BoseEinstein Condensation and Superfluidity

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Trento PhD course on Ultracold atomic Fermi
gases (February-March 2008)
Lecture 2
BEC gases and the role of the interaction the
Gross-Pitaevskii equation
Sandro Stringari
University of Trento
CNR-INFM
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1-body density matrix and long-range order
(Bose field operators)
Relevant observables related to 1-body density
- Density
- Momentum distribution
In uniform systems
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If n(p) is smooth function
If
Smooth function
Off-diagonal long range order (Landau,
Lifschitz, Penrose, Onsager)
Example of calculation of density matrix in
highly correlated many-body system liquid
He4 (Ceperley, Pollock 1987)
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Long range order and eigenvalues of density
matrix
BEC occurs when . It is then
convenient to rewrite density matrix by
separating contribution arising from condensate
For large N the sum can be replaced by integral
which tends to zero at large distances.
Viceversa contribution from condensate remains
finite up to distances fixed by
size of
BEC and long range order consequence of
macroscopic occupation of a single-partice state.
Procedure holds also in non uniform and finite
size systems.
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Diagonalization of 1-body density matrix in a
small droplet of liquid He4 at T0
(Lewart, Pandharipande and Pieper, Phys. Rev. B
(1988))
In bulk condensate fraction is 0.1. In the
droplet the fraction is larger because of
surface effects. Condensate density is close to
0.1 in the center of the droplet. It increases
and reaches 1 at the surface.
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ORDER PARAMETER
Diagonalization of 1-body density matrix permits
to identify single-particle wave functions
. In terms of such functions one can write
field operator in the form
If one can use Bolgoliubov
approximation (non commutativity
inessential for most physical
properties within 1/N approximation).
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Bogoliubov approximation is equivalent to
treating the macroscopic component of the field
operator as a classical field (true also in
liquid helium)
From field operator to classical field
thermal and quantum fluctuations
with
Usually fluctuations are small in
dilute gases at T0 field operator is
classical object (analogy with classical limit of
QED, see later). In helium quantum fluctuations
are instead always crucial
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Order parameter

• Complex function
• Defined up to a constant phase factor
• S determines superfluid velocity
• (irrotationality, see Lecture 6)
• Corresponds to average
  where average means
  
• 
  
• For stationary configurations
• time evolution is hence fixed by chemical
  potential

Chemical potential fundamental parameter in
Bose-Einstein condenstates. Fixes time evolution
of the phase
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Behaviour of BEC in non interacting gas
(grand canonical ensemble)
value of is fixed by normalization condition
BEC starts when chemical potential takes minimum
value, so close to (
) that occupation number of i0 state becomes
large and comparable to N
If for igt0 one can
replace with and occupation number of
i-state does not depend any more on N
Mechanism of BEC
number of atoms out of the condensate
depends only on T (not on N)
Condition fixes value of critical
temperature
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3D gas in harmonic potential
BEC starts at
If one can transform sum
into integral (semiclassical approximation) Integ
ration yields ( increases with T,
independent of N) Condition then
yields
and
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CONDENSATE FRACTION (Jila 96) EXPERIMENTAL
EVIDENCE FOR PHASE TRANSITION
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ROLE OF INTERACTIONS ON BECSOME QUESTIONS

• Do interactions modify shape of order parameter?
• Do interactions reinforce or weaken BEC?
• Do they enhance or decrease critical temperature?
• Can BEC be fragmented? (more than one s.p. state
  with macroscopic occupancy?)

