breaking) and the high-momentum aˆ†aˆ†aˆaˆ → 4(cid:104)aˆ†aˆ(cid:105)aˆ†aˆ+(cid:104)aˆ†aˆ†(cid:105)aˆaˆ+(cid:104)aˆaˆ(cid:105)aˆ†aˆ†
quadratic dispersion, w√ith crossover scale set by the cor- −2(cid:104)aˆ†aˆ(cid:105)(cid:104)aˆ†aˆ(cid:105)−(cid:104)aˆ†aˆ†(cid:105)(cid:104)aˆaˆ(cid:105),
relationlengthξ =1/ 2mgn. Thecorrespondingcoher- ≈ 4n aˆ†aˆ−2n2, (2.16a)
ence factors defining U are given by d d
k aˆ†aˆaˆ → 2(cid:104)aˆ†aˆ(cid:105)aˆ≈2n aˆ, (2.16b)
d
1(cid:18)ε (cid:19) 1(cid:18)ε (cid:19) aˆ†aˆ†aˆ → 2aˆ†(cid:104)aˆ†aˆ(cid:105)≈2n aˆ†, (2.16c)
u2 = k +1 , v2 = k −1 . (2.11) d
k 2 E k 2 E
k k
where we kept the depletion density n = (cid:104)aˆ†aˆ(cid:105) and ne-
d
The ground state is a vacuum of Bogoluibov quasi- glected “anomalous” averages (e.g., (cid:104)aˆaˆ(cid:105) = 0) and high
particles, αˆ |gs(cid:105) = 0, with the energy density E = order correlators (e.g., (cid:104)aˆ†aˆaˆ(cid:105)=0) that we expect to be
k gs
V−1(cid:104)gs|Hˆ|gs(cid:105) given by subdominant.
WiththeseweapproximateHˆ andHˆ byalinearand
3 4
1 1 (cid:88) quadratic forms
E = gn2− E n , (2.12a)
gs 2 V k k
(cid:90)
k(cid:54)=0 Hˆ = g [Ψ aˆ†aˆ†aˆ+Ψ∗aˆ†aˆaˆ]→δHˆ , (2.17)
2πa (cid:20) 128 (cid:21) 3 0 0 1
= sn2 1+ √ (na3)1/2 , (2.12b) r
m 15 π s
where
where the T =0 momentum distribution function (cid:90)
δHˆ = g (2Ψ n aˆ†+h.c.), (2.18)
1 0 d
n = (cid:104)gs|aˆ†aˆ |gs(cid:105)=|v |2 ≈ C/k4, (2.13) r
k k k k k→∞
and
with Tan’s contact C = ∂E /∂a−1 = 16π2n2a2[(1 +
gs s s
36√4π(na3s)1/2] and Hˆ = g (cid:90) aˆ†aˆ†aˆaˆ→δHˆ +δHˆ , (2.19)
4 2 0 2
r
(cid:20) (cid:21)
4πan 32
µ = 1+ √ (na3)1/2 . (2.14) where
m 3 π s
δHˆ = −gVn2, (2.20a)
Theinteraction-drivencondensatedepletion,n ≡n− 0 d
d (cid:90)
nc is given by δHˆ = 2g n aˆ†aˆ. (2.20b)
2 d
r
n = 1 (cid:88)n ≈ √8 (cid:0)na3(cid:1)1/2n, (2.15)
d V k 3 π s The grand-canonical Hamiltonians now take the follow-
k(cid:54)=0 ing forms: Hˆ ≈Hˆ(cid:48) +Hˆ(cid:48) +Hˆ(cid:48), where
0 1 2
and provides an important measure of the validity of the Hˆ(cid:48) =Hˆ +δHˆ ,
Bogoluibov approximation that neglects the difference 0 0 0
(cid:90) (cid:104) g (cid:105) (2.21a)
between n and nc. = Ψ∗((cid:15)ˆ−µ)Ψ + n2−gn2 ,
Clearly, for a large gas parameter, na3 (cid:29)1 the deple- 0 0 2 c d
s r
tion is substantial and must be accounted for. Although
there is no currently available systematic analysis in this
Hˆ(cid:48) =Hˆ +δHˆ ,
nonperturbativelimit,aswewillshowinsubsequentsec- 1 1 1
tions, an uncontrolled self-consistent method, akin to a =(cid:90) (cid:2)aˆ†((cid:15)ˆ+gn +2gn −µ)Ψ (cid:3)+h.c., (2.21b)
spherical, large-N model[35,36,59,67,68]capturesim- c d 0
r
portant qualitative physics in this resonantly interacting
regime.
