Ultracool Gases far from Equilibrium - Max Planck Society

Ultracool Gasesfar from Equilibrium

Thomas Gasenzer

Institut für Theoretische Physik Ruprecht-Karls Universität Heidelberg

Philosophenweg 16 • 69120 Heidelberg • Germany

email: [email protected]: www.thphys.uni-heidelberg.de/~gasenzer

BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer

Overview Lecture 1: Introduction

Ultracold gases & their dynamics, mean-field theory

Lecture 2: Non-equilibrium QFTFunctional approach, 2PI effective action.

Lecture 3: Far-from-equilibrium DynamicsThermalization. Turbulence & critical dynamics.

Lecture 2Nonequilibrium QFT

2.1 Real-time path integral


From classical to quantum dynamics

tfin

tini

Classical dynamics of from S.[] = 0.


tfin

tini




tfin

tini



Quantum dynamics of = from variation of an effective action, [] = 0


QM transition amplitude:

Z.[J.] = ∫D e i S.[]/ℏ.

D = d(x)

Path Integral Approach

tfin|tini

x=xini

xfin



tini

tfin


Effective Action



tini

tfin

Generating functional:

Z.[J.] = ∫D e i S.[]/ℏ.– i ∫J.

.=.i lnZ. =.Z.-1∫D e i S.[]/ℏ

J J.=.0


Effective Action


Quantum dynamics of = from variation of an effective action, []/ = –.J :

Z.[J.] = ∫D [ – ]e i .[]/ℏ.+ i ∫J.

= e i [[]/ℏ – ∫J.]

tini

tfin


Z.[J.] = ∫D e i S.[]/ℏ.– i ∫J.

.=.i lnZ. =.Z.-1∫D e i S.[]/ℏ

J J.=.0


Effective Action



Z.[J.] = e i [[]/ℏ + ∫J.]

[] = –.i.ℏ.ln.Z.[J.] – ∫J.

tini

tfin

Legendre transform


Z.[J.] = ∫D e i S.[]/ℏ.– i ∫J.

.=.i lnZ. =.Z.-1∫D e i S.[]/ℏ

J J.=.0


Effective Action



Z.[J.] = e i [[]/ℏ + ∫J.]

[] = –.i.ℏ.ln.Z.[J.] – ∫J.= S[] –.i./2 Tr.ln.G + .

tini

tfin

Gaussian integration


Z.[J.] = ∫D e i S.[]/ℏ.– i ∫J.

.=.i lnZ. =.Z.-1∫D e i S.[]/ℏ

J J.=.0

2.2 Schwinger-Keldysh


Initial value problems...

t ∣O∣t = t0.∣U†(t)O U(t)∣t0. = Tr[ρ(t0) U

†(t)O U(t) ]

G.ab(x,y) = Tr[ρ(t0) U†(x0)a(x)U(x0−y0)b(y)U(y0) ] – disc.

...require the Schwinger-Keldysh closed time path (CTP):

e.g., for x0 > y0

CTP C :

QFT

y0 x0


Initial value problems...

t ∣O∣t = t0.∣U†(t)O U(t)∣t0.

= ∫D0D0 ρ[0,0]∫D'D' O e i (S[ ]–S[ ])/ℏ

...require the Schwinger-Keldysh closed time path, now in the Path Integral:

QFT

2.3 2PI effective action


Dynamical Field Theory

[(t,x),†(t,y)] = (x – y) (Bose)

{(t,x),†(t,y)} = (x – y) (Fermi)

Our focus: time dependence of the lowest-order “correlation” functions:

G.ab(x,y) = T a(x)b(y) x = (t,x)

(2-point correlation function,

2-time Green function,

Single-particle density matrix, pair function)

QFT


2PI Effective Action (-Functional)[Luttinger, Ward (60); Baym (62); Cornwall, Jackiw, Tomboulis (74)]

Generating functional for (connected) Greens functions (2 sources)

with classical action S, e.g.,


1PI Effective Action

Consider for the first the 1PI effective action (1 source) for

To 1-loop order (Gaussian integral):

“mass” shift


2nd Legendre transform... ...yields the 2PI effective action:

where we used the K-dependence of J():


2nd Legendre transform... ...yields the 2PI effective action and stationarity conditions:

Plugging in the 1-loop 1PI effective action, i.e.

gives, summing beyond 1-loop terms in :


2PI Effective Action (in summary)[Luttinger, Ward (60); Baym (62); Cornwall, Jackiw, Tomboulis (74)]

closed for Gaussian initial conditions (only , G ≠ 0 @ t = 0)

(@ J = 0, K = 0)


2PI Effective Action... ...now reads:

Variation w.r.t. G gives the Schwinger-Dyson equation:

self energy

i.e.:

1PI only ⇒ 2PI only



2PI 1PI⇔



includes daisies, superdaisies, ladders, &tl


Dynamic EquationsFrom the stationarity condition , one obtains, in leading order,the Gross-Pitaevskii equation:


Dynamic EquationsFrom the stationarity conditions , one obtainsthe HFB dynamic equations:

Note:


Dynamic Equations (⊇ Kadanoff-Baym)

From the stationarity conditions , one obtainsthe full dynamic equations:


Numerical Demand

x0

y0

in memory

nth order Runge-Kuttapropagation in t=x0

Gij(x0,y0;x,y)

i(x0;x)

16 x 16 spatial grid 1000 x 1000 temporal grid N x N index grid, N= 2⇒ ~16 GB RAM

e.g.

