Download Degenerate Fermi Gases

Degenerate Fermi Gases
Alessio Recati
CNR-INO BEC Center/
Dip. Fisica, Univ. di Trento (I)
BEC Group (28 members):
Theory
Experiment
4 UNI staff
3 CNR staff 2 CNR staff
7 PostDoc
7 PhD
3 Master
2 Master
Trento
Algiers
Temperature
Temperaturescale
scale
Ultra-cold Gases
1012-1015 at/cm3
Bosons and Fermions
“Bosons like
staying together,
in the same state”
“Fermions hate
staying together,
in the same state”
Bose-Einstein statistics (1924)
Fermi-Dirac statistics (1926)
Dilute gases: 1995, JILA, MIT
Dilute gases: 1999, JILA
IDEAL
FERMI GASES
Non-interacting Gas at T=0
The Hamiltonian of a non-interacting gas in second quantization
(although we could discuss everything in terms of particle occupation number)
Occupation number: for fermions
Single-particle energy states
Homogeneous case
GROUND-STATE: fill up the momentum space till the Fermi momentum simply defined by
The Fermi
energy is
In this case the chemical potential is just the Fermi energy:
Non-interacting Gas at T=0
The momentum distribution is
But what is the GROUND-STATE structure in real space?
Density
Pair correlation function
Non-interacting Gas at T=0
HARD WALLS: the GROUND-STATE structure in real space
Density
n(x)/N
Friedel
Oscillations:
N=3, 10, 100
x/L
Excitations at Fixed N - particle-hole excitation
k+q
k
;
The excitation energy
of the p-h excitation is linear
for small |q|
Since the excitation spectrum
depends on the actual state of
the system its measurement (if
possible)
is a powerful way to
get an insight in the system's state
1p-1h
continuum
T=0 harmonically trapped Fermi gas
Experimentally the atoms are trapped by means of electromegnetic fields properly tailored
Typically the confining potential is harmonic-like
Cooling & Trapping neutral cold gases
1. Laser light pressure
2. Electric and/or magnetic confinement: harmonic traps
3. Evaporative cooling (a.k.a. cup of coffee cooling)
From C. Salomon (ENS)
Cooling & Trapping neutral cold gases
1. Laser light pressure
2. Electric and/or magnetic confinement: harmonic traps
3. Evaporative cooling (a.k.a. cup of coffee cooling)
From C. Salomon (ENS)
T=0 harmonically trapped Fermi gas
Experimentally the atoms are trapped by means of electromegnetic fields properly tailored
Typically the confining potential is harmonic-like
T=0 harmonically trapped Fermi gas
Experimentally the atoms are trapped by means of electromegnetic fields properly tailored
Typically the confining potential is harmonic-like:
=
E
For large N it looks like
a simple function,
the Friedel oscillations
are almost irrevelant
F
T=0 harmonically trapped Fermi gas
Thomas-Fermi (TF) or Local Density Approximation (LDA)
For large particle number N we can assume that the system is locally uniform,
i.e.,
Normalizing to N one finds the (global)
chemical potential to be:
x
T=0 harmonically trapped Fermi gas
Thomas-Fermi (TF) or Local Density Approximation (LDA)
For large particle number N we can assume that the system is locally uniform,
i.e.,
Normalizing to N one finds the (global)
chemical potential to be:
GENERALLY: (for many purposes) one studies the equation of state for an homogeneous gas
and apply LDA to study the trapped system,
where different (homogeneous) phases can coexist
Time-Of-Flight (TOF) measurement
n(x)
Open the trap and
observe the cloud
after a certiain tof:
get the
Momentum distribution!
n(p)
with
p=mr/t
Fermi see vs. Bose-Einstein condensate
2001
ENS
(Paris)
104 7Li atoms, in thermal equilibrium with
104 6Li atoms in a Fermi sea
Quantum degeneracy: T= 0.28 mK = 0.2(1) TC= 0.2 TF
E
F
Observing the Fermi Surface
Nicely the
gas expand
like a cube
Collective oscillations of a Trapped Gas
Collective Oscillations:
exp. (MIT '97): ω=1.57ω z
teory (HD):
ω=1.58ω z
ideal gas:
ω=2 ω z
Collective oscillations of a Trapped Gas
Both the ground state and the dynamics of the gas (Bose or Fermi or Boltzmann)
can be obtained via variational principles on a proper action.