Hˆ(cid:48) =Hˆ +δHˆ ,
2 2 2
(cid:90) (cid:104) gn (cid:105)
B. Generalization for large scattering length = aˆ†((cid:15)ˆ+2g(nc+nd)−µ)aˆ+ 2c(aˆaˆ+aˆ†aˆ†) .
r
(2.21c)
To extend the analysis to a large na3 we need to ac-
s
count (even if approximately) for the nonlinear compo- Above, for simplicity, we have defined n ≡ |Ψ |2 and
c 0
nents of the Hamiltonian, Hˆ neglected in the Bogolui- n ≡ (cid:104)aˆ†aˆ(cid:105) and in Eqs. (2.21a),(2.21b),(2.21c) have dis-
3,4 d
bovmodel. Tothisend,inthespiritofvariationaltheory cardedthe”anomalousaverage”termm˜ ≡(cid:104)aˆaˆ(cid:105)tosatisfy
or a spherical model [68], we replace these nonlinear op- the constraint of Goldstone theorem, which requires a
erators by their “best” approximation in terms of opera- gaplessexcitationspectrum. Thisamountstothewidely
tors up to a quadratic order in fluctuation field aˆ. Using used Popov approximation [69].
7
Following what was done in the last subsection, we fix The solution to Eq. (3.2) is illustrated in Fig. 2. We
the chemical potential µ by requiring Hˆ(cid:48) =0 find that the self-consistency constraint suppresses con-
1
densate depletion, leading to a higher condensate frac-
((cid:15)ˆ+g|Ψ0|2+2gnd)Ψ0 =µΨ0. (2.22) tion nc than the Bogoluibov approximation for the same
strength of the interaction parameter k a . We also ob-
n s
For a uniform system, this gives
servethat,asexpectedthecorrectiontoBogoluibovthe-
oryfromtheself-consistencyconditiongrows(fromzero)
µ=gn +2gn . (2.23)
c d withincreasinggasparameterk a ,therebyavoidingthe
n s
spurious transition to a vanishing condensate state ap-
Thus we obtain the grand-canonical Hamiltonian
pearing in the Bogoluibov theory.
(cid:90) (cid:104) g (cid:105)
Hˆ = aˆ†((cid:15)ˆ+gn )aˆ+ n (aˆaˆ+aˆ†aˆ†) −E
c 2 c 0
r
(cid:20) (cid:21) IV. DYNAMICS FOR SHALLOW QUENCH
(cid:88) 1
= ((cid:15) +gn )aˆ†aˆ + gn (aˆ†aˆ† +aˆ aˆ ) −E ,
k c k k 2 c k −k k −k 0
k(cid:54)=0 We now turn to nonequilibrium dynamics following a
(2.24) changeinthescatteringlengtha fromitsinitialvaluea
s i
tothefinalvaluea ,ascanberealizedexperimentallyin
where E0/V = g2n2c+2gncnd+gn2d. It exhibits the stan- aFeshbachresonanftBosegasbyachangeintheexternal
dard Bogoluibov form with gapless spectrum, but also magneticfield[56]. Hereweassumethechangeisinstan-
approximately accounts for a potentially strong deple- taneous(suddenquench),allowinganalyticalanalysis. In
tionthroughthecondensatedensitync replacingthefull this section, we focus on shallow quenches characterized
density n as the self-consistently determined parameter by both na3 (cid:28) 1 and na3 (cid:28) 1, so that the Bogoluibov
i f
of the Hamiltonian. approximation remains rigorously valid.
