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Massive minimal coupled scalar field from a 2-component
Bose-Einstein condensate
ESF COSLAB Network Conference
present
ed at
by
August 28th - September 4th 2005
Smolenice, Slovakia
Silke Weinfurtner, Matt Visser and Stefano Liberati
What I am going to talk about.
Excitations in Bose-Einstein condensates:
sound waves in a 2-component BEC
Interpretation of massless and massive
classical scalar fields in curved space-time.
Dispersion relation for coupled sound waves in a
2-component BEC in the hydrodynamic limit
Application as an Analogue Model for
Quantum Gravity Phenomenology:
Talk on Friday: Stefano Liberati (11:00)
2-component Bose-Einstein condensation.
Bose-Einstein condensation in experiment
gas of bosons, e. g. 87Rb (Eric Cornell) or 23Na (Wolfgang Ketterle)
extremely low densities, 1015 atoms/cm3
very cold temperature, T1K
Bose-Einstein condensation in theory
nearly all atoms occupy the ground state
non condensed atoms are neglected
microscopic system can be replaced by a classical mean-field, a macroscopic wave-function
2-component Bose-Einstein condensation.
Interactions in a coupled 2-component BEC
low-energy elastic collisions within each species, UAA and UBB
UBB
low-energy elastic collisions between the the two species, UAB
transitions between the two species
Kinematics is given by 2 coupled Gross-Pitaevskii equation
many-body Hamiltonian
time-dependence via Heisenberg equation of motion
replacing field operators by classical fields
UAA
UAB
2-component Bose-Einstein condensation.
Gross-Pitaevskii equations
Macroscopic wave functions
2-component Bose-Einstein condensation.
From the GPE to a pair of coupled wave equations
Physical interpretations:
mass-density matrix
background velocity
this equation represents kinematics of sound waves in the 2-component BEC
a small (in amplitude) perturbation in 2-component BEC results in pair of coupled sound waves
coupling matrix
this description holds for low and high energetic perturbations
interaction matrix + quantum pressure term
contains the modified interactions due to the external coupling
2-component Bose-Einstein condensation.
Fine tuning of the interactions via the external coupling field :
the external laser field modifies the interactions
~
~
UBB
UAA
~
UAB
the sign of can be positive or negative ( additional trapping frequency ), e.g it is
possible to make the modified XX or XY interactions zero:
~
UAA
~
UBB
2-component Bose-Einstein condensation.
Beyond the hydrodynamic limit
the quantum potential has to be taken into account
the quantum potential term (here in flat space-time) can be absorbed in the redefinition
of the interaction matrix between the atoms (effective interaction matrix)
this term gets relevant at wave length comparable to the healing length
a change to momentum space shows the effective interaction is k-dependent
We are in the hydrodynamic limit if the wave length of the perturbations is much smaller then the
healing length!
2-component Bose-Einstein condensation.
The role of different initial phases
contribution to mass term
damping terms
for the model
The 2-component BEC as an Analogue Model for Gravity.
Sound waves in a 1-component BEC
can be treated as an Analogue Model
for Gravity for massless particles.
The idea was to do the same with
our 2-component BEC, hoping
that we would get additional terms
in the wave equation, which can
be identified as the mass of the
phonon-modes..
How to continue:
decoupling of the phonon modes on the level on the wave equation.
the two independent wave equations can be treated in the same way as a 1-component system
for each mode it is possible to assign a mass and space-time geometry
forcing the two space-times to be equal by adding a mono-metricity condition
Klein-Gordon equation for massive phonon modes.
Decoupling the wave equation onto the two eigenstates
B1
The system is in an eigenstate, if:
the perturbed phases are in-phase
the perturbed phases are in anti-phase
A1
Klein-Gordon equation for massive phonon modes.
The two decoupled wave equations can be written as two scalar fields in
curved space-times:
in-phase mode
anti-phase mode
the in-phase mode represents a massless scalar field
the anti-phase mode represents a massive scalar field
the two effective metrics are different, due to different speeds of sound:
Klein-Gordon equation for massive phonon modes.