Interval...

Supplementary slides

2.4 2PI truncations


Approximations to Γ2

E.g. 4 interaction (=0) :

Loop expansion [Cornwall, Jackiw, Tomboulis PRD 10 (74) 2428]

Literature: e.g. lecture notes by J.Berges hep-ph/0409322 (AIP Conf. Proc.)


2PI 1/N Expansion[Berges, NPA 699 (02) 847; Aarts, Ahrensmaier, Baier, Berges, & Serreau, PRD 66 (02) 45008]

N

2PI eff. action = O(N )-singlet ⇒ these irreducible O(N )-invariants only:

with

N



N

tr(G)2/N ∝ N tr(G2)/N ∝ N 0

2PI eff. action = O(N )-singlet ⇒ these irreducible O(N )-invariants only:

with since (n) = 0, n ≥ 3

⇒ each trace ∝ N , each vertex ∝ 1/N, e.g. О(g) graphs:

N

...

b

a

a

aa

b

b

b



upto

NLO

Vertex resummation bare vertex

2.5 Equilibration of a 1D Bose gas


density info in F = statistical correl. function

Near-equilibrium dynamics: Damping

p p

n n

Exponential damping

n - neq ~ exp(-Γ t)


Far-from-equilibrium dynamics

Thermal equilibrium: Loss of information about prior evolution.

Only a few conserved quantities persist.


Conserved quantities


Observables

We set

a(t,x) = 0 (is always one solution)

and calculate the time dependence of:

n(t,p) = ∫dr G11(t,r; t,0) eipr (no. of particles with mom. p)

QFT


Equilibration of a 1D Bose gas

initialGaussian

final Bose-Einstein

Number of particles n(t,p) with momentum p:

[TG, J. Berges, M. Seco & M.G.Schmidt, PRA 72 (05); J. Berges & TG, PRA 76 (07)]


Temperature appears

[J. Berges & TG, PRA 76 (07)]

flat ≙ temperature

[also: Berges & Cox (01), Berges (01), Berges, Borsanyi, & Serreau (03), Berges, Borsanyi, & Wetterich (04)]

from fit of n to


Equilibration of a 1D Bose gasNo. of particles n(t,pi) with momentum pi:


lowest

highest momentum

•••


No. of particles n(t,pi) with momentum pi:


Equilibration of a 1D Bose gas

Far from equilibrium(no fluct.-dissip. rel.)

Near equilibrium (fluct.-dissip. rel.)




mean-fieldapprox.:no scattering•

••

+ ...


Elastic scattering of classical point-like particles in one dimension

Before collision:

p − p

After collision:

− p p




...

Quantum vs. classicaldynamics


Initial value problem

t ∣O∣t = t0.∣U†(t-t0) O U(t-t0)∣t0.

= Z-1∫DD O 0 e i (S[ ]–S[ ])/ℏ

Schwinger-Keldysh closed time path:


c


2.–.2

=.(.–.)(.+.)

=:.~

Quadratic action (QM Harm. Osc.): S.[]~∫dt{(∂t)2 – 2}:


c


2.–.2

=.(.–.)(.+.)

=:.~


Consider QM Harm. Osc.:

S.[] – S.[] ~ –∫dt{ (∂t2 + 2) – (∂t

2 – 2) }

~ –∫dt (∂t2 + 2)

~


c


2.–.2

=.(.–.)(.+.)

=:.~


Path integral:

∫DD O [0,0] e i (S[]–S[])/ℏ

~ ∫DD O [0,0] exp[ –∫dt (∂t2 + 2)/ℏ]

~~ ~

[J. Berges, TG, PRA (07). ClPI goes back to Hopf (50), cf. also Phythian (75), DeDominicis et al. (76), Janssen et al. (76), Chou et al. (85), Blagoev et al. (01), Polkovnikov (03)]


Classical Path Integral

Path integral evaluates to classical solution:

∫DD O [0,0] e i (S[]–S[])/ℏ

~ ∫DD O [0,0] exp[ –∫dt (∂t2 + 2)/ℏ]

~~ ~





∫DD O [0,0] e i (S[]–S[])/ℏ

~ ∫DD O [0,0] exp[ –∫dt (∂t2 + 2)/ℏ]

~ ∫D O W [0,0] [(∂t2 + 2)]

~~ ~





∫DD O [0,0] e i (S[]–S[])/ℏ

~ ∫DD O [0,0] exp[ –∫dt (∂t2 + 2)/ℏ]

~ ∫D O W [0,0] [(∂t2 + 2)]

~~ ~

Not with interactions!g (4-4) = g (3+3)~ ~





∫DD O [0,0] e i (S[]–S[])/ℏ

~ ∫DD O [0,0] exp[ –∫dt (∂t2 + 2 + g2)/ℏ]

~ ∫D O W [0,0] [(∂t2 + 2 + g2)]

~~ ~

Not with interactions!g (4-4) = g (3+3)~ ~

Quantum vertex



Classical vs. Quantum diagrams(2PI 1/N )

Interactions:g (4-4) = g (3+3)~ ~


...provides in a clear way the relation:

Classical statistical evolution...

The path integral...

...vs quantum statistical evolution:

t

x

P(x)

?

initialclassicalprobabilitydistribution


Classical vs. quantum evolution [J. Berges, TG, PRA 76, 033604 (07)]

Ultracool Gases far from Equilibrium - Max Planck Society

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