Atomic Quantum Gases far from Thermal Equilibriumbec10/Gasenzer_Lecture1.pdf · Thomas Gasenzer...

Atomic Quantum Gasesfar from Thermal 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

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 · 9. August 2010 · Lecture 1 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.


Thanks & credits to......my work group in Heidelberg:

Cédric Bodet Matthias KronenwettBoris Nowak Denes SextyMartin TrappeJan Zill

Alexander Branschädel ( Karlsruhe)Stefan Keßler ( LMU München)Christian Scheppach ( Cambridge, UK)Philipp Struck ( Konstanz) Kristan Temme ( Vienna)

...my collaboratorsJürgen Berges (Darmstadt) • Hrvoje Buljan (Zagreb) • Keith Burnett (Oxford/ Sheffield) • David Hutchinson (Otago) • Paul S. Julienne (NIST Gaithersburg) • Thorsten Köhler (Oxford/UCL) • Otto Nachtmann (Heidelberg) • Markus K. Oberthaler (Heidelberg) • Jan M. Pawlowski (Heidelberg) • Robert Pezer (Sisak) • David Roberts (Oxford/Los Alamos) • Janne Ruostekoski (Southampton) • Michael G. Schmidt (Heidelberg) • Marcos Seco (Santiago de Compostela)

LGFG BaWue

€€€...

ExtreMe Matter Institute

Lecture 1Introduction to Dynamicsof Ultracold Quantum Gases


ULTRAcool...

... atoms @ nanokelvins –

trapped only a few mm away from

glass cell @ room temperature

(vacuum of 10-12 Torr, i.e. 10-15 bar,or 10-10 Pa,

≈ atmospheric pressure on the moon)


Bose-Einstein condensation

Bose-Einstein condensation (BEC)

bimodaldistribution

Experimental picture after free expansion of the trapped cloud:

dens

ity


Degenerate Fermi gases

[Truscott et al., Science 291, 2570 (01)]

[D. Jin, Phys. World (Apr 02)]

Pauli blocking


Quantum Nonequilibrium Dynamics of Ultracold Atomic Gases:

Theory?

• Mean-field theories [Gross-Pitaevskii, Hartree-Fock(-Bogoliubov)] • Kinetic/hydrodynamic approaches (close to equilibrium)

[Quantum Boltzmann, ZNG, …]

• (Semi-)classical simulations [Truncated Wigner Approximation, Positive P, ...]• Exactly solvable models [Lieb & Liniger, Luttinger, Girardeau, McGuire, Gaudin, Minguzzi, Buljan, ...] • Quantum-Information inspired: tDMRG, MPS/PEPS

[Vidal, Schollwöck, Kollath, White, Feiguin, Manmana, Muramatsu, Wolf, Cirac, ...]• Quantum Monte Carlo, stochastic Quantisation [Mak, Egger, Stoof; Berges, Sexty & Stamatescu, ...]

• Functional QFT: 2PI effective action, Kadanoff-Baym, RG methods

1.1 BEC theory


How to describe a condensate?

Cloud of atoms (density profiles) Matter wave (interference pattern)



[A. Gretzki]

Particles Waves or?

Go for both!


Coherent states

Superposition

ξ

n

α 2n

n!

â|⟩ = |⟩

= Poisson-distribution:

p(n) =


Coherent states in phase space

â|⟩ = |⟩

X ~ Re()

P ~ Im()

⟨X⟩ ~ ⟨ â + â† ⟩ ~ ⟨E(x,t)⟩

Gaussian w/ width: ~ â†â − â†â

Wigner function

||=n1/2

Rotation w/HO freq. ω


⟨ x

⟩ ~

⟨ E

⟩

Coherent state

Source: http://www.gerdbreitenbach.de/gallery/

Wigner function

P ~ Im()

X ~ Re()


(Min. uncertainty) coherent states

Nature, 387, 471 (1997)

X ~ Re()

P ~ Im()


Nonclassical states in phase space

Nature, 448 (2007)



( )



( )

Cooper pairs, ...


There is a subtlety...

Two (generally inequivalent) alternatives:

(⇔ coherent state)


...in describing a condensate:



Penrose-Onsager criterium:

which is called Off-Diagonal Long Range Order (ODLRO)

compatible with mean field:


...in describing a condensate:



Penrose-Onsager criterium, with 0:

(⇔ shift condensate density into the connected part, i.e., assume number state)

also = 0

takes into accountnumber conservation

1.2 Mean-field dynamics


Gross-Pitaevskii dynamicsMany-body Hamiltonian:



Gross-Pitaevskii, i.e. Classical Field Equation for “matter waves”, from vNE:

⇒




typical scattering length:typical bulk density:

⇒ diluteness parameter: (GPE valid for ⇔ small condensate depletion)

⇒

a ≃ 5 nmn ≃ 1014 cm−3

na3≃ 10−5 na3 ≪ 1


Low-energy scattering

internuclear distance r

pote

ntia

l en

ergy

V(r

)

= angle between p and r



internuclear distance r

pote

ntia

l en

ergy

V(r

)


for low p = k0: s-wave amplitude:“effective

range”

∈ ℝ





range”

∈ ℝ





range”

∈ ℝ(forget about the

black curves)


Gross-Pitaevskii hydrodynamicsGPE:

Hydrodynamic ansatz: Density :

Velocity field:

Obtain continuity and (quantum) Euler equations:

(irrotational ideal liquid = superfluid) e.o.s:


Gross-Pitaevskii hydrodynamics

30°init. twist: density / phase



[M. Davis (01)]

GPE:


Velocity field:

Obtain continuity and (quantum) Euler equations:

(irrotational ideal Example: The Scissors mode: liquid = superfluid)


Gross-Pitaevskii hydrodynamicsGPE:


Velocity field:

Nonlinear dynamics: Vortex generation – superfluid flow:

[Anglin & Ketterle (02)]

[Abu-Shaeer et al. (01)]

1.3 Beyond Gross-Pitaevskii




⇒


Beyond Gross-Pitaevskii

Hartree-Fock-Bogoliubov (HFB) from and factorisation:

analogously for 3rd-order correlation function.


Beyond Gross-Pitaevskii

Hartree-Fock-Bogoliubov (HFB) from and factorisation:

analogously for 3rd-order correlation function.

⇒ Closed system of coupled e.o.m. for (x) = x ñ(x,y) = x

†y − x*y

m(x,y) = xy − xy

~


Hartree-Fock-Bogoliubov(HFB) from and factorisation:

⇒ Closed system of coupled e.o.m. for and ñ, m:

mean-field approximation

~

Interval

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