ENERGY
FUNCTIONAL:
Kinetic
en. density
LDA:
GROUND STATE:
Ext ptential
en. density
Number of particle
fixed
Collective oscillations of a Trapped Gas
Both the ground state and the dynamics of the gas (Bose or Fermi or Boltzmann)
can be obtained via variational principles on a proper action.
Action:
Multi-parameter wf describing the mode
we are interested in, e.g., breathing, dipole...
Dynamics
the sloshing mode ideal gas:
the brething mode ideal gas:
due to the possibility of elastic deformation of the Fermi sphere
(i.e., not the relation with the density one has at equilibrium)
If there is interaction the equation of state changes and so do the mode frequencies.
Indeed the collective mode frequency measurements are – in cold trapped gases a precise and sensitive tool to investigate the equation of state of the system
INTERACTING
FERMI GASES
Interaction: s-wave scattering length
At low density and temperature the 2- body
interaction is conveniently described by an
effective contact potential
which reproduces the low-energy behaviour
of the microscopic potential
V(x-x')
s-wave scattering length
i) a>0 : positive scattering & a Bound State (D=3)
ii) a<0 : negative scattering & NO Bound State (D=3)
Due to Pauli principle only fermions in different internal states can – at this level- interact
Interaction: s-wave scattering length
[MIT, Nature 392, 151 (1998)]
[Innsbruck, PRL 93, 123001 (2004
Interaction: s-wave scattering length
Cold Atoms: possibility of tuning the scattering length
[MIT, Nature 392, 151 (1998)]
[Innsbruck, PRL 93, 123001 (2004)]
Interaction: the Hamiltonian
Free gases
V(q)
Depending on the parameters interaction can change completely
the possible states of the system, leading to an extremely rich physics.
To deal with an interacting Fermi system a huge amount of analitical and numerical
methods have been developed.
In the next a flavour of (very recentely realized/studied in the context of cold-gases):
- Polaron physics (in the spirit Landau quasi-particles)
- BCS-BEC crossover
- Polarized Fermi gas
- Itinerant Ferromagnetism
Interaction: the “easiest case” of N+1
A single blu-down atom (impurity) interacts with the
many red-up ones (non-interacting).
The energy to first order in perturbation theory for an
impurity at rest:
Clearly if it moves we have to add its kinetic energy, BUT being in a bath
its inertia is modified.
One has to calculate the system's energy at finite momentum p and expand
to lowest order in p2 :
Its coefficient can be identified with an effective mass
Interaction: the “easiest case” of N+1
A single blu-down atom (impurity) interacts with the
many red-up ones (non-interacting).
The energy to second order in perturbation theory
where the small parameter is
and p
It behaves like a particle with a certain chemical potential (or “binding energy”) and
an effective mass:
a quasi-particle, aka a FERMI-POLARON.
A few of them would form a non-interacting Fermi gas of Polaron (Fermi energy and so on...)
Interaction: the “easiest case” of N+1
One can better, e.g., summing a proper class of diagrams with green function formalism or
rely on Monte-Carlo calculations.
Get more insight using a variational approach based on the simple 1p-1h excitations:
p
Variational
parameter
+Σ
q
k
p+q-k
The ground state is found by minimizing the energy
with respect to the variational parameters.
Interaction: the “easiest case” of N+1
It behaves like a particle with a certain chemical potential
(or “binding energy”) and an effective mass:
a quasi-particle, aka a FERMI-POLARON.
A few of them would form a non-interacting Fermi gas of
Polaron (Fermi energy and so on...)
ATTRACTIVE CASE
Notice than nothing special happens at UNITARITY, i.e.,
Infinity
Interaction: the “easiest case” of N+1
A single blu-down atom (impurity) interacts with the
many red-up ones (non-interacting).
It behaves like a particle with a certain chemical potential
(or “binding energy”) and an effective mass:
a quasi-particle, aka a POLARON.
A few of them would form a non-interacting Fermi gas of
Polaron (Fermi energy and so on...)
