basic concepts of cold atomic gases -...

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Basic concepts of cold atomic gases

Franco Dalfovo

Lecture #2

ECT* Workshop January 2015

Ψ0 (r, t) = e−iµ t/Ψ0 (r)

By inserting this

into the GP equation

i ∂∂tΨ0 (r, t) = −

2∇2

2m+Vext (r)+ g Ψ0 (r, t)

2%

&'

(

)*Ψ0 (r, t)

one finds the stationary GP equation:

−2∇2

2m+Vext (r)+ g Ψ0 (r)

2$

%&

'

()Ψ0 (r) = µΨ0 (r)

Stationary GP

It gives the ground state of the condensate and all possible stationary states (vortices, solitons, etc.)

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Stationary GP: BEC in a box

Example: 1D box of size L and hard walls. Solution of Schrödinger equation for free particles: GP equation with

Ψ0 = 2n sin(π z / L)

average density

n(z)

ideal gas GP

0>a

Ψ0 →1nΨ0 , z→ z

ξwhere ξ =

2

2mgn=

18πan

healing length

−d 2

dz2Ψ0 (z)+Ψ0

3(z) =Ψ0 (z)The GP equation becomes:

If one can use the boundary conditions: Ψ0 (0) = 0,Ψ0 (∞) =1L >> ξ

and the solution is:

Ψ0 (z) = n tanh z2ξ

In order to stress the role of interaction in GP, let us rescale the units:


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ξ =2

2mgn=

18πan

healing length

Ψ0 (z) = n tanh z2ξ


crucial parameter characterizing the interaction

If GP predictions differ significantly from those of an ideal gas ! L >> ξ

Vext =12mωho

2 r2

Noninteracting ground state: Ψ0 (r)∝ exp(−r2 / aho

2 )

aho = /mωho depends on ħ

Role of interactions:

Using and as units of lengths and energy, and GP equation becomes

[− ∇2 + r 2 +8π (Na / aho ) Ψ2 (r)] Ψ(r) = 2 µ Ψ(r)

If

If 1/ >>hoaNa

1/ <<hoaNa Noninteracting ground state

Thomas-Fermi limit (a>0)

hoa hoω Ψ = N −1/2aho−3/2Ψ0

normalized to 1

Thomas-Fermi parameter

Stationary GP: harmonic trap

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Na / aho >>1If

Ψ0 (r)2= n(r) = 1

g[µ −Vext (r)]

and thus

)()()()(2 00

20

22

rrrr Ψ=Ψ⎥⎦

⎤⎢⎣

⎡Ψ++

∇− µgV

m ext

Thomas-Fermi density profile

In an isotropic harmonic potential the density is an inverted parabola with radius R = aho(15Na / aho )

1/5

The chemical potential is fixed by the normalization to N: µ = gn(0) = (1 / 2)ωho(15Na / aho )

2/5

The Thomas-Fermi limit implies: Na / aho >>1

µ >> ωho , R >> aho , R >> ξ

Stationary GP

from noninteracting to Thomas-Fermi:

Large effects due to interaction at equilibrium; good agreement with experiments

non interacting

non interacting

wave function

exp: Hau et al, 1998

hoaNa /

GP

−2∇2


2$

%&

'

()Ψ0 (r) = µΨ0 (r)

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Note: Thomas-Fermi regime is compatible with diluteness condition

Gas parameter in the center of the trap na3 = µa3 / g = 0.1(N1/6a / aho )

12/5

Thomas-Fermi if Dilute gas if Na / aho >>1 N1/6a / aho <<1

example: a / aho =10−3, N =106

Na / aho =103 N1/6a / aho =10

−2

Gross-Pitaevskii theory is not perturbative even if the gas is dilute (role of BEC)!


