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Fermion-Fermion and Boson-Boson Interaction at low Temperatures

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### Fermion-Fermion and Boson-Boson Interaction at low Temperatures

Seminar “physics of relativistic heavy Ions”

TU Darmstadt, WS 2009/10

BCS – state: Long range attractive interaction between fermions

BEC: 87Rb above, at and below TC

W. Ketterle, M. W. Zwierlein in Ultracold Fermi Gases, Proceedings of the International School of Physics ”Enrico Fermi”, Course CLXIV, Varenna, 20 - 30 June 2006

Anderson et al., 1995, Science 269, 198

November 12, 2009 | Christian Stahl | 1

Outline

- Cold trapped Fermions and Bosons

• BEC and BCS – states

- Tuning atomic interactions: Feshbach resonances

• Short review on scattering theory

• Resonance scattering

• Interaction potentials between (alkali) atoms

- Gas instabilities close to Feshbach resonances

• Atom loss due to inelastic collisions

November 12, 2009 | Christian Stahl | 2

Cold trapped Bosons

Assume N non-interacting bosonic atoms trapped in a harmonic potential

Single Particle Energies are

Particle number is given by

Symmetry of the trap fixes the symmetry of the problem

November 12, 2009 | Christian Stahl | 3

In the thermodynamic limit N→∞ and taking out the lowest state N0

assuming N0→ 0 at the Transition Temperature for BEC yields

This gives the fraction of the condensed atoms below TC

where

November 12, 2009 | Christian Stahl | 4

Data: dilute 87Rb gas, Ensher et al., 1996, Phys. Rev. Lett. 77, 4984

Deviations of the experimental results from the prediction are due to finite size (particle number ≈ 40k at TC)

First order correction:

lower TC for small N

November 12, 2009 | Christian Stahl | 5

Below TC the ground state has a macroscopic occupation

→ Bose-Einstein-Condensation

Cold trapped Bosons

Shape of the cloud is Gaussian (spherical Trap)

condensed atoms

thermal cloud (non-condensed gas)

5000 non-interacting Bosons at T=0.9TC in the model discussed above and assuming classical Boltzman distribution in the thermal cloud

Dalfovo et al., 1999, Rev. Mod. Phys 71, 463

November 12, 2009 | Christian Stahl | 6

- Inclusion of Interactions
- For cold and dilute gases the main interactions are low energy two-body collisions
- Main contribution comes from s-Wave scattering
- → More in the next Chapter…
- Interaction is characterized by the Scattering Length a

(a>0 → repulsive Interaction, a<0 → attractive Interaction)

- Interatomic Potential V(r’-r) can be described by the parameterization

where

November 12, 2009 | Christian Stahl | 7

- Number of particles in ground state is expressed as
- where is a field with the meaning of an order parameter (“condensate wave function”)
- is governed by the Gross-Pitaevskii-Equation
- which is valid for
- scattering length a much smaller than average particle distance
- N0 >> 1
- Zero Temperature (all particles in condensate state)

November 12, 2009 | Christian Stahl | 8

Solution depends on ratio of kinetic and interaction energy

( is the oscillator length of the confining potential)

Repulsive interaction

Attractive interaction

Figures from Dalfovo et al., 1999, Rev. Mod. Phys. 71, 475

November 12, 2009 | Christian Stahl | 9

Onset of quantum degeneracy is the same for bosons and fermions

But the consequence is different!

Bosons → phase transition to BEC

Fermions → Multiple occupation of a state is forbidden → ?

Ideal Fermi gas

Classical gas

De Marco et al., 2001, Phys. Rev. Lett. 86, 5409

November 12, 2009 | Christian Stahl | 10

For Fermions, the Fermi distribution function has to be used (with local density approximation):

Introduce the single particle energy

and the single particle density of states for the harmonic oscillator

Then, the Particle number for a given spin species fixes the chemical potential

November 12, 2009 | Christian Stahl | 11

Density and Momentum distributions are

where

and

is defined via the Fermi Energy (= the chemical potential at T=0)

Density distribution is similar to that of bosons (Radius changes with for Fermions and for Bosons)

Reason: Bosons → repulsive Interactions, Fermions → quantum pressure

Momentum distribution: Width increases with N for Fermions and is independent of N for non-interacting

Bosons

(T=0)