BUT for a>0 there is a bound state, whose energy goes like -1/a2 and size like a, thus at some
point our impurities should couple with a red-up atom...and become BOSONS!
Polaron
Molecule
Due to the Fermi see the molecular state
appears when its size is smaller than
the interparticle distance:
Interaction: the “easiest case” of N+1
RadioFrequency Wave
Binding energy obtained via RF-frequency measurement in Zwierlein's group
(MIT, 2009)
Molecular side
Unitarity
Oscillation of a Polaron in the trapped case
Write the polaron Energy as:
Supposing the bath (red-up Fermi atoms) is not “essentially” modified by the interaction
with the impurity and that it is well described by LDA, one can write
an effective single-particle Hamiltonian for the impurity:
Which gives a renormalized
dipole oscillation frequency:
Spin-dipole
mode
E.g., AT UNITARITY
Momentum relaxation time
Untill now we assumed that the (quasi-particle) polaron is well defined for each value
of its momentum. This is generally not the case!
At finite momentum it will have a finite lifetime due to the collisions with the red-up particles
We consider the momentum relaxation of an homogeneous highly polarized Fermi gas.
The minority component have a mean momentum k with respect to the majority one:
total momentum per unit volume
Decaying time of the collective modes
{
Collisionless regime: possible to see the dipole mode
Hydrodynamic regime: the dipole mode overdamped
:
MIT regime
:
INTERACTING
FERMI GASES:
Superfluidity
Attractive Interaction
For a>0 there is a bound state and at some point our impurities couple with a red-up atom
...and become BOSONS
BUT in the balanced case (equal number of red and blue atoms) a similar effect is
present for any value of the interaction also for small negative scattering length:
COOPER PAIR FORMATION and BCS-superfluidity
In 3D in vacuum an attractive potential has to be “attractive enough” to bound a pair of
(red-blue) atoms. In 2D and 1D any attractive potential make it due to the different
density of states.
Well, if we consider (Cooper problem) 2 atoms on top a frozen Fermi sea
also in 3D any attractive potential allows for a state with an energy smaller than
the Fermi Energy itself!
Attractive Interaction
Cooper problem:
Schroedinger Equation
continuum
Since from one also gets
calculate as:
Non-perturbative result no matter
how weak is the interaction
the size of the pair can be easily
Much larger
than the
interparticle distance
BCS vs Bose-Einstein Condensation
Note on finite T: Except for very weak coupling (BCS) pairs/molecules
form and condense at different temperature, T* and Tc
BCS vs Bose-Einstein Condensation
Molecular Bose-Einstein condensation
from a fermionic gas
[JILA, Innsbruck, MIT, ENS, RICE,
2003]
Vortex lattice on the BCS-BEC crossover [MIT, 2005]
Court. M. Zwierlein
BCS vs Bose-Einstein Condensation
Landau critical velocity for a system to give rise to energy dissipation
Due to pair breaking
Due to phonon excitation
(as in a BEC (Pitaevskii lecture))
Superfluid fermions at unitarity
The only scales at unitarity are the Fermi energy and the temperature.
The thermodynamic properties have an “universal” form.
In particular at T=0
energy density, pressure, chemical potential are proportional
to the ones of an ideal Fermi gas with a density equal
to the superfluid one.
The universal parameter (via Montecarlo & Experiments)
S
Collective Oscillation of a SF Fermi-gas
Both the ground state and the dynamics of the gas
can be obtained via variational principles on a proper action.
Action:
Multi-parameter wf describing the mode
we are interested in, e.g., breathing, dipole...
Dynamics
the sloshing mode SF gas:
The (axial) brething mode (elongated) SF gas:
due to coherence which assures local equilibrium
(i.e., the relation with the density one has at equilibrium)
If there is interaction the equation of state changes and so do the mode frequencies.
Indeed the collective mode frequency measurements are – in cold trapped gases a precise and sensitive tool to investigate the equation of state of the system
Fermi gas at Unitarity : expansion
Hydrodynamic equation for a superfluid
or a perfect (collisional) fluid
At Unitarity one finds same expansion for T<Tc<<T F
and T close to TF,
but different from a wealy interacting Fermi gas
Strongly Interacting 6Li
gas T = 10-7 K
[Duke, Science (2002)‫]‏‬
Fermi gas at Unitarity : expansion
At Unitarity one finds same expansion for T<Tc<<T F and T close to TF,
but different from a wealy interacting Fermi gas
Moment of Inertia
Red—normal fluid
Blue—superfluid
y2
x2
Fermi gas at Unitarity : a perfect fluid?