Time-dependent Gross-Pitaevskii equation

This equation can be ü  Numerically solved (GP simulations)

ü  Linearized for small oscillations (Bogoliubov equations)

ü  Rewritten in terms of density and velocity (T=0 hydrodynamics)

i ∂∂tΨ0 (r, t) = −

2∇2


2%

&'

(

)*Ψ0 (r, t)

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Time-dependent Gross-Pitaevskii equation Numerical integration. Example: a BEC oscillating in a trap + optical lattice. Onset of instabilities.

time

Time-dependent Gross-Pitaevskii equation Linearization for small oscillations

Ψ0 (r, t) = e−iµ t[Ψ0 (r)+uj (r)e

−iω jt + vj*(r)eiω jt ]Ansatz:

Zero-order in u and v: −2∇2


2$

%&

'

()Ψ0 (r) = µΨ0 (r)

First-order in u and v:

Bogoliubov equations !

ω juj = −2

2m∇2 +Vext −µ + 2gn0

#

$%

&

'(uj + gΨ0

2vj

−ω jv j = −2


#

$%

&

'(vj + gΨ0

2uj

u and v are Bogoliubov quasiparticle amplitudes. ħω are quasiparticle energies. n0 is the ground state density: n0 (r) = Ψ0 (r)

2

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Time-dependent Gross-Pitaevskii equation Linearization for small oscillations

Ψ0 (r, t) = e−iµ t[Ψ0 (r)+uj (r)e

−iω jt + vj*(r)eiω jt ]Ansatz:

Zero-order in u and v: −2∇2


2$

%&

'

()Ψ0 (r) = µΨ0 (r)

First-order in u and v:

Bogoliubov equations !

ω juj = −2


#

$%

&

'(uj + gΨ0

2vj

−ω jv j = −2


#

$%

&

'(vj + gΨ0

2uj

Note: the same equations can be also derived diagonalizing quantum Hamiltonian using Bogoliubov transformations. interacting particles à noninteracting quasiparticles

are real, unless (ωi −ωi*) dr( ui

2− vi

2 )∫ = 0 dr ui2= dr∫ vi

2∫⇒

orthogonality and normalization dr(uiuj* − vivj

*)∫ = δij

iω⇒

If , with solution of Bogoliubov eqs., then the energy change with respect to equilibrium is:

δE = ω j dr uj2− vj

2( )∫Condition of energetic stability ω j dr( uj

2− vj

2)∫ > 0

jjj vu ω,,Ψ0 (r, t) = e−iµt (Ψ0 +uje

−iω jt + vj*eiω jt )

0>Eδ

occurrence of complex solutions dynamic instability

Some properties of u and v:

Bogoliubov equations

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u,v∝ eiq⋅r

ω 2 = 2 q2

2m!

"#

$

%&

2

+ q2c2

λ = 2π / q

ξ = / 2mgn0 = / 2mc

Solutions of Bogoliubov equations in a uniform gas:

Bogoliubov dispersion law

Wavelength of the oscillation:


to be compared with the healing length

c = gn0 /mwith

ε

q

cq

q2/2m

phonons

single-particles

ξ-1

u,v∝ eiq⋅r

ω 2 = 2 q2

2m!

"#

$

%&

2

+ q2c2



Experiments (Davidson et al.):


c = gn0 /mwith

ε

q

cq

q2/2m

phonons

single-particles

ξ-1

Rev. of Mod. Phys. 77, 187 (2005)

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u,v∝ eiq⋅r

ω 2 = 2 q2

2m!

"#

$

%&

2

+ q2c2




c = gn0 /mwith

In nonuniform systems: numerical solutions

ω juj = −2


#

$%

&

'(uj + gΨ0

2vj

−ω jv j = −2


#

$%

&

'(vj + gΨ0

2uj


This equation can be ü  Numerically solved (GP simulations)

ü  Linearized for small oscillations (Bogoliubov equations)

ü  Rewritten in terms of density and velocity (T=0 hydrodynamics)

i ∂∂tΨ0 (r, t) = −

2∇2


2%

&'

(

)*Ψ0 (r, t)

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Time-dependent Gross-Pitaevskii equation Rewritten in terms of density and velocity

i ∂∂tΨ0 (r, t) = −

2∇2


2%

&'