November 12, 2009 | Christian Stahl | 12

- For Fermions: Occurrence of Superfluidity only due to Interactions
- Assume weak attractive Interaction (a<0):
- • Formation of bound states with exponentially small pairing energy ~
- • Pairs are very large, much larger than the inter-particle distance
- (Cooper, 1956)
- • Pairs show bosonic character
- • Fermionic Superfluidity below
- • Suppression-Factor can easily be about 100
- Superfluidity of electron-gas → Superconductivity
- (Bardeen, Cooper, Shrieffer 1957)

W. Ketterle, M. W. Zwierlein in Ultracold Fermi Gases, Proceedings of the International School of Physics ”Enrico Fermi”, Course CLXIV, Varenna, 20 - 30 June 2006

November 12, 2009 | Christian Stahl | 13

Interactions in dilute gases at low temperatures

Relevant length scales:

- inter-atomic Potential R0

- thermal wavelength

- inverse Fermi wave vector kF-1 or average Particle Distance d for Bosons, respectively

In cold dilute gases:

Interactions take place in form of two-body collisions. The cross section is given by

and the scattering amplitude f can be expanded on Legendre Polynomials („partial waves expansion“)

November 12, 2009 | Christian Stahl | 14

- Low energy regime, that is with the wave vector of the scattered wave
- Centrifugal Barrier strongly suppresses interaction for l≠0:
- → All partial waves with l≠0 are negligible!
- → S-Wave scattering
- S-Wave scattering amplitude is independent of the scattering angle (S-Wave is spherical)
- Going to zero momentum, we get
- the
- a shows the kind and strength of the interaction of the scattering particles

( is the phase shift)

S-Wave scattering length a

November 12, 2009 | Christian Stahl | 15

Expanding the scattering amplitude in a yields

with „effective Range“ of the potential

→ Description becomes independent of the potentials details

Strength and kind of Interaction is expressed by the scattering length a:

0<a<R: Repulsion a<0: Attraction a>R: Deep Attraction

(no bound states) (bound states)

Figures: J. J.Sakurai: Mordern Quantum Mechanics

November 12, 2009 | Christian Stahl | 16

Is there a Way to control the Detuning

→ control the Scattering Length?

Feshbach Resonances

- Assuming 3D square-well potential and (broad Resonance):
- Two-body bound States exist when Potential Depth exceeds ,
- (ER stems from the Energy Uncertainty of the Particle confined in Potential-Well of Size R)
- Binding Energy is
- where V-Vn is the Detuning
- Occurrence of a new bound State → diverging a = Phase Shift is π/2 (Resonance)

November 12, 2009 | Christian Stahl | 17

- Controling the detuning
- Scattering of two atoms with spins s1 and s2
- Relative Orientation of the spins is crucial for the interaction:

If no coupling: Scattering in Triplet potential VT(r)

But: Hyperfine interaction potential is not diagonal

in S = s1 + s2

Singlet state

Triplet state

Antisymmetric, thus coupling singlet (antisymmetric)

and triplet (symmetric) state

November 12, 2009 | Christian Stahl | 18

Singlet bound state

Figures: W. Ketterle, M. W. Zwierlein in Ultracold Fermi Gases, Proceedings of the International School of Physics ”Enrico Fermi”, Course CLXIV, Varenna, 20 - 30 June 2006

Feshbach Resonances

Singlet state

Initial triplet configuration can be scattered into a singlet bound state, if incoming energy and bound state energy match

Triplet state

Singlet and Triplet state have different

magnetic moment moments

→ relative Energy can be tuned by magnetic Field

November 12, 2009 | Christian Stahl | 19

- Coupling of unbound tripled state („open channel“) and bound singlet state („closed channel“)
- Both modeled by spherical well potentials and assuming only one bound state |m>
- Continuum of plane waves of relative momentum k in the open channel are denoted |k>
- Without coupling |m> and |k> are eigenstates of the free Hamiltonian H0
- Coupled state is described by
- Hamiltonian is given by H=H0 +V, the only non-zero Matrix Elements of V are

Controllabledetuning via

B-Field

Figures: W. Ketterle, M. W. Zwierlein in Ultracold Fermi Gases, Proceedings of the International School of Physics ”Enrico Fermi”, Course CLXIV, Varenna, 20 - 30 June 2006

