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
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Cold trapped Bosons
Assume N non-interacting bosonic atoms trapped in a harmonic potential
Symmetry of the trap fixes the symmetry of the problem
Single Particle Energies are
Particle number is given by
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Cold trapped Bosons
In the thermodynamic limit N→∞ and taking out the lowest state N0
where
assuming N0→ 0 at the Transition Temperature for BEC yields
This gives the fraction of the condensed atoms below TC
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Cold trapped Bosons
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:
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lower TC for small N
Cold trapped Bosons
Below TC the ground state has a macroscopic occupation
→ Bose-Einstein-Condensation
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
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Cold trapped Bosons
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
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Cold trapped Bosons
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)
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Cold trapped Bosons
Solution depends on ratio of kinetic and interaction energy
(
Attractive interaction
Figures from Dalfovo et al., 1999, Rev. Mod. Phys. 71, 475
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is the oscillator length of the confining potential)
Repulsive interaction
Cold trapped Fermions
Onset of quantum degeneracy is the same for bosons and fermions
Ideal Fermi gas
Classical gas
But the consequence is different!
De Marco et al., 2001, Phys. Rev. Lett. 86, 5409
Bosons → phase transition to BEC
Fermions → Multiple occupation of a state is forbidden → ?
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Cold trapped Fermions
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
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Cold trapped Fermions
Density and Momentum distributions are
(T=0)
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
Reason: Bosons → repulsive Interactions, Fermions → quantum pressure
for Bosons)
Momentum distribution: Width increases with N for Fermions and is independent of N for non-interacting
Bosons
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Cold trapped Fermions
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)
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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
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“)
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Feshbach Resonances
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)
(
is the phase shift)
Going to zero momentum, we get
the
S-Wave scattering length a
 a shows the kind and strength of the interaction of the scattering particles
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Feshbach Resonances
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
Figures: J. J.Sakurai: Mordern Quantum Mechanics
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a<0: Attraction
(no bound states)
a>R: Deep Attraction
(bound states)
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)
Is there a Way to control the Detuning
→ control the Scattering Length?
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,
Feshbach Resonances
Controling the detuning
 Scattering of two atoms with spins s1 and s2
 Relative Orientation of the spins is crucial for the interaction:
Singlet state
If no coupling: Scattering in Triplet potential VT(r)
But: Hyperfine interaction potential is not diagonal
in S = s1 + s2
Triplet state
Antisymmetric, thus coupling singlet (antisymmetric)
and triplet (symmetric) state
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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
Triplet
state
Singlet and Triplet state have different
magnetic moment moments
→ relative Energy can be tuned by magnetic Field
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
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Feshbach Resonances
 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
Controllable
detuning 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
 Hamiltonian is given by H=H0 +V, the only non-zero Matrix Elements of V are
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Feshbach Resonances
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
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Feshbach Resonances
 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
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
 Positive values of a correspond to bound Dimers
→ Bose Gas

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Continuous connection between
Fermi Superfluidity and Bose-Einstein-Condensation
Feshbach Resonances
Fraction of bound Dimers in the state
(|m> is the bound state)
(In the discussed approach of two coupled Square-Wells)
molecule fraction α²
condensate fraction
Fig.: Numerical thermodynamic Calculations by Williams et al., 2004, New J. of Phys. 6, 123
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Feshbach Resonances
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)
with k=0,574
Roberts et al., 2001,
k=0.459±0.012 ±0.054 (exp.,
Phys. Rev. Lett.86, 4211)
November 12, 2009 | Christian Stahl | 24
fig.: Romans et al., 2004, Phys. Rev. Lett. 93, 020405
Gas instabilities
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)
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Gas instabilities
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
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Gas instabilities
System of a fermionic Molecule (composite of a Boson and a Fermion)
colliding with
a) distinguishable (Fermions)
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)
b) Bosons
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.
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)
figures from Zirbel et al., 2008, Phys. Rev. Lett. 100, 143201
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Gas instabilities
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
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Gas instabilities
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
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Gas instabilities
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
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Summary
 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)
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References (excerpt)
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
A. 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
Reserve
critical temperature in a fermionic system
fig. from S. Giorgini et al., 2008, Rev. mod. Phys. 80, 1215
Figures: Mayer-Kuckuck, Kernphysik
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Reserve
Koetsier et al., 2009, Phys. Rev. A 79, 063609
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Reserve
Manybody-Hamiltonian for Bosons
BEC
where
and
small perturbation
, replace
November 12, 2009 | Christian Stahl | 35
by