(

)*Ψ0 (r, t)

n= Ψ02

Ψ0 = n eiSvS = /m( )∇S

Write with density

velocity

and insert into

∂∂tn+∇⋅ (vSn) = 0

m ∂∂tvS +∇

12mvS

2 +Vext + gn−2

2m n∇2 n

%

&'

(

)*= 0

Time-dependent Gross-Pitaevskii equation Rewritten in terms of density and velocity

n= Ψ02

Ψ0 = n eiSWrite

velocity

∂∂tn+∇⋅ (vSn) = 0

m ∂∂tvS +∇

12mvS

2 +Vext + gn−2

2m n∇2 n

%

&'

(

)*= 0

These look like hydrodynamic equations, except for quantum pressure

quantum pressure

density with

vS = /m( )∇S

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What is quantum pressure in terms of energy density:

GP kinetic energy:

Ekin =2

2mdr ∇Ψ0∫

2=m2

drvS2n∫ +

2

2mdr ∇ n( )

2

∫

n= Ψ02

quantum pressure

energy of the condensate flow

To be compared with mean-field energy density ≈ gn

vS = /m( )∇S


∂∂tn+∇⋅ (vSn) = 0

m ∂∂tvS +∇

12mvS

2 +Vext + gn−2

2m n∇2 n

%

&'

(

)*= 0

Hydrodynamic equations are obtained when quantum pressure is negligible, i.e., if during the oscillation the density varies over distances λ such that

2 /mλ 2 << gn λ >> ξor

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∂∂tn+∇⋅ (vSn) = 0

m ∂∂tvS +∇

12mvS

2 +Vext + gn−2

2m n∇2 n

%

&'

(

)*= 0

∂∂tn+∇⋅ (vSn) = 0

m ∂∂tvS +∇

12mvS

2 +Vext + gn$

%&

'

()= 0

Hydrodynamic equations of a superfluid at T=0 vs is the superfluid velocity. The velocity field is irrotational! There is no viscosity in these equations! Note: Planck constant has disappeared !

λ >> ξ

Superfluidity

“[…] from a modern point of view, superfluidity is not a single phenomenon but a complex of phenomena” [A.J. Leggett, Rev. Mod. Phys. 73, 307 (2001)]

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Superfluidity

Landau criterion for superfluidity

Case #1: Fluid at rest in the presence of moving walls or impurities, at T=0.

v The dynamics in the moving frame is governed by pv ⋅−= 0HH

The fluid can absorb energy and momentum only through the creation of elementary excitations.

At T=0 excitations can be created only if

vc =min pε(p)p

≠ 0 cvv ≤

ε(p)− v ⋅p < 0

If the fluid remains at rest for

Superfluidity


Case #2: Fluid moving in the presence of walls or impurities at rest, at T=0.

v Similar arguments as before

vc =min pε(p)p

≠ 0 cvv ≤if the current will not decay for (persistent current)

The fluid can absorb energy and momentum only through the creation of elementary excitations.

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Superfluidity


In a dilute (weakly interacting) BEC, the spectrum of Bogoliubov quasiparticles is

The system is superfluid: it flows without dissipation below a critical velocity!!

vc =min pε(p)p

= c

Superfluidity


Case #3: fluid at rest in a rotating bucket

Dynamics in rotating frame governed by H = H0 −Ωlz

Fluid at rest if where

Ω

CΩ<Ω

energy of elementary excitation ≡)(lε≡l angular momentum of elementary excitation

ΩC =minlε(l)l

Landau criterion for superfluidity 0≠ΩC

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Superfluidity


Case #3: fluid at rest in a rotating bucket

Ω

•• •• •……..

The superfluid does not follow the rotation of the bucket for small Ω, but at higher Ω it can lower its energy by nucleating vortices!

Quantized vortices in BEC (Dalibard et al.,

2001)

Quantized vortices in

superfluid helium (Packard et al.

1979)

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Superfluidity and quantized vortices

A BEC behaves as an irrotational superfluid, as a consequence of

n= Ψ02Ψ0 = n eiS

vS = /m∇S

The velocity field is the gradient of a scalar

∇× vS = 0

d∫ ⋅vS = 0For any closed path in a simply connected geometry

with

No rotation!