No general answer Effects depend on
dimensionality, sign of interaction, nature of
trapping (harmonic, double well,periodic..)
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Equation for order parameter can be derived
starting form equation for field operator
Replacing with classical field
in interaction term
requires proper procedure if V contains short
range components
Simple procedure is applicable (in 3D) if i)
range of force and s-wave scattering length a are
much smaller than distance d between
particles ii) temperature is sufficiently
low iii) only macroscopic variations of
are considered (variation along distances
much larger than a)
Only low energy two-body scattering properties
are relevant for describing the many-body
problem.
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IN PARTICULAR - scattering length a is the only
relevant interaction parameter. - Equation for
order parameter is properly derived by replacing
V with effective potential
where is relevant coupling
constant of problem.
Equation for order parameter becomes
(Gross-Pitaevskii)
density
- diluteness (quantum
fluctuations negligible) - low temperature
(thermal fluctuations negligble)
assumptions equivalent to treating field
operator like a classical field
(density
coincides with condensate density)
For non dilute gases and/or finite T one has

(density does not coincide with

total density, see Lecture 4)
condensate
total
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• Gross-Pitaevskii (GP) equation for order
  parameter plays role
• analogous to Maxwell equations in classical
  electrodynamics.
• Condensate wave function represents classical
  limit of
• de Broglie wave (corpuscolar nature of matter
  no longer important)

Important difference with respect to Maxwell
equations GP contains Planck constant
explicitly. Follows from different dispersion
law of photons and atoms
from particles to waves
photons
atoms
particle (energy)
wave (frequency)
GP eq. is non linear (analogy with non linear
optics) GP equation often called non linear
Schroedinger equation Equation for order
parameter is not equation for wave function
(also equation for field operator is non linear).
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• Gross-Pitaevskii equation admits several types of
  solutions
• Stationary solutions (ground state, vortices,
  solitons)
• Time dependent solutions
• - small amplitude oscillations (elementary
  excitations)
• - large amplitude solutions (e.g. expansion)

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Example 1. BEC in a box (hard wall boundary
conditions)
Non interacting ground state
L size of box
Wave function vanishes at the border
ideal gas
GP theory
healing length
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BEC in box. Role of interactions.
Gross-Pitaevskii equation can be rewritten in
dimensionless form
where
healing length (agt0)
Gross-Pitaevskii eq.
If one can use boundary
conditions
Solution
Healing length crucial parameter
characterizing the interaction If
the system can be considered uniform (except
near the boundary)
Interactions deeply modifying predictions of
ideal gas
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Time dependent Gross-Pitaevskii equation
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From Gross-Pitaevskii to Bogoliubov equations
Elementary excitations of the condensate can be
studied in all regimes (not only in HD) by
solving linearized GP equation with ansatz
After linearization
Bogoliubov equations
Bogoliubov equations have been derived in the
framework of a theory for a classical field (time
dependent Gross-Pitaevski theory)
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Some properties of Bogoliubov equations
i) is real, unless
(occurrence of complex solutions dynamic
instability)
ii)
(orthoganility) if

• For each solution
• there exist another solution with
• (the two solutions describe the same
  physical oscillation)

iv) If
, with
solution of Bogoliubov eqs., energy change
with respect to equilibrium is given by
Condition of energetic stability
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Solution of Bogoliubov equations in uniform matter
In uniform matter one finds solutions of the
form and dispersion law
Bogoliubov dispersion law
Wave length of the oscillation Healing length
If quantum pressure is
negligible and (phonon)

If quantum pressure dominates
and (free particle)
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Density response function in uniform matter
By adding density dependent perturbation of the
form
the linearized solutions of time dependent GP
eqs. can be written as with u, v fixed by
Response function is defined by relationship
In our case we find
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Simple algebraic result for density response
function

• - Poles of response function coincide with
  Bogoliubov dispersion law
• Static response given by compressibility sum
  rule
• at large frequency exact f-sum rule result
• imaginary part dynamic structure factor
• from which one extracts result
• for static structure factor
  
• (S(q) directly related to density
  fluctuations)

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Piano del corso
Lecture 2. BEC gases and the role of the
interaction The Gross-Pitaevskii equation
Lecture 3. Bose-Einstein condensates in harmonic
traps
Lecture 4. Ideal Fermi gas and the role of the
interaction
Lecture 5. BEC-BCS crossover and the Bogoliubov
de Gennes equations
Lecture 6. Interacting Fermi gases in harmonic
traps
Lecture 7. Superfluidity and hydrodynamic
behavior
Lecture 8. Rotating Fermi superfluids
Lecture 9. Spin polarized Fermi gases
Lecture 10. Fermi gases in periodic potentials