He near λ
λαµ βδ αpoint
QGP simulations
String theory limit
Unbalanced Fermi gases:
From Polaron to Superfluid Physics
P=1 Normal Liquid
P=0
Superfluid
Phase Transition
[Phase Transition to a normal phaase for large magnetic field
B. S. Chandrasekhar (1962), A. M. Clogston (1962)]
Recent Experiments on imbalanced Fermi gases at unitarity
1: Collective mode
Measurement of the (axial) breathing mode of the majority component in an
elongated (cigar-shaped) trap, reducing the amount of minority atoms
P
Recent Experiments on imbalanced Fermi gases at unitarity
2: Vortices
MIT, Science 311, 492 (2006)
Recent Experiments on imbalanced Fermi gases at unitarity
3: Condensate Fraction
BEC
Unitarity
BCS
[MIT, Phys. Rev. Lett. 97, 030401 (2006)]
Normal phase of polarized Fermi gas at unitarity
Assumption:
at high polarization homogeneous phase,
NORMAL FERMI LIQUID: consider a very dilute mixture of
spin-↓ atoms immersed in non-interacting gas of spin-↑ atoms
Energy expansion for small concentration
...
Non interacting gas
single-particle energy
quantum pressure
of a Fermi gas of quasi-particles
with an effective mass
Superfluid-Normal phase coexistence at unitarity
Quasi-particles interaction
Values using FN-QMC
A = 0.99(2)
m*/m = 1.09(3)
B = 0.14
x=1: EN=1.12(2)
[S. Pilati and S. Giorgini,
Phys. Rev. Lett. 100, 030401 (2008)]
Critical concentration xc:
PSF = PN
SF
N with
xc=0.44
Phase Separation
xc=0.44
Coexistence line
x=1: ES=0.84(2)
Exploring Phase diagram in the Trap: LDA
LDA :
Decreasing outward
Constant also inside the trap
x
By the total
number of atoms
SuperFluid
N
Fully
Polarized
By the imbalance
Critical imbalance
Exploring Phase diagram in the Trap: LDA
LDA :
Decreasing outward
Constant also inside the trap
x
N
SF
FP
FP
N
SF
Normal phase of polarized Fermi gas at unitarity: TRAP
Density profiles
3D density
Density Jump
Superfluid
“Cooper pair
condensate”
Normal liquid:
POLARON
GAS
Normal phase of polarized Fermi gas at unitarity: TRAP
1) Critical Polarization (IN TRAP): PC = 0.77
Outside the unitarity regime
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SF
N
SF
[S. Pilati and S. Giorgini, Phys. Rev. Lett. 100, 030401 (2008)]
INTERACTING
FERMI GASES:
Ferromagnetism
Repulsive Interaction
Let us consider the energy to the first order in the interaction
Which P minimizes the energy?
a) Without interaction: P=0, i.e., better to have same number of up and down
atoms: paramagnetic phase)
b) With interaction:
{
P=0;
i.e., better to have more up (+) or down (-) atoms: ferromagnetic phase
The system prefers to have separates in 2 islands when the majority of
atoms are either the blue or the red ones
a.k.a: STONER INSTABILITY
Repulsive Interaction
Let us consider the energy to the first order in the interaction
Paramagnet
Which P minimizes the energy?
a) Without interaction: P=0, i.e., better to have same number of up and down
atoms: paramagnetic phase)
b) With interaction:
{
P=0;
Ferromagnet
i.e., better to have more up (+) or down (-) atoms: ferromagnetic phase
The system prefers to have separates in 2 islands when the majority of
atoms are either the blue or the red ones
a.k.a: STONER INSTABILITY
Susceptibility
An important quantity in a spin-like system is its susceptibility,
which tell us how easy is to polarise a the system.
For the repulsive interaction close to P=0 one has:
Thus:
which diverges at the critical value
Susceptibility
An important quantity in a spin-like system is its susceptibility,
which tell us how easy is to polarise a the system.
