Bose Einstein Kondensation und die Gross-Pitaevski

Bose Einstein Kondensation und
die Gross-Pitaevski Gleichung
Sebastian Diehl
Institut für Theoretische Physik, TU Dresden
verwandte Vorlesungsfolien: http://iqoqi006.uibk.ac.at/users/c705291/
verwandtes Vorlesungsskript: I. Boettcher, J. Pawlowski, S. Diehl,
Nucl. Phys. B Proceedings supplement, arxiv:1204.4394 (2012)
Introduction: Cold Atoms vs. Condensed Matter
Q: What can cold atoms add to many body physics?
•
New models of own interest
- Bose-Hubbard model
- Strongly interacting continuum systems: BCS-BEC Crossover; Efimov effect
- long range interactions other than Coulomb; SU(N) Heisenberg models with variable N...
•
Quantum Simulation: clean/ controllable realization of model Hamiltonians which are
- less clear to what extent realized in condensed matter
- extremely hard to analyze theoretically
•
e.g. 2d FermiHubbard model
Nonequilibrium Physics, time dependence:
- Condensed matter: thermodynamic equilibrium and ground state physics.
- Cold atoms: e.g. creation of excited many-body states, study their dynamic behavior
•
Scalability:
- Study crossover from systems with few- to (thermodynamically) many degrees of freedom:
Change of concepts?
•
Integrate quantum optics concepts and many-body theory, coherent-dissipative
manipulation, non-equilibrium stationary states
Vorschau und Motivation
•
Vergleiche:
•
Ein-Teilchen Schrödinger Gleichung: linear in der Wellenfunktion
i~@t '(x, t) =
•
•
~2
4 + V (x) '(x, t)
2m
Gross-Pitaevski Gleichung: nichtlineare Gleichung fuer makroskopische “Wellenfunktion”
i~@t '(x, t) =
•
'(x, t)
~2
4 + V (x) + g'⇤ (x, t)'(x, t) '(x, t)
2m
Behauptung: beschreibt Ensemble ultrakalter schwach wechselwirkender Atome
Ziele:
•
•
•
•
Erinnerung Bose-Einstein Kondensation: makroskop. Besetzung eines Quantenzustands
Typische Skalen in ultrakalten Quantengasen, effektiver Hamiltonian
Diskussion notwendiger Näherungen, Herleitung
Physikalische Implikationen: Nichtlinearität & makroskopische Phasenkohärenz
BEC: Statistische Mechanik nicht-wechselwirkender Bosonen
•
zweite Quantisierung:
-
Vernichtungs- (Erzeugungs-) operator für Mode q :
aq (a†q )
-
nichtverschwindende Kommutatorrelation:
[aq , a†q0 ] = (q
-
•
n̂q = a†q aq
Besetzungsoperator:
Nicht-wechselwirkende Bosonen im freien Raum: Besetzungszahl-Darstellung,
grosskanonisches Ensemble
chemisches Potential
H = Hkin
•
µN̂ =
Z
q
q2
2m
µ a†q aq
Statistische Eigenschaften beschrieben durch freie Energie:
U = kB T log Z,
Z = tr exp
1
kB T
(Hkin
µ0
µN̂ )
Temperatur
•
q0 )
Chemisches Potential stellt mittlere Teilchenzahl ein: Zustandsgleichung
N=
@U
= hN̂ i =
@µ
Z
q
ha†q aq i
=V
Z
dd q
(2⇡~)d
Bose-Einstein
Verteilung
1
1
2
q
(
e kB T 2m
µ)
1
BEC: Makroskopische Besetzung eines Quantenzustands
•
reduziere T und betrachte Verhalten vonµ (n fix, 3D)
T
N
n=
=
V
T=0
Z
=
µ
d3 q
(2⇡~)d
1
1
2
q
(
e kB T 2m
3
µ/kB T
(T
)g
(e
)
3/2
dB
µ)
1
Polygamma
Funktion
zn
g (z) =
n
n=1
• µ verschwindet bei endlichem T!