Forshallowquenches,thesystemiswellapproximated
by Hamiltonian (2.7) with g and g for the initial and
i f
III. SELF-CONSISTENT ANALYSIS FOR
final Hamiltonians, respectively, with corresponding Bo-
STRONGLY INTERACTING GROUND STATE
goluibovquasi-particlebases(αˆ ,αˆ†)priortothequench
k k
and (βˆ ,βˆ†) post the quench. Focussing on a sudden
Before turning to our main focus of nonequilibrium k k
quench, the two sets of bases are related to the atomic
post-quench dynamics, we examine the ground state
basis (aˆ ,aˆ†) via a pseudo-unitary transformations
properties of a strongly interacting resonant Bose gas, k k
crahmareatcetrernizae3d(cid:29)by1.aTlahrigseresgciamtteerliinesgbleenyognthd athnedsgcaospepao-f (cid:18) aˆk (cid:19) = (cid:18)u(cid:48)k vk(cid:48)(cid:19)(cid:18) αˆk (cid:19), (4.1a)
s aˆ† v(cid:48) u(cid:48) αˆ†
the standard Bogoluibov theory. Nevertheless we expect −k k k −k
tobeabletotreatitqualitativelycorrectly(evenifquan- Φˆ (0) = U (0−)Ψˆ (0−), (4.1b)
k k k
titatively uncontrolled) by taking into account the large
depletionn−n >0throughtheHamiltonian(2.24)and and
c
tchonesseerlvf-actoionnsistency condition through the atom number (cid:18) aˆk (cid:19) = (cid:18)uk vk(cid:19)(cid:18) βˆk (cid:19), (4.2a)
aˆ†−k vk uk βˆ−†k
1 (cid:88)
n = nc+ V (cid:104)aˆ†kaˆk(cid:105), (3.1a) Φˆk(0) = Uk(0+)Ψˆk(0+), (4.2b)
k(cid:54)=0
where
= n + √8 (cid:0)n a3(cid:1)1/2n , (3.1b)
c 3 π c s c (cid:115) (cid:18) (cid:19) (cid:115) (cid:18) (cid:19)
1 (cid:15) +ng 1 (cid:15) +ng
u(cid:48) = k i +1 , v(cid:48) =− k i −1 ,
where in the second line we calculated the depletion by k 2 E k 2 E
ki ki
diagonalizing(2.24)asinSec.IIAoftheconventionalBo-
(4.3a)
goluibovtheory,butwithn replacingn. Suchtreatment
c (cid:115) (cid:115)
(cid:18) (cid:19) (cid:18) (cid:19)
is quite close in spirit to the self-consistent Hartree-Fock 1 (cid:15) +ng 1 (cid:15) +ng
u = k f +1 , v =− k f −1 ,
approximations, and the BCS and other mean-field gap k 2 E k 2 E
kf kf
equations.
