Transcript Vortices in isotropic and anisotropic condensates

Dipolar Quantum Gases: Bosons and Fermions
Han Pu 浦晗
Rice University, Houston, TX, USA
• Dipolar interaction in quantum gases
• Dipolar BEC
• Rotons and charge density wave
• Dipolar Fermions
11/14/2007
NSU, Singapore
Current cold atom research at Rice Univ.
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Hulet group (exp.)
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Killian group (exp.)
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BEC (Anderson localization, soliton)
Fermi superfluid (Feshbach resonance, BEC-BCS crossover)
Strongly correlated fermions in optical lattice
Ultracold plasma
Neutral Sr (photoassociation, condensation)
Pu group (th.)
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BEC (vortex and vortex lattice, spinor condensate, dipolar condensate,
coupled atom-molecule condensate)
Fermi gas (dipolar, BEC-BCS crossover, Fermi superfluid)
Bose-Fermi mixture
Quantum simulation (optical lattice systems)
Quantum optics (interaction between light and cold atoms)
http://rlup.rice.edu
Two-body collisions at ultracold temperature
(current paradigm)
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Dominated by the s-wave (isotropic)
Short ranged (~1/R6)
Effective interaction --- pseudopotential
4 2 a
V (r  r ') 
 (r  r ')
m
2 2



1

ˆ
H   dr (r)  
 Vext (r)  (r)   dr '  dr  (r)  (r ')V (r  r ') (r ') (r)
2
 2m

Gross-Pitaevskii Equation
 2 2

2
i
 (r)   
 Vext (r)  c0  (r)  (r)
t
 2m

Dipolar system
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Atoms: magnetic dipole moment
Polar molecules: large electric dipole moment
Besides the short-range interaction, atoms or molecules
may interact with each other via long-range anisotropic
dipole-dipole interaction.
Dipolar interaction between two atoms
z
μ1
Vd (r1  r2 ) 

μ2
0 1  2  3(1  r)( 2  r)
3
4
r1  r2
0  2 1  3 cos 2 

3
4
r1  r2
r1  r2
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(Polarized dipole)
Long ranged (~1/R3)
Anisotropic
 2 2


1  3cos2 
2
2
i
 (r)   
 Vext (r)  c0  (r)  cd  dr
 (r)  (r)
3
t
2
m

r r


Computational method
 2 2


1  3cos2 
2
2
i
 (r)   
 Vext (r)  c0  (r)  cd  dr
 (r)  (r)
3
t
2
m

r r


Two problems:
1)
How to calculate efficiently the convolution integrals.
2)
Dipolar interaction looks singular.
(1  3cos 2  )
Veff (r )  d
r3
2
One solution:
Convolution theorem → Fourier Transform (FFT).
Fourier transform of dipolar interaction is non-singular.
4 d 2
Veff (k ) 
(3cos 2   1)
3
Chromium BEC realized
Dipolar effects on expansion dynamics
  6 B
a  100aB
cd c0  3  10 2
c0 tunable via Feshbach Resonance
Griesmaier et al., Phys. Rev. Lett. 95, 150406 (2005)
Stuhler et al., Phys. Rev. Lett. 95, 150406 (2005)
c0 decreased by a factor of ~5
Vortex States in Dipolar condensate
(and its connection with roton)
Yi and Pu, PRA 73, 061602(R) (2006)
Vortex studies in BEC
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Formation and decay
Multiply charged vortices
Tkachenko and Kelvin mode excitation
Vortex state in combined harmonic and quartic trap
Vortex in spinor condensate
Vortex rings
Scissors mode excitation
Vortex lattice Bragg spectroscopy
Quantum melting
Vortex pinning
(Ketterle, MIT)
Quasi-2D dipolar condensate
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Trapping potential

1
Vtrap ( x, y, z )  m2 x 2  y 2   2 z 2
2
Trap aspect ratio:  1
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Wave function decomposition
( x, y, z, t )   ( x, y, t ) ( z)

Axial wave function:  ( z )   
 