der Zustandsgleichung notwendig
• Bose und Einstein (1925): qualitative Modifikation
Z
1. Modifikation:
N=
ha†q=0 aq=0 i
+V
Eq
d3 q
(2⇡~)3 nB (q)
N ⇠V !1
2. thermodyn. Limes:
=> q = 0 Mode makroskopisch besetzt: klassische Behandlung
aq=0 ⇡ a0
•
kritische Temperatur:
de Broglie Wellenlänge
and
n
dB
3
dB
a0 , a⇤0 ⇠ N 1/2
q
= g3/2 (1) = ⇣(3/2) ⇡ 2.612
= (2⇥
2
/mkB T )1/2
d=n
1/3
mittlerer
Teilchenabstand
BEC: Makroskopische Besetzung eines Quantenzustands
•
reduziere T und betrachte Verhalten vonµ (n fix, 3D)
T
N
n=
=
V
T=0
Z
=
µ
d3 q
(2⇡~)d
1
1
2
q
(
e kB T 2m
3
µ/kB T
(T
)g
(e
)
3/2
dB
µ)
1
Polygamma
Funktion
zn
g (z) =
n
n=1
• µ verschwindet bei endlichem T!
der Zustandsgleichung notwendig
• Bose und Einstein (1925): qualitative Modifikation
Z
1. Modifikation:
N=
2. thermodyn. Limes:
ha†q=0 aq=0 i
+V
d3 q
(2⇡~)3 nB (q)
Eq
N ⇠V !1
=> q = 0 Mode makroskopisch besetzt: klassische Behandlung
aq=0 ⇡ a0
•
bei T = 0:
and
3
dB
a0 , a⇤0 ⇠ N 1/2
/ T 3/2 ! 0
=) N =
⇤
a0 a0
q
a⇤0 a0
N 1
vollständiges Kondensat,
keine thermische Dichte
T /Tc
1
-
The operator nq = a†q aq measures the particle number in the momentum mode q. The total particle
P †
number is N̂ = q aq aq .
Role of Dimension: No qBEC in 2D
The dispersion relation is Eq = !ω =
2
2m
. This is the energy of a single particle in the mode q.
• Lower T and study the behavior of µ at fixed n (2D):
• Lower T and study the behavior of µ at fixed n (3D):
T Tc the equation of state has no solution
• At a finite T , µ hits zero: below this
➡ EoS without condensate can be
= 0 and Einstein (1925): Equation below T needs modification due to macroscopic
• TBose
c fulfilled for all T:
d
=
3
➡ No BEC in (homogeneous) 2d
d=2
space
µ
•
Reason: Infrared (low momentum) divergence due to smaller phase space
d2 q
d2 q
⇤n̂q ⌅ =
q2
2
(2⇥)
(2⇥)2 e( 2m
1
µ)/kB T
1
⇥
low momenta,
small mu
•
dqq
q2
µ
logarithmic
divergence
Remark: More general result
-
No spontaneous breaking of continuous symmetries at finite temperature
(Mermin-Wagner Theorem)
-
quasi long range order possible: Kosterlitz-Thouless transition
Effektiver Hamiltonian für kalte Atomgase
•
Bisher: Freie Teilchen im homogenen Kontinuum:
H0 = H
µN̂ =
Z
nach Fourier Transformation
•
q
2m
aq =
Z
µ aq =
eiqx/~ ax ;
x
Z
x
~2 4
2m
a†x
[ax , a†y ] = (x
µ ax
y)
Jetzt: Realistischere Beschreibung für (Alkali) kalte atomare Gase:
-
Fallenpotential: lokale Dichte erfährt lokale potentielle Energie
Htrap =
-
-
Längenskala:
losc
V (x)n̂x , V (x) = 12 m
xp
= 2/(m!)