(4.3b)
In the dimensionless form for nˆ = n /n, the self-
c c
consistency equation reduces to define Bogoluibov transformations for Hamiltonians Hˆ
i
(with interaction g ≡g(0−)) before and Hˆ (with inter-
1−nˆ −λnˆ3/2 =0, (3.2) i f
c c action g ≡ g(0+)) after the quench, respectively. The
f
√ √
where λ = 8(na3)1/2/(3 π) ≡ 8(k a )3/2/(3 π), with corresponding excitation spectra are
s n s
kn ≡ n1/3 the mometum scale set by the boson density (cid:112) (cid:113)
n. Eki = (cid:15)k2+2ngi(cid:15)k, Ekf = (cid:15)k2+2ngf(cid:15)k, (4.4)
8
and the two quasi-particle bases are related by Having derived the evolution of the atomic fields
Φˆ (t) = (aˆ (t),aˆ†(t)), we can now compute the basic
(cid:18) βˆ (cid:19) (cid:18) αˆ (cid:19) k k k
k = U−1(0+)U (0−) k , atomic bilinear correlator (supressing the momentum k
βˆ−†k k k αˆ−†k argument on the right hand-side):
(cid:18) (cid:19)(cid:18) (cid:19)
= csoinshh∆∆θθkk csoinshh∆∆θθkk αˆαˆ−†kk , (4.5) Ckij(t,t(cid:48)) = (cid:104)Φˆ†i(t)Φˆj(t(cid:48))(cid:105),
= (cid:104)Ψˆ† (0−)U† (t)U (t(cid:48))Ψˆ (0−)(cid:105),
m mi jn n
with
= U† (t)N U (t(cid:48)), (4.12)
(cid:20) (cid:18) (cid:19)(cid:21) mi mn jn
1 1 E E
∆θ = cosh−1 kf + ki . (4.6)
k 2 2 E E in terms of the pre-quench (t = 0−) quasi-particle occu-
ki kf
pation matrix
Wetaketheinitialstate|0−(cid:105)tobethegroundstateof
the pre-quenched Hamiltonian Hˆi [61], and thus a vac- Nmn = (cid:104)Ψˆ†m(0−)Ψˆn(0−)(cid:105), (4.13a)
uum of αˆ quasi-particles, αˆk|0−(cid:105) = 0. At finite temper- (cid:32) (cid:104)αˆ†αˆ (cid:105) (cid:104)αˆ†αˆ† (cid:105) (cid:33)
ature this generalizes to Bose-Einstein distribution of αˆ = k k k −k , (4.13b)
(cid:104)αˆ αˆ (cid:105) (cid:104)αˆ αˆ† (cid:105)
quasi-particle occupation, −k k −k −k mn
(cid:18)n (0−) 0 (cid:19)
1 = k , (4.13c)
(cid:104)αˆ†αˆ (cid:105) = . (4.7) 0 n (0−)+1
k k 0− eEki/T −1 −k mn
(cid:18) (cid:19)
0 0
Because experiments probe physical observables ex- = , for T =0, (4.13d)
0 1
pressed in terms of atomic operators, we need to com- mn
pute time evolution of Φˆk(t)=(aˆk(t),aˆ†k(t)). Using free from which physical observables, such as the momen-
post-quench evolution of βˆ quasi-particles tum distribution function, structure function, RF spec-
troscopy signal, and many others can be obtained. We
(cid:18) βˆk(t) (cid:19)=(cid:18)e−iEkft 0 (cid:19)(cid:18) βˆk(0) (cid:19)≡T (t)Ψˆ (0+), turn to their computation in the following subsections.
βˆ† (t) 0 eiEkft βˆ† (0) k k
−k −k
(4.8)
the relation (4.5), together with the simplicity of matrix A. Time of flight: momentum distribution function
elements of αˆ quasi-particles in the pre-quench ground
state (vacuum of αˆ ), we find Time of flight measurements, where a gas is released
k
from its trap and its density profile is measured at long
(cid:18) aˆk(t) (cid:19) = U (0+)(cid:18) βˆk(t) (cid:19), (4.9a) times, is one of the central experimental probes dating
aˆ† (t) k βˆ† (t) back to the realization of BEC in dilute alkali gases [70,
−k −k
(cid:18) βˆ (0) (cid:19) 71]. A straightforward analysis demonstrates [11], that
= U (0+)T (t) k , (4.9b) at long times the density profile is proportional to the
k k βˆ† (0)
−k momentum distribution function. At T = 0, that is our
(cid:18) (cid:19)
αˆ (0) main focus here, we obtain
= U (0+)T (t)U−1(0+)U (0−) k ,
k k k k αˆ† (0)
−k n (t) = (cid:104)0−|aˆ†(t)aˆ (t)|0−(cid:105)=C11(t,t),
(4.9c) k k k k
Φˆ (t) = U (t)Ψˆ (0−)=R (t)U (0−)Ψˆ (0−), = |(uke−iEkftsinh∆θk+vkeiEkftcosh∆θk)|2,
k k k k k k
(4.9d) = (cid:15)k+gin+ 2gf(cid:15)k(g+f2−ggfin)n2 sin2(Ekft) − 1, (4.14)
(cid:112)
2 (cid:15) ((cid:15) +2g n) 2
where the matrix k k i
at t=0 reducing to the ground-state momentum distri-
R (t) = U T (t)U−1, (4.10a)
ij il lm mn butionEq.(2.13),asexpectedbycontinuityofevolution.