The fine tuning for the decoupling the wave equations:
the densities
and interactions
within each condensate are equal
The two speed of sounds are:
the mono-metricity condition must be
which requires the fine tuning
Within this fine tuning the eigenfrequency of the anti-phase (massive) mode is:
Klein-Gordon equation for massive phonon modes.
About the mass of the phonon mode..
phonon mass is proportional to the laser-coupling , therefore you
need a permanent coupling
it is possible to calculate the general expression for the mass of the phonon modes
Klein-Gordon equation for massive phonon modes.
About the fine tuning in terms of possible space-times..
the effective metric obtained by our calculations are the same one gets for a single BEC
(c 2 v 02 ) v 0x v 0y v 0z
v
1
0
0
0x
gab
v 0y
0
1
0
v
0
0
1
0z
in principle the 2-component BEC Analogue Model is possible to reproduce all the
configurations in the same way as in the simple BEC: e.g. Schwarzschild Black Hole, FRW
and Minkowski space-time.
Note: For example, in the case of FRW where one changes the scattering length through an
external potential, also the fine-tuning would have to be re-adjusted!
Sound waves in a moving fluid.
Supersonic and subsonic region…
c 2 v 02 0
c 2 v 02 0
v0 0
horizon
fluid velocity
fluid at rest
Dispersion relation for uniform condensate.
Changing into momentum space leads to the dispersion relation:
Note: The change to momentum space is only exact, if the densities are uniform
and the background velocity is at rest ( Minkowski space-time ).
We recover perfect special relativity for the decoupled phonon modes in the
hydrodynamic limit.
Decoupled sound waves in a 2-component BEC in fluid at rest.
high energetic perturbations
low energetic perturbations
v0 0
v0 0
fluid at rest
fluid at rest
The first step towards an Analogue Model for QGP.
Alternative route to obtain the dispersion relation
The 2-BEC Analogue Model presents a massive and massless
scalar field. We also know from condensed matter physics, that
for high energy modes the Lorentz invariance will be broken.
The idea is know to look at Minkowski space-time ( uniform
density and zero background flow ) and calculate the
dispersion relation for the two coupled modes in the
hydrodynamic limit.
How to continue:
change the wave equation to position space
the dispersion relation
the modes have to fulfill the generalized Fresnel equation
in the hydrodynamic limit - for low energy - we want to recover special relativity
Dispersion relation for high energy phonon modes.
The wave equation for a uniform background at rest reduces to:
for a uniform condensate is constant it is possible to introduce:
it is useful to introduce
after changing in momentum space we get the dispersion relation
the modes have to fulfill the generalized Fresnel equation
Dispersion relation for high energy phonon modes.
The dispersion relation is given by:
again, in the hydrodynamic limit we want to recover special relativity:
the following fine tuning is necessary to obtain LI in the hydrodynamic limit:
in terms of physical parameter the constraints are:
Conclusion and Outlook.
The kinematics for sound waves in a coupled 2-component BEC is analogue
to a massive minimal coupled scalar field embedded in curved-space time.
The external coupling is crucial in order to obtain a massive phonon mode.
The transition rate can be used to tune the system.
For an arbitrary 2-component system the decoupling on the level of the wave
equation (physical acoustics) puts strong tuning parameter onto the system.
The dispersion relation obtained from the two Klein-Gordon equations is
Lorentz invariant, therefore we recovered perfect special relativity.
For a uniform condensate at rest it is possible to calculate the dispersion
relation without decoupling the phonon modes first.
In the hydrodynamic limit we can recover perfect special relativity with milder
constraints, as for the physical acoustics.
We know how we have do modify our theory for high energy modes (wave
length comparable to the order of the healing length of the condensate).
This model is a suitable object to study Quantum Gravity Phenomenology.
Thank you for your attention.