physikalische Observable: Streulänge a
8⇡~2
g=
a
m
Unser universeller effektiver Hamiltonian:
H=
Z
x
a†x
~2 4
2m
V (x)
x
2 2
Lokale Zweiteilchen-Wechselwirkungen
Z
Z
g
g
Hint =
y)n̂x n̂y =
a†x2 a2x + quadratisch
s (x
2 x
x,y 2
-
•
q
a†q
2
g †2 2o
µ + V (x) ax + ax ax
2
d
contact interaction
Microscopic Origin of the Interaction Term
model potential with
same scattering length
Example: Na atom (Alkali atoms)
• Microscopic scattering physics: Lennard-Jones (LJ) potential
- 1/r12 hard core repulsion: repulsion of electron clouds rrep = O(aB )
- 1/r6 attraction: van der Waals (induced dipole-dipole interaction)
true interatomic
potential U(x)
x
rvdW = (50...200)aB for alkalis: typ. order of magnitude for interaction length scale
The Sodium atom has one valence electron outside a closed shell.
2
2
of the
LJ type
potentials at
• General
The
groundproperties
state has
configuration
1slow
2senergies:
2p6 3s.
- isotropic s-wave scattering dominates; the scattered wave function behaves asymptotically as (x)
We order the interactions according to the importance:
- a is the scattering length. Knowledge of this single parameter is sufficient to describe low
energy scattering!
- within Born approximation, it can be calculated as
Electron
- nucleus and electron - electron interaction
interatomic potential
U (x) system:
Wave function of the atom as a
aBorn
many electron
x
➡ very
unℓsame
(r) scattering length!
different interaction potentialsΨmay
have
the
=Φ
Y (θ, φ)
core
r
ℓm
a/x
The Model Hamiltonian as an Effective Theory
H=
⇧
x
•
•
⇤ †
ax
⇥
2m
⇥
⌅
2
µ + V (x) ax + gn̂x
Efficient description by an effective Hamiltonian with few parameters.
For ultracold bosonic alkali gases, a single parameter, the scattering length a, is
sufficient to characterize low energy scattering physics of indistinguishable particles :
Effective interaction
2
8⇡~
g=
a
m
•
A typical order of magnitude for the scattering length is
•
The validity of the model Hamiltonian is restricted to length scales
•
For bosons, we must restrict to repulsive interactions a > 0 (else: bosons seek solid
ground state, collapse in real space)
•
So far: microscopic description; now: many body scales!
a = O(rvdW ), rvdW (50...200)aB
l & rvdW
Skalenhierarchie verdünnter ultrakalter Gase
Wechselwirkung:
Streulänge
a
phys. Bedeutung:
•
•
•
mittlerer
Teilchenabstand
⌧ d=n
Gültigkeit des eff. Hamiltonians
verdünntes Gas
de Broglie
Wellenlänge
1/3
•
•
⌧
dB
Oszillatorlänge
d. Falle
1/2
⇠T
⇡ losc
Quanten-Degeneriertheit
ultrakaltes Gas
schwache Wechselwirkung:
Wechselwirkungsenergie
a · n = (a/d) · d
2
typische Zahlwerte
length
aB = 5.3 ⇥ 10
2
nm
scattering length
interparticle sep.
de Broglie w.l.
(0.05 ... 0.2)10^3
(0.8 ... 3)10^3
(10 ... 40)10^3
a/aB
d/aB
dB /aB
Bohr radius
trap size
losc /aB
(3 ... 300)10^3
Gross-Pitaevski Gleichung: Herleitung
•
Heisenberg Bewegungsgleichung für den Feldoperator:
➡
~ 4
2m
2
i~@t ax (t) = [ax , H] =
•
µ + V (x) + ga†x ax ax
Nichtlineare partielle OperatorDifferentialgleichung...
Vereinfachungen:
Wechselwirkung:
Streulänge
•
Oszillatorlänge
d. Falle
mittlerer
Teilchenabstand
•
verdünntes Gas:
➡ geringe Quanten-Fluktuationen
➡ erwarte Skalierung:
•
Formalisierung:
-
Operatorentwicklung:
aq=0 ⇡
p
Idealisierung: T=0:
➡ keine thermische Fluktuationen
➡ keine WW, homog.:
N
vollst. Satz orthog.