November 12, 2009 | Christian Stahl | 20

- Bound state has a size R far smaller than deBroglie-Wavelength
- For low-Energy scattering we can assume up to an cut-off
- (ER stems from the energy uncertainty of the particle confined in Potential-Well of size R)
- Effects of the coupling
- Resonance Position is shifted by
- Two-Particle-Energy of the bound state is shifted downwards

• far from Resonance

with

• close to the Resonance

- At the Feshbach-Resonance, a diverges as

with

November 12, 2009 | Christian Stahl | 21

- By the application of an magnetic field, a can be tuned from positive to negative values
- At the resonance, a is divergent (“unitarity regime”)

→ System is at the same time dilute in the sense that

R<<d

and strongly interacting in the sense that

a>>d

→ Gas is expected to show universal behavior independent

of the details of the inter-atomic potential, that is all length

scales associated with it disappear. For Example, at unitarity

with

- Negative and small values of a correspond to a Fermi Gas

→ becomes strongly interacting for large a → higher TC

- Positive values of a correspond to bound Dimers

→ Bose Gas

Fig.: W. Ketterle, M. W. Zwierlein in Ultracold Fermi Gases, Proceedings of the International School of Physics ”Enrico Fermi”, Course CLXIV, Varenna, 20 - 30 June 2006

Continuous connection between

Fermi Superfluidity and Bose-Einstein-Condensation

November 12, 2009 | Christian Stahl | 22

molecule fraction α²

Feshbach Resonances

Fraction of bound Dimers in the state (|m> is the bound state)

(In the discussed approach of two coupled Square-Wells)

Fig.: Numerical thermodynamic Calculations by Williams et al., 2004, New J. of Phys. 6, 123

November 12, 2009 | Christian Stahl | 23

For Bosons the situation is similar, but in some ways more difficult:

- Going from negative to positive a and vice versa, a real Phase transition occurs between

a) a phase where only molecule BEC-states exists (far on the molecule side of the Feshbach Resonance)

b) a phase where both molecule and atomic BEC-states exists

Existence of the phase transition (in contrast to the smooth

crossover in the Fermi case) is due to the breakdown of the

normal-phase symmetries in the annihilation operators

and

that do not occur for Fermions.

- For attractive Interaction (a<0) the BEC is not stable

(collapse due to high density). Critical value k is function of

(theoretical by Dalfovo et al., 1996,

Phys. Rev. A 53, 2477 and others)

k=0,574

k=0.459±0.012 ±0.054

with

(exp., Roberts et al., 2001,

Phys. Rev. Lett.86, 4211)

fig.: Romans et al., 2004, Phys. Rev. Lett. 93, 020405

November 12, 2009 | Christian Stahl | 24

Atom Loss due to inelastic Collisions

Dimers of size ~a formed in a Feshbach Resonance are usually in a highly excited rotovibrational state.

In collisions, they can fall into deeper bound states of size R0 (=Interaction Range) releasing Energy

in the order causing the colliding Atom/Dimer that gains this Energy to leave the trap

a

Dimer releases Energy

R0

E > VHO

E<<VHO

Collisions can occur between Dimers and Dimers as well as Dimers and Atoms with the scattering lengths

for Atom/Dimer collisions and for Dimer/Dimer collisions

causing loss rates

for Atoms and for Dimers

Loss coefficients αXY depend on the collision partner (atom/dimer)

and on symmetry (distinguishable / indistinguishable constituents)

November 12, 2009 | Christian Stahl | 25

Loss Rates in Dependence of Scattering Length a close to Resonance (large a)

Simplest case: Dimer consists of 2 Fermions differing only in their Hyperfine state

Collission with a Fermion identical to one of the Dimer’s constituents

αad~a-3.33 and αdd~a-2.55 (Petrov et al., 2004, Phys. Rev. Lett. 93, 090404)

→ Fermi-Fermi-Molecules are stabilized close to a Feshbach Resonance

Bosonic Case: System of a Dimer consisting of 2 Bosons and and a bare bosonic Atom,

two of the particles are identical

αad~a and αdd ~as with s>0 (D’Incao et al., 2006, Phys. Rev. A 73, 030702, Giorgini et al., 2008, Rev. Mod. Phys 80, 1215)

→ Boson-Boson-Molecules are destabilized close to a Feshbach Resonance

November 12, 2009 | Christian Stahl | 26

System of a fermionic Molecule (composite of a Boson and a Fermion)

colliding with

a) distinguishable (Fermions) b) Bosons c) indistinguishable Fermions

Decay rate is further suppressed for indistinguishable Fermions. 1/e Lifetime in this case (Bosons are 87Rb and Fermions 40K) is around 100ms (see inlet)

For large a: Loss rate is enhanced as expected due to bosonic attraction.