However, if the system is not simply connected (e.g., it has a hole), than one can choose a path such that

Quantized vortex!

d∫ ⋅vS =m

d ⋅∫ ∇S = k hm

•

Ψ0 = n eiS

ΔS = 2kπ, k = 0,±1,±2,... This condition follows from the single-valuedness of the function


quantized circulation!


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However, if the system is not simply connected (e.g., it has a hole), than one can choose a path such that

or quantized circulation in a toroidal geometry

d∫ ⋅vS =m

d ⋅∫ ∇S = k hm

Ψ0 = n eiS

ΔS = 2kπ, k = 0,±1,±2,... This condition follows from the single-valuedness of the function


quantized circulation!


Vortex lattice in BEC (JILA, 2002)

Abrikosov lattice in type II superconductors


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Many experiments on quantized vorticity in BECs in the last 15 years! A lot of interesting physics: vortex nucleation, vortex arrays, fast rotations and Lowest Landau Level regime, quantum turbulence, BKT transition in 2D, Tkachenko waves, Kelvin modes, persistent currents in toroidal traps, etc.

For a review, see A.Fetter, Rev. Mod. Phys. 81, 647 (2009)


Recent results in Trento: solitonic vortices in BEC of sodium.


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∂∂tn+∇⋅ (vSn) = 0

m ∂∂tvS +∇

12mvS

2 +Vext + gn$

%&

'

()= 0

BEC and superfluid hydrodynamics

Hydrodynamic equations of a superfluid at T=0

From GP equation, neglecting quantum pressure:

Can be rewritten in the form

∂∂tn+∇⋅ (vSn) = 0

m ∂∂tvS +∇

12mvS

2 +Vext +µ(n)$

%&

'

()= 0

local chemical potential

In this form they are more general !

∂∂tS = − 1

2mvS

2 +Vext +µ#

$%

&

'(

this Euler equation is equivalent to the equation for the phase:

∂∂tn+∇⋅ (vSn) = 0

m ∂∂tvS +∇

12mvS

2 +Vext +µ(n)$

%&

'

()= 0

These equations can be obtained, independently of GP, starting from the equation for the bosonic field operator in uniform systems, imposing Galilean invariance, and using a local density approximation for a slowly varying order parameter.

ü  Equations are classical (do not depend on Planck constant). ü  Velocity field is irrotational (role of the phase). ü  Condensate density does not enter HD eqs. ü  HD valid for macroscopic phenomena (length scales >> healing length) ü  HD applicable to both Bose and Fermi superfluids. ü  HD equations depend on equation of state µ(n) (sensitive to quantum correlations, statistics, dimensionality, ...). ü  HD equations can be linearized for small oscillations.

Hydrodynamic eqs of superfluids at T=0

In this context, n is the total density and the superfluid velocity is vS =m∇S

BEC and superfluid hydrodynamics

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Fermions

Ultracold fermions

•  Condensation is only possible for BOSONS. •  FERMIONS behave differently, due to Pauli.

7Li (BEC)

6Li (Fermi see)

(Salomon, ENS, 2001)

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(Rice, 2001)

Observing quantum statistics

BEC Degenerate Fermi gas

if one can use semiclassical approximation

f (r, p) = 1exp[(p2 / 2m+Vext (r)−µ) / kBT ] +1

normalization can be written as:

N =drdp(2π)3∫∫ f (r, p) = 1

2(ωho )3 dε∫ ε 2

exp[(ε −µ) / kBT ]+1

hoBTkN ω>>>> ,1

At T=0:

µ→ EF ; f (r, p)→Θ(p2 / 2m+Vext −EF )

EF = ωho(6N )1/3

Vext =12m ωx

2x2 +ωy2y2 +ωz

2z2!" #$

distribution function

Same dependence as BEC critical temperature kBTc = 0.94 ωhoN

1/3

Ultracold fermions Ideal fermions in a trap

step function

One gets

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Ultracold fermions

kBT<< kBTc = 0.94 ωhoN1/3 kBT<< kBTF = EF =1.82ωhoN

1/3

0.0

0.5

1.0

1.5

2.0

2.5

Opt

Ical

dep

th

radIus arb

T/T = 0.2F

classical gas

TF profile

T/TF=0.77

T/TF=0.27

T/TF=0.11

EF

T=0

kbTF

(Regal et al., JILA)

Ultracold fermions

Density profile

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Ultracold fermions

Another consequence of Pauli exclusion principle:

Fermions of the same atomic species and in the same spin state do not interact in s-wave scattering !!