Basisfunktionen
ax =
0 (x)aq=0
+
X
q (x)aq
=
0 (x)aq=0
+ ax
q6=0
-
Skalierungen (thermodyn. Limes N ~V):
q (x) ⇠ V
1/2
de Broglie
Wellenlänge
aq=0 ⇠ V 1/2
aq6=0 ⇠ V 0
aq=0 =
p
N
Gross-Pitaevski Glg.: “Makroskopische Wellenfunktion”
•
Heisenberg Bewegungsgleichung für den Feldoperator:
~ 4
2m
2
i~@t ax (t) = [ax , H] =
•
•
➡
µ + V (x) + ga†x ax ax
Nichtlineare partielle OperatorDifferentialgleichung...
Vereinfachungen:
ax (t) = '(x, t) + ax (t) ⇡ '(x, t)
⇠1
⇠ V 1/2
'(x, t) :=
0 (x, t)a0
Führende Terme in Heisenberg Glg.:
i~@t '(x, t) =
~2
4
2m
µ + V (x) + g'⇤ (x, t)'(x, t) '(x, t)
Gross-Pitaevski Gleichung:
Bewegungsgleichung der Kondensatamplitude
•
Eigenschaften:
-
klassische Feldgleichung (vgl. klass. Elektrodynamik vs. QED): Materiewelle
-
i.a. nichtlineare Gleichung -> reichhaltiger als Schrödinger Gleichung
“klassisch” im Sinne von makroskopischer Besetzung, nicht
~=0
g = 0: erhalte formal lineare Schrödinger Gleichung -> erwarte quantenmechanisches Verhalten:
“makroskopische Wellenfunktion” '(x, t)
Nichtlinearität und Vortexlösung
•
Gross-Pitaevski Gleichung:
i~@t '(x, t) =
•
~2
4
2m
µ + V (x) + g'⇤ (x, t)'(x, t) '(x, t)
Vortex Lösung (l = 1)
Wechselspiel Quantenmechanik und Nichtlinearität:
quantisierte Vortexlösung
-
uniformer Fall V(x) = 0, suche statische,
zylindersymmetrische Lösung ohne z-Abhängigkeit.
Ansatz:
'(x, t) = '(r, ) = f (r)e
-
f (r)
Eindeutigkeit der Wellenfunktion:
GP Gleichung:
0=
2
f
f” +
2m
r
`
=
i`
2mgn
Heilungslänge
exp. Daten:
ganzzahlig
Dalibard Group (2000)
⇥2 f ⇥
r2
µf + gf 3
-
grosse Distanzen: konstante Lösung, bestimmt chem. Potential
-
separierende Skala: “Heilungs”-Länge, Längenskala der WW-Energie
kleine Distanzen: Kondensatamplitude muss wegen
Zentrifugalbarriere verschwinden (Ursprung: Phasenquantisierung)
a
d
1
⇠=d p
4 ⇡
r
d
a
losc
T
r
Ausblick: Limitationen der Gross-Pitaevski Beschreibung
• Generische Skalenhierarchie und mögliche Verletzung
Wechselwirkung
Teilchenabstand
de Broglie
Wellenlänge
Oszillatorlänge Falle
Feshbach Resonanz
or
Optische Gitterkonstante
• Feshbach Resonanz: Verletzung a/d << 1 möglich: Dichtes degeneriertes System
• Optische Gitter: neue Längen- und Energieskala:
•
•
Gitterkonstante = Wellenlänge des Lichts: hohe Dichte (“Füllung“) möglich
Gittertiefe: Kinetische Energie stärker erniedrigt als potentielle (WW) Energie: “starke Korrelationen”
• NB: Trotz der Verletzung der generischen Skalenrelation bleibt die Beschreibung durch
universelle, einfache Hamiltonians korrekt!