(cid:18) (cid:19)
= (cosEkft)Iij +isinEEkfkft (cid:15)k−+gfgnfn −(cid:15)kg−fngfn , aRsetˆsc=alitn/gt m≡omtnegnt,uwmeoabstkˆai=ntkhξe≡mokm/(cid:112)en2tummngdfisatrnidbuttiimone
0 f
(4.10b) in terms of dimensionless variables as
evolves the initial Bogoluibov spinor (u (0−),v (0−))→ (cid:113)
(uk(t),vk(t)), and k k n (tˆ)= [kˆ2+σ+ 2kˆ(12−+σ2)sin2(tˆ kˆ2(kˆ2+2))] − 1,
U (t) = U (0+)T (t)U−1(0+)U (0−), (4.11a) kˆ 2(cid:113)kˆ2(kˆ2+2σ) 2
k k k k k
= (cid:18)uke−iEkft vkeiEkft(cid:19)(cid:18)cosh∆θk sinh∆θk(cid:19). (4.15)
vke−iEkft ukeiEkft sinh∆θk cosh∆θk where the initial-to-final scattering length ratio, σ ≡
(4.11b) a /a characterizes the “depth” of the quench.
i f
9
The column momentum distribution n˜ (tˆ) ≡ n
kˆ k
(cid:82) dkˆznkˆ(tˆ) is a more experimentally relevant quantity 10
that we plot at different times in Fig. 7. We observe
8 thermalstate
3.0 t=0.1 quenched
t=0.3
t=0.5 6 state
2.5
t=1.0
t=10
2.0 t=∞ 4
1.5 2
1.0 adiabaticstate
0 kξ
0.0 0.2 0.4 0.6 0.8 1.0
0.5
FIG.8: (Coloronline)Quenchedsteady-statemomentumdis-
0.2 0.4 0.6 0.8 1.0 1.2 1.4 tribution function nsks following a scattering length quench
a = 0.1a → a (thick black curve), as compared to the
i f f
FIG.7: (Coloronline)Timeevolutionofthecolumnmomen- groundstatemomentumdistributionataf (dash-dottedred)
tum distribution defined in the text following a scattering and the corresponding Bogoluibov thermalized distribution
length quench from 0.1af →af, referring to Eq. (4.15) using (dotted blue) at temperature T =0.45ngf.
Bogoluibov approximation. It illustrates the initial narrow
momentum distribution (lowest curve) evolving to a much
broadermomentumdistribution(highestcurve), correspond- A simpler measure of the post-quench dynamics is the
ing to a pre-thermalized steady state. Here momentum and evolution of the condensate depletion, obtained from the
√
time are rescaled with ξ ≡1/ 2mng and t ≡1/(ng ), re- momentum distribution function, n (t), (4.14),
f 0 f k
spectively. Thegreyregionindicatesarangeofmomentanot
resolved in JILA experiments, due to initial inhomogeneous (cid:88) (cid:90) d3k
n (t) = n (t)=V n (t),
real space density profile and finite trap size. d k (2π)3 k
k
that starting with a narrow BEC peak, the column mo- = n0dFd(σ,t), (4.17)
mentum distribution function quickly broadens and de- √
where n0 = 8/(3 π)(na3)1/2 is the ground-state deple-
velops a large momentum tail. The momentum distribu- d f
tion for a =a .