Just a (almost) free degenerate Fermi gas…

Ultracold fermions

Another consequence of Pauli exclusion principle:

Fermions of the same atomic species and in the same spin state do not interact in s-wave scattering !!

Just a (almost) free degenerate Fermi gas…

BUT what about a mixture of two spin states or two species ? Ø  s-wave scattering is possible and dominates at low temperature

Ø  s-wave scattering length can be tuned thanks to Feshbach resonances

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Ultracold fermions

A dilute mixture of two spin states or two species, interacting via a contact interaction (low-energy s-wave scattering):

H int = g dr Ψ↑+(r)Ψ↓

+(r)Ψ↑(r)Ψ↓(r)∫ g = 4π 2a /mwith

If a < 0 atoms can form bound pairs (bosons) and undergo BCS superfluidity

Pairing between spin-up and -down atoms in momentum space (Cooper pairs). Order parameter characterizing the long range order of two-body density matrix. Quasi-particle excitation spectrum has a gap: BCS critical temperature:

BCS theory

Δ(r) = Ψ↑(r)Ψ↓(r)

ε(p) = Δgap2 +[p2 / 2m)−µ]2

Tc = 0.28TF exp −π

2kF a

"

#$$

%

&''

Ultracold fermions

kF | a |<<1a < 0

Note: this can be very small if kF|a| is small !

Note: this prefactor contains Gorkov and Melik-Barkhudarov corrections to BCS (renormalization of scattering length due to screening effects)

Same physics of weak coupling superconductors!

½ of the energy required to break a pair

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Ultracold fermions




If a < 0 atoms can form bound pairs (bosons) and undergo BCS superfluidity If a > 0 atoms can form bound molecules (bosons) and undergo BEC.

BEC of molecules

Ultracold fermions

a > 0

When the scattering length a is positive, the interaction can produce bound molecules (bosons) of size a. If kF|a| is small, the size of molecules is much smaller than the average distance between them (gas of bosonic dimers). By solving the two-body scattering problem one finds the molecular binding energy: At low T, molecules form a BEC. The critical temperature for a gas of bosons of mass 2m (molecules) at density n is directly related to value of Fermi energy of fermions of mass m at the same density. In a uniform gas: In a harmonic trap: Critical temperature for superfluidity is much higher in BEC than in BCS side where it is exponentially small.

kF | a |<<1

E = − 2

ma2

TBEC = 0.2TF TBEC = 0.5TF

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Ultracold fermions





Ultracold fermions





In both cases one gets deep modifications of many-body wave function. The ideal Fermi gas is no longer a proper starting point.

No perturbative theories

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Ultracold fermions





The scattering length can be tuned at will when the atomic species exhibits Feshbach resonances. BCS-BEC crossover

degenerate Fermi gas

molecular BEC

Fermions Bosons

na3=0.04

na3=0.28

na3= kF|a| = ∞

kF|a|=6

BCS-BEC crossover 6Li atoms @ Innsbruck

Ultracold fermions

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First theoretical approach developed by Leggett (1980). Nozieres and Schmitt-Rink (1985) generalized the gap equation of BCS theory to include the whole resonance region. Theory predicts (Randeria,1993): -  critical temperature and equation of state as a function of dimensionless the parameter -  formation of molecules with energy on the BEC side -  BEC of molecules interacting with scattering length

0.4

0.2

0

0 1-1

BEC

BCS

T C /

T F

-1/kFa 0

akF22 /ma

1<<akF

aaM 2=

Theory of the BCS-BEC crossover

At unitarity the system is strongly correlated but its properties do not depend on the value of scattering length a (not even on the sign of a)!