For a<1000a0: Loss rate rises, maybe due to the bound Boson being effectively distinguishable from the free one.

Expected Scaling is a-1, measured exponent is

-0.97±0.16 for large a. (Which was done with respect to the molecule size, see inset)

figures from Zirbel et al., 2008, Phys. Rev. Lett. 100, 143201

November 12, 2009 | Christian Stahl | 27

System of a fermionic Molecule (composite of a Boson and a Fermion)

colliding with

a) distinguishable (Fermions) b) Bosons c) indistinguishable Fermions

Bosonic gas is instable due to collisions close to Resonance

Fermionic gas is stabilized close to Resonance

figures from Zirbel et al., 2008, Phys. Rev. Lett. 100, 143201

November 12, 2009 | Christian Stahl | 28

Mass-dependence of loss rates

Considering a system with 2 identical Bosons or 2 Identical Fermions

plus one distinguishable Atom with different mass mX

Excerpt form Table I in D’Incao et al., 2006, Phys. Rev. A 73, 030702, figure from the same paper

November 12, 2009 | Christian Stahl | 29

- Mass-dependence of Dimer loss rates due to Dimer-Dimer collisions in a Fermion system
- For the Dimer-Dimer-Relaxation, the mass ratio is crucial, too:
- For m1/m2=M/m > 12.3, the exponent s in the power law changes sign
- →Stabilization of the Fermion-Fermion Dimers is lost
- For M/m > 13.6 the universal description in terms of a is lost due to Dominance of short-Range Physics

figure from Petrov et al., 2005, J. Phys. B 38, S645

November 12, 2009 | Christian Stahl | 30

- Both Bosons and Fermions show quantum effects when cooled down below
- Bosons macroscopically occupy the ground state → Bose-Einstein-Condensation
- Fermions form Cooper-Pairs and exhibit superfluid behavior
- Interactions at low energy are characterized by the scattering length a
- Variation of a can be obtained by varying an external B-Field → Feshbach-Resonances
- Feshbach-Resonances allow for a continuous transition between weakly and deeply bound states
- → Creation of Fermion-Fermion pairs with bosonic character → BEC
- Bose-Einstein- and Fermi-Statistics cause stability of fermionic gases and instability of bosonic gases
- close to the resonance (diverging scattering length a)

November 12, 2009 | Christian Stahl | 31

- Sections I+II
- S. Giorgini et al., 2008, “Theory of ultrocold atomic Fermi gases”, Rev. mod. Phys. 80, 1215
- W. Ketterle, M. W. Zwierlein, 2006: “Making, probing and understanding ultracold Fermi gases” in Ultracold Fermi Gases, Proceedings of the International School of Physics ”Enrico Fermi”, Course CLXIV, Varenna, 20 - 30 June 2006
- F. Dalfovo et al., 1999: “Theory of Bose-Einstein condensation in trapped gases”, Rev. Mod. Phys. 71, 463
- Section III
- Koetsier et al., 2009: “Strongly interacting Bose gas: Noziéres and Schmitt-Rink theory and beyond”, Phys. Rev. A 79, 063609
- D. S. Petrov et al., 2005: Diatomic molecules in ultracold Fermi gases—novel composite bosons, J. Phys. B 38, S645
- J. J. Zirbel et al., 2009: Collisional Stability of Fermionic Feshbach Molecules, Phys. Rev. Lett. 100, 143201
- J. L. Roberts et al., 2001: Controlled Collapse of a Bose-Einstein Condensate, Phys. Rev. Lett. 86, 4211

November 12, 2009 | Christian Stahl | 32

critical temperature in a fermionic system

fig. from S. Giorgini et al., 2008, Rev. mod. Phys. 80, 1215

Figures: Mayer-Kuckuck, Kernphysik

November 12, 2009 | Christian Stahl | 33

Manybody-Hamiltonian for Bosons

BEC

where and

small perturbation

, replace by

November 12, 2009 | Christian Stahl | 35

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