tion approaches a pre-thermalized steady-state n˜ss from s f
k
high momenta, with momenta k > kpth(t) taking time F (σ,t) = (σ)3/2+ 3√1−σArccos(√σ)
tpth(k)≈1/Ekf to pre-thermalize [61]. Thus we obtain d 2
√
(cid:113) 3 2(cid:90) (1−σ)cos(2ty(cid:112)y2+2)
t (kˆ)=1/ kˆ2(kˆ2+2), (4.16) − ydy (4.18)
pth 2 (y2+2)(y2+2σ)1/2
consistent with experiments [56] scaling as 1/k and 1/k2
is the nonequilibrium depletion enhancement factor
at small and large momenta, respectively.
above the corresponding ground state, that interpolates
The steady-state momentum distribution, nss for a
k between σ3/2 (giving the initial depletion at t = 0− for
a = 0.1a → a is plotted in Fig. 8 and compared
i f f a =a ) and the asymptotic depletion
to the ground state n for the same a as well as s i
k f
thermal state nk at finite temperature. We observe Fss(σ)≡F (σ,t→∞)=σ3/2+ 3√1−σArccos(√σ)
that this steady-state momentum distribution lies above d d 2
the ground state one, indicating that even in the long (4.19)
time limit the post-quench system remains in the ex-
cited states, as required by energy conservation. How- of the pre-thermalized state, plotted in Fig. 10.
ever, it also differs significantly from the correspond- As is clear from the asymptotics of Fd(σ,t) defined by
ing finite-temperature thermal-equilibrium distribution, (4.19) and illustrated in F√ig. 9, the depletion fraction
nTk = (u2k + vk2)(cid:104)αˆk†αˆk(cid:105)0− + vk2 = 1/(eEkf/T − 1) + monotonically increases as t over a characteristic time
vk2coth(Ekf/2T), demonstrating that even in the long 1
time, stationary state limit the system is only pre- tpth ≈ ng , (4.20)
thermalized. This is expected because of the quadratic, f
fullyintegrableformoftheBogoluibovHamiltonian. The approaching its asymptotic pre-thermalized value, that
latterguaranteestheabsenceofscatteringoftheBogolui- is always higher than that of the ground state with the
bov quasi-particles βˆ , with a conserved momentum dis- same scattering length a = a . The quenched steady-
k s f
tribution function, that is directly related to the initial state depletion enhancement, Fss(σ) monotonically in-
d
distribution by (4.5). creasing with decreasing σ (deeper quench), reaching a
10
n /n B. Bragg spectroscopy: structure function
d
0.15 A two-time structure function S (t,t(cid:48)) =
q
(cid:104)δnˆ(−q,t)δnˆ(q,t(cid:48))(cid:105) is another central probe of the
quenchedstate nonequilibrium dynamics of degenerate atomic gases.
0.10 It can be measured via Bragg spectroscopy through
a stimulated two-photon transitions [72], and via a
correlation function of a measured density excitation,
0.05 δnˆ(q,t) at momentum q and time t [38]. The former
thus allowed measurements of the excitation spectrum
adiabaticstate
of a strongly interacting (85Rb) BEC, near unitarity
t/t (na3 (cid:29) 1), demonstrating a large deviation from the
0 s
1 2 3 4 5
Bogoluibov and Lee-Huang-Yang (LHY) prediction of
Sec. (IIA).Thelattertechniquewasusedtocharacterize
FIG. 9: (Color online) Post quench dynamics of the conden-
dynamics of a Feshbach-resonant Cesium gas, following
sate depletion fraction as a function of rescaled time in units
a shallow quench in its scattering length [38].
