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Sound velocity and multibranch Bogoliubov - Anderson modes of a Fermi superfluid along the BEC-BCS crossover. Tarun Kanti Ghosh Okayama University, Japan In collaboration with Prof. K. Machida Ref.: Physical Review A 73 , 013613 (2006) + unpublished results. Outline of this talk: part-I.

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Sound velocity and multibranch Bogoliubov -Anderson modes of a Fermi superfluid along the BEC-BCS crossover

Tarun Kanti Ghosh

Okayama University, Japan

In collaboration with Prof. K. Machida

Ref.: Physical Review A 73, 013613 (2006) + unpublished results

outline of this talk part i
Outline of this talk: part-I
  • Difference between bosons & fermions
  • What is Bose-Einstein condensation (BEC) & Bardeen-Cooper-Schriffer state (BCS) ?
  • Two-component Fermi gases
  • Brief introduction of scattering theory & Feshbach resonance
  • BEC-BCS crossover
outline of this talk part ii
Outline of this talk: part-II
  • Hydrodynamic equations of motion in the crossover regime
  • Sound velocity along the crossover
  • Comparison with ongoing experimental result at Duke univ.
  • Dynamic structure factor calculation and discussion of Bragg spectroscopy to analyze multibranch Bogoliubov-Anderson spectrum
bose einstein vs fermi dirac
Bose-Einstein vs Fermi-Dirac



harmonic trap


S. N. Bose

A. Einstein

E. Fermi

P. A. M. Dirac



experimental signature of fermi pressure
Experimental signature of Fermi pressure





Boltzmann distribution

high temperature: classical gas

-: bosons

+: fermions

intermediate temperature

very low temperature: effect

of Fermi pressure due to Pauli


Truscott et al. Science 2001


Inter particle distance d

density n

wave-particle duality

Many particle system can be

described by a SINGLE


why many alkali atoms are bosons
Why many alkali atoms are bosons?

Alkali atoms

All alkali atoms have only one electron in the outer “s” shell


Rubidium Rb

electronic spin: S=1/2

nuclear spin: I=3/2

Total spin: I+S= 1 or 2


Sodium Na

hence it behaves like a bosons


Lithium Li

First experimental observation of BEC

condensate is much dilute compared

to air

E. A. Cornell et al., Science 1995.


Bardeen-Cooper-Schriffer (BCS) state

bare electron-electron

interaction is repulsive

Phonon mediated

exchange interaction

induces attractive

interaction between

two electrons

Binding energy

Critical temperature


Trapped atomic Fermi gases



Duke Univ. -- J. E. Thomas

MIT Cambridge -- W. Ketterle

ENS Paris -- C. Salomon

Rice Univ. -- R. Hulet

Innsbruck Univ. – R. Grimm

JILA Bouldar -- D. Jin

ETH Zurich -- T. Esslinger

LENS Florence -- M. Inguscio

basic scattering theory without spin degrees of freedom
Basic scattering theory (without spin degrees of freedom)

Lennard Jones potential

Model potential


distance r


Basic length scale:

~ 1-10 nm

van der Waals potential



& Smith

Scattering length

  • we have to exploit the presence of hyperfine state to make large
  • scattering length and hence a bound state of two atoms
spin dependent atom atom interaction
Spin dependent atom-atom interaction

total spin of two valence electrons is either 1 (triplet state ) or 0 (singlet state)

Spin dependent atom-atom interaction:

Spin Hamiltonian:

Hyperfine interaction

Zeeman energy

why two component fermi gas
Why two-component Fermi gas?
  • At low temperature, s-wave scattering contribution is large, but it does not
  • arise between identical fermions, it can occur between atoms with different
  • values of
  • Consider two hyperfine state of with equal number N,
  • say |1/2,1/2> & |1/2,-1/2>

“spin up”

“spin down”

feshbach resonance
Feshbach Resonance

Scattering length

D: coupling between

two channels



  • Many molecular bound state in S channel
  • Continuum energy in T channel falls within the bound state energy in S channel
  • Energy difference between T and S channels can be tuned by magnetic field
  • When total energy of two colliding atoms in T is close to the bound state
  • energy in S, the effective scattering length becomes very large and two
  • colliding atoms in T channel forms a bound state in S channel

when a > o, binding energy of a pair of atoms:


Space-Time diagram for Feshbach resonance




Bound state




These molecules are weakly bound but very stable

1 msec – 20 sec !!!

Long life time:

bec bcs crossover
BEC-BCS Crossover

Regal & Jin

PRL 2000



Scattering length:




  • The bound state in interacting Fermi gases are bosonic in nature, hence
  • can Bose condense, just like a Bose atoms can
  • From two-component Fermi system, one can go from molecular BEC to
  • BCS state through the strongly interacting regime (unitarity regime) by
  • changing external magnetic field

So far what we have learned?