Attractive case:
In the normal phase the susceptiblity
should decrease
In the superfluid phase the susceptibility
should instead goes to zero
(reflecting the gap in the spin excitation spectrum)
Relative number (spin) fluctuations in V
V
At “large enough” temperature T
& volume cell V
thermodynamic relation:
MIT measured the reduction of the susceptibility in the attractive (superfluid) regime.
On the repulsive branch the susceptibility should diverge if there exists a
ferromagnetic transition.
Interesting the quantum fluctuations (within Fermi-Landau liquid theory) also diverge
but much more slowly[4] (logarithmically in
)
[4] A.R. and S. Stringari, PRL 106, 080402 (2011)
(Not) The END...
...more...
...more...
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More exotic phases (polarized Superfluid, FFLO, Sarma...)
More than 2 species (analogies with color superfluidity?)
Bose-Fermi mixtures
Include disorder and noise
Cold and Dipolar Molecules
Low dimensional systems
Antiferromagnetic order: Néel transition
Quantum Hall effect
Cold gases on atom chip
Cold gases in Cavity QED
Trapped ions
Quantum Information/Computation
Cold gases in optical lattices and Hubbard models
Why
Why are
are Cold
Cold Gases
Gases interesting?
interesting?
• Diluteness: atom-atom interactions described by 2-body and 3-body physics
At low energy: a single parameter, the scattering length
• Comparison with theory: Gross-Pitaevskii, Bose and Fermi Hubbard models, search for
exotic phases,…
• « Simplicity of detection »
New way to address some pending questions in the physics of
interacting many-body systems
Link with condensed matter (high Tc superconductors, magnetism in lattices),
nuclear physics, high energy physics (quark-gluons plasmas), astrophysics…
Experimental tunability of almost all the parameters which
enter in the physics of the system under study!
Why
Why are
are Cold
Cold Gases
Gases interesting?
interesting?
Experimental tunability of almost all the parameters which
enter in the physics of the system under study!
Control of sign of interaction and of trapping parameters:
weakly and strongly interacting systems
• 1D, 2D, 3D geometry & (optical) lattice
• access to time dependent phenomena & out of equilibrium situations
• and more…
●
Towards quantum simulations with cold atoms
« a’ la Feynman »,i.e., the first idea of quantum computation
Number fluctuations in a cell V
V
At “large enough” temperature T
& volume cell V
thermodynamic relation (TR):
In general we have deviations from it.
In particular at T=0 we have quantum fluctuations which scales
differently with N (e.g., [1,2])
EXP: Ideal Fermi gas: ETH (T. Esslinger), MIT (W. Ketterle)
Interacting Fermi gas: MIT (W. Ketterle)
Decaying time of the collective modes
We consider the momentum relaxation of an homogeneous highly polarized Fermi gas.
The minority component have a mean momentum k with respect to the majority one:
total momentum per unit volume
p
p'
p-q
p' + q
Decaying time of the collective modes
{
Collisionless regime: possible to see the dipole mode
Hydrodynamic regime: the dipole mode overdamped
:
MIT regime
:
Why
Why are
are Cold
Cold Gases
Gases interesting?
interesting?
• Diluteness: atom-atom interactions described by 2-body and 3-body physics
At low energy: a single parameter, the scattering length
• Comparison with theory: Gross-Pitaevskii, Bose and Fermi Hubbard models, search for
exotic phases,…
• « Simplicity of detection »
New way to address some pending questions in the physics of
interacting many-body systems
Link with condensed matter (high Tc superconductors, magnetism in lattices),
nuclear physics, high energy physics (quark-gluons plasmas), astrophysics…
Experimental tunability of almost all the parameters which
enter in the physics of the system under study!
Why
Why are
are Cold
Cold Gases
Gases interesting?
interesting?
Experimental tunability of almost all the parameters which
enter in the physics of the system under study!
Control of sign of interaction and of trapping parameters:
weakly and strongly interacting systems
• 1D, 2D, 3D geometry & (optical) lattice
• access to time dependent phenomena & out of equilibrium situations
• and more…
●
Towards quantum simulations with cold atoms
« a’ la Feynman »,i.e., the first idea of quantum computation
The End