of pre-thermalization timescale t = (cid:126)/(ng ) = m/(4πa n(cid:126))
0 f f
With current experiments in mind, for simplicity we
(solidblackcurve),followingascatteringlengthquenchfrom
0.1a →a with k a =0.1 (where k ≡n1/3), as compared focus on the equal-time t = t(cid:48) structure function (non-
f f n f n
trivial for nonequilibrium dynamics),
tothegroundstatedepletionatk a (dashedredline). Fora
n f
typical85Rbexperimentwithn=5×1012cm−3,a =1100a
f 0
(here a0 =5.29×10−11m is the Bohr radius), t0 ≈360µs. Sq(t) = (cid:104)δnˆ(−q,t)δnˆ(q,t)(cid:105),
1 (cid:90)
= eiq·(r−r(cid:48))(cid:104)ψˆ†(r,t)ψˆ(r,t)ψˆ†(r(cid:48),t)ψˆ(r(cid:48),t)(cid:105) ,
V c
r,r(cid:48)
≈ S0(t)+δSB(t), (4.21)
q q
minimum at σ = 1 (no quench), and exhibiting a maxi-
where
mum at σ = 0, corresponding to initially noninteracting
gasoraquenchdeepintounitaryregime,wherea →∞. (cid:104)
f S0(t) = n (cid:104)aˆ†(t)aˆ (t)(cid:105)+(cid:104)aˆ (t)aˆ† (t)(cid:105)
We note, however, that the latter strongly-interacting q c q q −q −q
resonant regime, clearly lies outside of the perturbative (cid:105)
+(cid:104)aˆ (t)aˆ (t)(cid:105)+(cid:104)aˆ†(t)aˆ† (t)(cid:105) , (4.22a)
Bogoluibov theory. We will treat this k a (cid:29) 1 non- −q q q −q
n f
perturbative regime in a subsequent section, using an = n (cid:2)C11(t)+C22(t)+C21(t)+C12(t)(cid:3),
c q q q q
approximate self-consistent treatment.
(4.22b)
and
Fdss(σ) δSqB(t) = V1 (cid:88)(cid:104)(cid:104)aˆ†k(t)aˆk(t)(cid:105)(cid:104)aˆk−q(t)aˆ†k−q(t)(cid:105)
k(cid:54)=0
2.5
(cid:105)
+(cid:104)aˆ†(t)aˆ† (t)(cid:105)(cid:104)aˆ (t)aˆ (t)(cid:105) ,
k −k k−q −k+q
2.0 = 1 (cid:88)(cid:104)C11(t)C22 (t)+C(12)(t)C(21) (t)(cid:105),
V k −k+q k −k+q
k(cid:54)=0
(4.23)
1.5
are, respectively the quadratic and quartic contribution
to S (t), both computed within the Bogoluibov approx-
q
σ imation.
0.2 0.4 0.6 0.8 1.0 Utilizing the Bogoluibov analysis of the nonequilib-
rium quenched dynamics from the previous subsection,
FIG. 10: (Color online) Quenched steady-state depletion en- (Eqs.(4.9), (4.10b), (4.11), (4.12), (4.13)) the leading
hancement factor Fdss(σ) above the corresponding ground quadratic contribution to SB(t) is given by [38]
state value as a function of σ = a /a following a quench q
i f
fernohmanaciem→enatf.FdsTs(h0e)tw=o3dπo/t4s c(oqpurernecshpionngdatontohne-inmtearxaimctuinmg S0(t) = S0(cid:34)1+ Eq2i−Eq2f sin2(E t)(cid:35), (4.24)
gas or quenching to unitarity) and minimum enhancement q q E2 qf
qf
Fss(1)=1 (no quench), respectively.
d
where as a check, at initial time S0(t = 0) and/or for
q
no-quench g = g above expression reduces to the pre-
i f