1>>akFUnitary regime when

The typical length scale of interaction becomes much larger than the size of the gas itself. It disappears from the description of the system.

All lengths disappear from the calculation of energy, chemical potential, thermodynamic functions, etc., except the interparticle distance, which is fixed by the total density of the gas n.

Universality


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1>>akFUnitary regime when

Universality All lengths disappear from the calculation of energy, chemical potential, thermodynamic functions, etc., except the interparticle distance, which is fixed by the total density of the gas n.

Example: The equation of state of a unitary uniform gas at T=0 must exhibit the same density dependence as the ideal Fermi gas (dimensionality arguments rule out different dependences). Thus

µ = (1+β) 2

2m6π 2n( )

2/3

00

≠

=

β

βwith

Ideal Fermi gas

Fermi gas at unitarity

Many-body calculations are needed to determine value of β. Possible comparison with experiments (see experiments on the equation of state at ENS and MIT) The equation of state can be used to determine density profiles, release energy and collective frequencies in Thomas-Fermi approximation.


Mean-field theory of the BCS-BEC crossover

Two-component Fermi gas as at T=0. Many-body Hamiltonian written in terms of fermionic operators Ψ↑

+, Ψ↓+, Ψ↑, Ψ↓

H = dr Ψσ+ (r) −

2∇2

2mσ

−µσ$

%&

'

()∫ Ψσ (r)

σ=↑,↓∑ +

12

drdr 'Ψ↑+∫∫ (r)Ψ↓

+(r ')V ( r− r ' )Ψ↓(r ')Ψ↑(r)

Approximation: include the interaction only in pairing correlations, treated at mean-field level. Ignore direct (Hartree) interaction terms proportional to the averages

s = r− r '

Ψ↑+ Ψ↑ Ψ↓

+ Ψ↓ these would give divergent terms at unitarity

One gets the BCS Hamiltonian

HBCS = dr Ψσ+ (r) −

2∇2

2mσ

−µσ$

%&

'

()∫ Ψσ (r)

σ=↑,↓∑ − dr∫ Δ(r) Ψ↑

+(r)Ψ↓+(r)− (1 / 2) Ψ↑

+(r)Ψ↓+(r)$

%'(+H.c.{ }

where Δ(r) =− ds∫ V (s) Ψ↓(r+ s / 2)Ψ↑(r− s / 2) order parameter

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HBCS = dr Ψσ+ (r) −

2∇2

2mσ

−µσ$

%&

'

()∫ Ψσ (r)

σ=↑,↓∑ − dr∫ Δ(r) Ψ↑

+(r)Ψ↓+(r)− (1 / 2) Ψ↑

+(r)Ψ↓+(r)$

%'(+H.c.{ }

Δ(r) =− ds∫ V (s) Ψ↓(r+ s / 2)Ψ↑(r− s / 2)

Crucial point: this Hamiltonian can be put in diagonal form by replacing particles with quasi-particles (Bogoliubov transformations) ! Ψ↑(r) = [ui (r)αi + vi

*(r)βi+ ]

i∑Ψ↓(r) = [ui (r)βi + vi

*(r)αi+ ]

i∑

HBCS =(E0 −µN )+ εi (αi+αi

i∑ + βi

+βi ) a gas of independent quasi-particles

The quasi-particle operators obey the anti-commutation rules αi,α j+{ }= βi, β j

+{ }= δij

The quasiparticle amplitudes obey dr∫ [ui*(r)uj (r)+ vi

*(r)vj (r)]= δij


Ψ↑(r) = [ui (r)αi + vi*(r)βi

+ ]i∑

Ψ↓(r) = [ui (r)βi + vi*(r)αi

+ ]i∑

The diagonalization of the Hamiltonian gives the equations for the quasi-particle amplitudes. In general, including the case of a Fermi gas in an external potential, these equations have the form

H0 (r) Δ(r)

Δ*(r) −H0 (r)

#

$

%%

&

'