• Beides führt zu starken Korrelationen, die nicht mehr durch GP Glg. beschrieben werden!
Bose and Einstein (1925): Equation below Tc needs modification due to macroscop
occupation of zero mode:
Quantum Fluctuations: Bogoliubov Theory
n=
•
†
⟨a0 a0 ⟩
+
dd q
(2π)d
homogeneous case: plane wave expansion of field
1
q^2
operator
e 2mkB T(
p p
p
0 (x)a
−
1 0 = 1/ V N0 = n0 )
X
p
iqx
- plausible: Bosons can populate
state
with
ax =singlenquantum
+
e
aqarbitrary number
0
-
•
†
macroscopic:
N
=
⟨a
a0 ⟩ = O(N ) ∝ V , i.e. q
extensive
0 phase
0 without
choice of zero
critical temperature:
loss of generalitydetermined by
3
Grand canonical Hamiltonian. Take
fluctuations into account to leading order:
nλthe
dB = g3/2 (1) = ζ(3/2) ≈ 2.612
-
zero order: homogenous mean field reproduced
linear terms: vanishes upon proper choice of the chemical potential (equilibrium condition) µ = gn0
quadratic part:
(2)
HBog
=
X
† q2
aq ( 2m
µ + 2gn0 )aq +
gn0
† †
(a
qa q
2
+ aq a
lectron - nucleus
and
electron
electron
interaction
q6=0
q)
!✓
◆
q
X
ave function of the
electron
system:
aq
1 atom† as a many
+ gn0
gn0
2m
=
(aq , a q )
+ const.
2
†
q
a q
2
gn0
+ gn0
2m
q6=0
unℓ (r)
Yℓm (θ, φ)
Ψ = Φcore
µ = gn0
r
2
th core wave function (filled shell) and u(r) the radial wave function of the valence ele
Quantum Fluctuations: Bogoliubov Theory
•
quadratic Bogoliubov Hamiltonian:
(2)
HBog
=
X
† q2
aq ( 2m
µ + 2gn0 )aq +
Eq
gn0
† †
(a
qa q
2
+ aq a
q)
q6=0
1X †
=
(aq , a
2
q6=0
•
q)
q
2m
+ gn0
gn0
gn0
q2
2m + gn0
!✓
aq
a† q
◆
+ const.
Discussion:
-
•
2
q
Off-diagonal terms: pairwise creation/annihilation out of the condensate
Coupling of modes with opposite momenta
Hamiltonian not diagonal in operator space: diagonalize to find spectrum
and elementary excitations
Remarks:
-
An equivalent approach linearizes Heisenberg EoM around homog. GP mean field
Validity: The ordering principle is given by the power of the condensate amplitude.
Bogoliubov theory is not a systematic perturbation theory in U but becomes good
at weak coupling
-
The operator aq creates a particle in the momentum mode q : aq |vac⟩ =
The operators
satisfy
Similarly, aq annihilates a-particle
in the mode
q. bosonic commutation relation
but with an index label
q).
†
′ ) (cf. ha
]
=
δ(q−q
- The operators satisfy bosonicoperators,
commutation relations,
[a
,
a
q
′
q
†
The
operator
n
=
a
a
measures
the particle n
q
q
q
operators, but with an index label q).
P †
†
number the
is N̂particle
= qa
q aq .