1/ 4

exp  z 2 2

Effective 2D dipolar interaction potential
2
1

3cos

Vdd2 D (    ')  cd  dz  ( z )
| r1  r2 |3
4
y
x
Dipoles polarized along z-axis
Isotropic (azimuthal symmetry in the xy-plane
Vortex state for z-polarized dipoles
Single vortex state:
 (  )   (  )ei
Higher rotating freq.
Vortex lattice: hexagonal
What causes the density ripples?
Vortex line as roton emitter?
calculated vortex core structure
for He II, at different pressure
Dalfovo, PRB 46, 5482 (1992)
Excitation spectrum of He II
Vortex structure vs. roton excitation in dipolar BEC
Single vortex state:
 (  )   (  )ei
Excitation spectrum
(homogeneous system)
Santos et al., PRL 90 250403 (2003)
Rotons can be ‘emitted’ by other impurities
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Vortex line, alien atoms, container wall
A qualitative plot of superfluid density
of HeII confined between two rigid walls.
--- Rasetti and Regge, 1978
Difficult to observe in superfluid helium as the
roton wavelength is only a few angstrons.
Quasi-2D BEC in a box
Like the vortex line, here the wall may become the roton emitter.
Roton instability induces charge density wave:
Square CDW → triangular CDW → collapse
Beyond quasi-2D
Charge density wave damped inside the bulk.
Dipolar BEC in 3D harmonic trap
Ronen, Bortolotti, Bohn
PRL 98 , 030406 (2007)
Conclusion for dipolar BEC
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Dipolar interaction: long-range, anisotropic,
 New ground state structure
 Roton excitation and CDW (crystallization,
supersolid?)
Dipolar fermions (spin polarized)
Miyakawa, Sogo and Pu, arXiv:0710.5223
Fermions vs. Bosons (T=0)
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Bosons
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Condensed (all atoms occupy the same state,
macroscopic wave function)
Symmetric many-body wave function: Hartree
direct energy
Real space and momentum space distribution:
Fourier transform
Fermions
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Pauli principle
Anti-symmetric many-body wave function: Hartree
direct and Fock exchange energy
Real space and momentum space distribution: no
direct relation
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Trapped non-interacting fermions: isotropic
momentum distribution.
Model
Hamiltonian
H     2 m  2  U  ri    12  Vd  ri  r j 
N
2
i 1
i j
Wigner Distribution Function
  r, r ' 
1
 2 
3
3
d
 kf
n  r     r, r  
n k  
1
 2 
3
1
 2 
3
 r+r2 ' , k  eik r-r '
3
d
 k f  r, k 
3
d
 r f  r, k 
N spin polarized (along z-axis) fermions
Interacting with each other via the
dipolar interaction
Total energy
Ekin   d k n  k 
3
2 2
k
2m
Etrap   d 3r n  r U  r 
EHartree 
1
2
3
3
d
rd
r ' n  r Vd  r - r '  n  r ' 

EFock   2 21


6
i  k -k ' r-r '
3
3
3
3
d
rd
r
'
d
kd
k
'
V
r
r
'
e
f


d


Goal: minimize the total energy
Strategy: treat the Wigner function variationally
 r+r2 ' , k  f  r+r2 ' , k '
Homogeneous case: Wigner function

f  r, k   f  k    k F2  1  k x2  k y2    2 k z2
k F   6 n f
2

1/ 3

: Fermi momentum for a noninteracting system
 : variational parameter, characterizing deformation of Fermi surface
 1: prolate Fermi surface
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   1: isotropic Fermi surface
  1: oblate Fermi surface
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Homogeneous case: energies
Etrap  0
EHartree  0
Ekin 
2 2
kF
V
5
2m
n f  2  2 
EFock    d3 V n 2f I  
2

I     d sin 

3cos 2 
 3 sin 2   cos 2 

1
0
Kinetic energy favors an isotropic Fermi surface (α =1);
Fock energy tends to stretch the Fermi surface along z-axis (α =0);
Competition b/w the two results in a prolate Fermi surface (0< α <1).
Inhomogeneous case: Wigner function
U  r   m2 r2  x 2  y 2   z2 z 2 

f  r, k    k F2  r   1  k x2  k y2    2 k z2

k F2  r   k F2   2    x 2  y 2    2 z 2 
1
 2  6
3
3
k f  r, k   N  k F   48 N 
rd
d

1/ 6
1/ 2
 : variational parameter, characterizing deformation in real space
 : variational parameter, characterizing scaling factor
(  1, shrinking in real space)
Inhomogeneous case: energies
Etrap  c1 N
4/3

1
2
0


2
 02
Ekin  c1 N 4 / 3  2  2 
,
0  

r 2 / 3
z
: trap aspect ratio
EHartree  c2 N 3/ 2  3/ 2 d 2 I   
EFock  c2 N 3/ 2  3/ 2 d 2 I  
Interaction energy is not bounded from below (dipolar interaction is partially attractive).
The system is not absolutely stable against collapse (λ → ∞).
A local minimum may exist: the system may sustain a metastable state.
Results (I): density profiles
Real space:
Momentum space:
Results (II): density profiles in real space
Solid lines: interacting
Dashed lines: non-interacting
Results (III): deformation
Momentum space
Real space
Results (IV): stability
Sufficiently large dipolar strength will collapse the system.
Conclusion and outlook
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Dipolar interaction deforms the quantum Fermi
gas in both real and momentum space.
Dipolar effects can be observed in TOF image.
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Ideal gas: isotropic expansion
Dipolar gas: anisotropic expansion
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Importance of the Fock exchange term.
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Near future:
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Collective excitation
Dipolar induced superfluid pairing
Collaborators:
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Hong Lu (Rice)
Su Yi (ITP, CAS)
Takahiko Miyakawa (Tokyo Univ. of Sci.)
Takaaki Sogo (Kyoto Univ.)