((

ui (r)vi (r)

#

$

%%

&

'

((= εi

ui (r)vi (r)

#

$

%%

&

'

((

Bogoliubov – de Gennes equations

with H0 (r) = −2∇2

2m+Vext (r)−µ

The order parameter can be obtained by means of a self-consistent procedure (a proper regularization of the interaction is needed):

Δ(r) = −g ui (r)i∑ vi*(r)

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H0 (r) Δ(r)

Δ*(r) −H0 (r)

#

$

%%

&

'

((

ui (r)vi (r)

#

$

%%

&

'

((= εi

ui (r)vi (r)

#

$

%%

&

'

((


H0 (r) = −2∇2

2m+Vext (r)−µ

Note: remarkable similarity with Bogoliubov equations for excitations (quasi-particles) in BECs

H0 (r) gΨ02

−gΨ0*2 (r) −H0 (r)

#

$

%%

&

'

((

ui (r)vi (r)

#

$

%%

&

'

((= εi

ui (r)vi (r)

#

$

%%

&

'

((

H0 (r) = −2∇2

2m+Vext (r)−µ + 2g Ψ0 (r)

2

Fermions

Bosons

ω juj = −2


#

$%

&

'(uj + gΨ0

2vj

−ω jv j = −2


#

$%

&

'(vj + gΨ0

*2uj

Note: some slides ago, we wrote the same eqs in this form:


Fermions Bosons vs.

Bogoliubov equations for BECs and Bogoliubov-de Gennes for fermions are two implementations of the same idea: Bogoliubov transformation, i.e. , diagonalization of the many-body Hamiltonian by replacing particle operators by quasi-particle operators.

one of the key concepts in many-body theories

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Fermions Bosons vs.

Bogoliubov equations for BECs and Bogoliubov-de Gennes for fermions are two implementations of the same idea: Bogoliubov transformation, i.e. , diagonalization of the many-body Hamiltonian by replacing particle operators by quasi-particle operators.

Important difference: Due to macroscopic occupation of a single state, in BEC the order parameter is obtained expanding the Hamiltonian in where . At zero-order one has the GP equation for . At first-order one gets the Bogoliubov equations for bosonic excitations. Conversely, in the BCS-BEC theory for fermions, there is no zero-order. The Bogoliubov-de Gennes equations give the order parameter itself, together with all fermionic excitations (but no bosonic excitations, such as phonons…). The ground state is the vacuum of quasi-particles.

)(ˆ)()(ˆ 0 rrr Ψ+Ψ=Ψ δ)(ˆ rΨδ)(0 rΨ


H0 (r) Δ(r)

Δ*(r) −H0 (r)

#

$

%%

&

'

((

ui (r)vi (r)

#

$

%%

&

'

((= εi

ui (r)vi (r)

#

$

%%

&

'

((


H0 (r) = −2∇2

2m+Vext (r)−µ

Fermions

The mean-field theory based on BdG equation gives the correct limit of free fermions in external potentials for a=0-

gives the correct GP equation for a BEC of molecules of mass 2m for a=0+

gives a smooth crossover from BCS to BEC, including unitarity. it is accurate enough for many purposes. It is misses important corrections to the ideal Fermi gas for small negative a. It gives the wrong scattering length for molecule-molecule interaction.

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H0 (r) Δ(r)

Δ*(r) −H0 (r)

#

$

%%

&

'

((

ui (r)vi (r)

#

$

%%

&

'

((= εi

ui (r)vi (r)

#

$

%%

&

'

((


H0 (r) = −2∇2

2m+Vext (r)−µ


n(r) = 2 | vi (r) |2

i∑

order parameter

density

Can be solved numerically by iterations


H0 (r) Δ(r)

Δ*(r) −H0 (r)

#

$

%%

&

'

((

ui (r)vi (r)

#

$

%%

&

'

((= εi

ui (r)vi (r)

#

$

%%

&

'

((


H0 (r) = −2∇2

2m+Vext (r)−µ


n(r) = 2 | vi (r) |2

i∑

order parameter

density

Example: a dark soliton in a uniform Fermi superfluid Antezza et al., PRA 76, 043610 (2007)