- The operator
number
in the momentum mo
Diagonalization in operator
space: nBogoliubov
transformation
q = aq aq measures
P †
q2
number is N̂ = q aq aq .- The dispersion relation is Eq = !ω = 2m . This is
Bogoliubov Quasiparticles
•
q
†
q
•
⇥
⇥
2
†
′
q
a
u
v
q
q
q
The
trafo
be
canonical,
[α
,
α
]=
δ( −particle
),
- ⇥ The⇥dispersion
relation is Eq = !ω = 2m . This is the energy ′of
a single
†
vq uq
a q
†
′
2
2
• [α
Lower
The trafo be canonical,
, α ′T
] =and
δ(
-
⇠
• At
a finiteofT ,µµathits
zero:
belowqthis Tc the e
• Lower T and study the
behavior
fixed
n (3D):
2
q
2
1 ⇠q
2
2
⇠
=
✏
+
gn
,
✏
=
+
1),
v
=
1)
(
q
q
0
q
E
=
⇠
✏
q
2M
q
2
E
q has
q no
• At a finite T q, µ hits zero: below this T the equation of state
c
Rewrite the Hamiltonian with new operators and do normal ordering
H=V
gn
2
2
1
2
X
(✏q + gn0
Eq ) +
q6=0
•
study
− ),the
thusbehavior
uq − vq =of
1. µ at fixed
The transformation
coefficients
are
chosen
make Tany
• Lower
T and
study
the
behavior
ofoff-diagonal
µ study
at fixed
(2D):
• toLower
and
thencontribution
behavior
ofinµ at fixed
terms of new operators vanish. They can be chosen real and evaluate to
u2q = 12 ( Eqq
•
=
⇥
X
Eq ↵q† ↵q
q6=0
quasiparticle dispersion
Result: Collection of harmonic oscillators. New operators:
elementary excitations on Bogoliubov ground state: quasiparticles
†
↵
- q creates a quasiparticle
qexcitation and ↵q |0Bog i =q0
-
Their dispersion is Eq =
q!0
2gn0 ✏q + ✏2q ! c|q|,
c=
gn0
m
At low momenta, this is linear and gapless
At high momenta, like free particles: quadratic
speed of sound
phonons
particles
Phonon Mode and Superfluidity
•
Landau criterion of superfluidity: frictionless flow
- Gedankenexperiment: move an object through a liquid with velocity v.
- Landau: the creation of an excitation with momentum p and energy ✏p is energetically
unfavorable if
✏p
v < vc =
p
➡ in
•
this case, the flow is frictionless, i.e. superfluidity is present
Weakly interacting Bose gas: Superfluidity through linear phonon excitation
✏p = c|p|, c =
•
q
gn0
m
! vc = c
Free Bose gas: No superfluidity due to soft particle excitations
phonons
p2
✏p =
! vc = 0
2m
➡ Superfluidity is due to linear spectrum of quasiparticle excitations
particles
Summary
•
The basic phenomenon of Bose-Einstein condensation is a statistical effect, not
driven by interactions.
•
Ultracold bosonic quantum gases realize a situation close to that: model
Hamiltonians with weak, local, repulsive interactions.
•
•
Important scale hierarchy:
•
GP mean field equation = nonlinear Schrödinger equation for macroscopic wave
function (hallmark: quantized vortices).
•
Bogoliubov theory encompasses quadratic fluctuations around GP mean field and
explains superfluidity through existence of phonon mode.
a
d
dB
In consequence, such systems are well described by relatively simple
approximations: at T=0, the physics is well understood in terms of GP equation +
quadratic fluctuations.
Idea of Landau Criterion
•
•
moving object
v
⌃0
Consider moving object in the liquid ground state of a system
Question: When is it favorable to create excitations?
•
ground state system
General transformation of energy and momentum under
Galilean boost with velocity v
•
⌃:
w/o moving object
E,
p
0
with moving ⌃
object
:
E0 = E
⌃
total system mass
pv + 12 M v2 ,
p0 = p
Mv
Energy and momentum of the ground state
w/o moving object
E0 ,
with moving object
⌃0 :
E00 = E0 + 12 M v2 ,
•
p0 = 0
p00 =
Mv
Energy and momentum of the ground state plus an excitation with momentum, energy
w/o moving object
⌃:
Eex = E0 + ✏p ,
with moving object
⌃0 :
0
Eex
= E0 + ✏ p
•
pex = p
pv + 12 M v2 ,
p0ex = p
Mv
Creation of excitation unfavorable if
0
Eex
E00
= ✏p
pv
✏p
|p||v| > 0
✏p
) v < vc =
p
p, ✏p