  • Take two different hyperfine states of fermions with equal number
  • Apply magnetic field and tune the scattering length accordingly
  • Interaction between two atoms can be either attractive or repulsive,
  • depending on the external magnetic field
  • For large repulsive interaction, tightly bound bosonic pairs will form and
  • condense at very low temperature
  • For attractive interaction, two different kind of fermions will form a loosely
  • bound ATOMIC COOPER pair
  • When magnitude of the scattering length is very large, the system behaves
  • like a free Fermi gas, since the scattering length drops out from the problem

Black box

Molecular BEC

Strongly interacting regime

BCS state

external magnetic field


Note that we do not need any

phonon mediated attractive

interaction, we have already

attractive interaction between

two alkali atoms


BCS regime

Pairing energy

Chemical potential

Size of the Cooper pair in coordinate space is larger than inter atom distance

Loosely bound pairs


Unitarity limit

Relevant length scale:

behaves like a free Fermi gas

How to measure b?

Fermi pressure stabilizes

the cloud against collapse,

similar to neutron star

high temperature superfluidity

a new kind of superfluid state



Molecular BEC regime

Chemical potential:

Molecular scattering length:

Petrov et al. PRL 2004

Molecular density:

Tightly bound pairs

are 2 component fermions really superfluid
Are 2-component fermions really superfluid?

Hallmark of superfluidity, be it bosonic or fermionic,

is the presence of quantized vortices

Ketterle et al., Nature 2005



Theoretical approaches

Eagles (1969) – Leggett (1980): BCS state at T=0, Cooper pairs molecules

Nozieres, Schmitt-Rink (1985) – Randeria et al.: finite T,

Simplest crossover theory

Qualitatively correct

Quantitatively wrong: in BEC regime

Unitarity limit

equation of state
Equation of State

Ground state energy per

particle along the crossover

MC:Giorgini et al. PRL 2005

Fit:Manini and Salasnich

PRA 2005


Chemical potential:

Manini & Salasnich PRA 2005

hydrodynamic equations of motion
Hydrodynamic Equations of Motion

Schrodinger equation of a Fermi superfluid along the BEC-BCS crossover

order parameter of the

composite bosons

Long cigar shaped trap:

superfluid velocity

Phase ()-density (n) representation:

Continuity equation:

Euler equation:


y << -1

y ~ 0

y >> 1

Power-law form of the

chemical potential:


Linearizing around the equilibrium:

Equilibrium density profile:


Energy spectrum:

Density fluctuation:

: Jacobi polynomial of order n

Dipole ( n=0, m =1):

Independent of interaction strength

It satisfy Kohn’s theorem


Matrix elements:

1) Sound velocity

2) Multibranch Bogoliubov-Anderson modes:

Each discrete radial modes are propagating along the symmetry axis

similar to electromagnetic wave propagation in a waveguide

sound velocity
Sound velocity

Sound velocity in non-uniform system:

Uniform system:


Smooth crossover

on resonance!



Comparison of sound velocity (in units of Fermi velocity)

Sound velocity in atomic (Bose/Fermi) system: mm/sec ~ cm/sec

Sound velocity in Helium 4 ~ 220 m/sec

Atomic systems are really dilute!!

sound excitation by a pulse of repulsive potential
Sound: Excitation by a pulse of repulsive potential

Slice of green

light (pulsed)


hold, release & image


Sound excitation:

Trapped atoms

speed of sound u 1 in the bec bcs crossover

Mean-field theory

of Ghosh & Machida

PRA 2006

Speed of sound, u1in the BEC-BCS crossover

system becomes very hot

during sound propagation

Also supports

multibranch bogoliubov anderson spectrum
Multibranch Bogoliubov-Anderson spectrum

BA modes are absent in usual electronic superconductors due to long-range


weight factors
Weight factors

Weight factors determine how many

modes are excited for a given value

of k

dynamic structure factors dsf
Dynamic structure factors (DSF)

Location of the peak determines the excitation energy

bragg spectroscopy
Bragg spectroscopy

z axis

Bragg potential:

Time duration of

the Bragg pulses:


bragg spectroscopy of a weakly interacting bec
Bragg spectroscopy of a weakly interacting BEC

Davidson et al. PRL 2003

Wizemann Institute of Science, Israel

future plans
Future plans
  • Finite temperature: superfluid + normal components,
  • study the first and second sound velocity
  • Unequal populations of two kind of hyperfine states.
  • Bose-Fermi mixture. i) Phase separation between superfluid
  • and normal component ii) Phase transition from superfluid to
  • normal component when the difference between two components
  • are increased. (Pauli limited phase transition)
  • Apply optical lattices into the fermionic superfluid and study the dynamical instability phenomena in this new kind of superfluid
  • Atom Lasers & Atom Chips
  • Quantum Hall effect in Graphene
  • Brief overview of current experiments on ultra-cold atomic gases
  • Mechanism of Feshbach resonance
  • BEC-BCS crossover
  • Compared predicted sound velocity with the ongoing experimental results
  • Complete excitation spectrum of an elongated Fermi superfluid along the crossover
  • Results of dynamic structure factors and Bragg spectroscopy to measure MBA modes