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H0 (r) Δ(r)

Δ*(r) −H0 (r)

#

$

%%

&

'

((

ui (r)vi (r)

#

$

%%

&

'

((= εi

ui (r)vi (r)

#

$

%%

&

'

((


H0 (r) = −2∇2

2m+Vext (r)−µ


n(r) = 2 | vi (r) |2

i∑

order parameter

density

As in the case of GP equation, they can be written as equations for the time evolution (time-dependent BsG equations)


H0 (r) Δ(r, t)

Δ*(r, t) −H0 (r)

#

$

%%

&

'

((

ui (r, t)vi (r, t)

#

$

%%

&

'

((= i

∂∂t

ui (r, t)vi (r, t)

#

$

%%

&

'

((

Time-dependent Bogoliubov – de Gennes equations

H0 (r) = −2∇2

2m+Vext (r)−µ

Δ(r, t) = −g ui (r, t)i∑ vi*(r, t)

n(r, t) = 2 | vi (r, t) |2

i∑Also these equations can be solved numerically.

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35


H0 (r) Δ(r, t)

Δ*(r, t) −H0 (r)

#

$

%%

&

'

((

ui (r, t)vi (r, t)

#

$

%%

&

'

((= i

∂∂t

ui (r, t)vi (r, t)

#

$

%%

&

'

((


H0 (r) = −2∇2

2m+Vext (r)−µ

Δ(r, t) = −g ui (r, t)i∑ vi*(r, t)

n(r, t) = 2 | vi (r, t) |2

i∑

Example: a soliton oscillating in a trapped Fermi gas

Scott et al., PRL 106, 185301 (2011) ) tim

e


H0 (r) Δ(r, t)

Δ*(r, t) −H0 (r)

#

$

%%

&

'

((

ui (r, t)vi (r, t)

#

$

%%

&

'

((= i

∂∂t

ui (r, t)vi (r, t)

#

$

%%

&

'

((


H0 (r) = −2∇2

2m+Vext (r)−µ

Δ(r, t) = −g ui (r, t)i∑ vi*(r, t)

n(r, t) = 2 | vi (r, t) |2

i∑

Important remark: time-dependent BdG include both bosonic and fermionic degrees of freedom!

time

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36


H0 (r) Δ(r, t)

Δ*(r, t) −H0 (r)

#

$

%%

&

'

((

ui (r, t)vi (r, t)

#

$

%%

&

'

((= i

∂∂t

ui (r, t)vi (r, t)

#

$

%%

&

'

((


H0 (r) = −2∇2

2m+Vext (r)−µ

Δ(r, t) = −g ui (r, t)i∑ vi*(r, t)

n(r, t) = 2 | vi (r, t) |2

i∑


time

Here a soliton becomes unstable because pair-breaking near the soliton (fermionic excitations) generates phonons (bosonic collective excitations).


H0 (r) Δ(r, t)

Δ*(r, t) −H0 (r)

#

$

%%

&

'

((

ui (r, t)vi (r, t)

#

$

%%

&

'

((= i

∂∂t

ui (r, t)vi (r, t)

#

$

%%

&

'

((


H0 (r) = −2∇2

2m+Vext (r)−µ

Δ(r, t) = −g ui (r, t)i∑ vi*(r, t)

n(r, t) = 2 | vi (r, t) |2

i∑


time

Here two solitons collide producing sound waves (bosonic excitations) as a result of inelastic processes induced by the dynamics of fermionic quasiparticles.

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Other theories for the BCS-BEC crossover

•  Effective GP equation (local chemical potential of Fermi gas replacing the nonlinear term of GP equation; only bosonic degrees of freedom; no pair-breaking)

•  Density Functional Theory beyond BdG (free parameters fixed from ab initio calculations)

•  Diagrammatic many-body approaches (T-matrix, etc.)

•  Exact asymptotic results at unitarity (Tan contact) •  and, of course, Monte Carlo simulations and others…

basic concepts of cold atomic gases -...

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