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Vortex configuration of bosons in an optical lattice
Congjun Wu
Kavli Institute for Theoretical Physics, UCSB
Collaborators: H. D. Chen, J. P. Hu, S. C. Zhang
Ref: Phys. Rev. A 69, 43609 (2004).
Boulder Summer School, July, 2004
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Motivation (I) : vortices in rotating traps
• Topological defect of the superfluid (SF) order parameter.
Time of flight image:
Weak coupling vortex: particle
density is suppressed in the core.
x
SF order
density
P. Engels, et al. PRL. 89, 100403 (2002)
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Motivation (II) : SF- Mott-Insulator transition
Superfluid
Mott insulator
M. Greiner et al., Nature (London) 415, 39 (2002).
• How about the vortex near the SF-Mott insulator (MI) transition?
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Rotating optical lattice through holographic method
S. Tung, V. Schweikhard, and E. A. Cornell, cond-mat/0607697
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Vortex pinning to the lattice potential
• Weak optical potential, square optical lattice.
V pin /   0.049
V pin /   0.084
V pin /   0.143
• How about in the strong coupling limit (near SF-MI) transition?


S. Tung, V. Schweikhard, and E. A. Cornell, cond-mat/0607697
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Bosons in optical lattices without rotation
• Lobes of commensurate MI phases.
• Particle-like, hole-like, and particle-hole symmetric SF phases.
 
H  t {a (ri )a(rj )  h.c.}
U

ri , r j


U 
   n(ri )   n(ri )( n(ri )  1)
ri
ri 2
N 3
n 1
N=2.05
N 2
N=2.00
M. P. A. Fisher et al., Phys. Rev. B 40, 546 (1989).
N=1.95
N 1
t /U
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Superfluid vortex with nearly Mott-insulating core
• Hole-like vortex (t/U=0.02) • particle-like vortex
à
®
®
d l A = 2 p,
W= h

IM
2 m L2
W= h
<N>=1.95

IM
2 m L2
(t/U=0.02)
<N>=2.05
C. Wu , et al., Phys. Rev. A, 69, 43609 (2004); cond-mat/0211457.
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Bosons in rotating optical lattices
  m 
• Vector potential for Coriolis force : A(r )    r ,

ri
 
  
H  t {a (ri )a(rj ) exp{i  dr  A(r )}  h.c.}

rj
ri , r j
 U
  n(ri ) 
2
ri



 n(ri )(n(ri )  1)
ri
Neglect the trapping potential and the centrifugal potential.
Josephson junction array in the magnetic field at low
temperatures.

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Mean field approximation: Guzwiller
• Decouple to single site problems:
a (ri )a(rj )  a (ri ) a(rj )  a (ri ) a(rj )  a (ri ) a(rj )
• The MF ground state wavefunction:


G

 ri  (ri )

• The spatially dependent SF order: a(ri )
• Valid at the small t/U.
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Competition between SF and MI phases
• teff/U is frustrated as approaching
the vortex core
à
®
®
d l A = 2 p,
• vortex core is more strongly
correlated than the bulk area.
• superfluid vortex with
Mott-insulating core.
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Evolution of the vortex particle density distribution
• Core particle density
approaches the nearest
integer number.
• Vortex core with
maximum particle density
in the hole-like case.
• Real space superfluidMott phase transition.
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Evolution of the vortex particle density distribution
U
N 2
N 1
t /U
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Strong to weak coupling vortex configuration
• Particle density profile along a cut passing the central plaquette.
t/U=
<N>=1.95
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Phase diagram with next nearest repulsion
• Charge density wave (CDW) and Super-solid phases appear at the half-integer
filling and small t/U.
• Map to the spin 1/2 Heisenberg model with the Ising or XY anisotropy.
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Vortex with the CDW core (“meron”)
• W/U=0.1, t/U=0.023 and <N>=1.5.
• Similar to the antiferromagnetic vortex core in underdoped
high Tc superconductors.
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Possible experiments
• Vortex core size on the optical lattice is about 2 microns.
Focus the probe laser beam to this size and scan the lattice to
determine the particle density distribution.
• Raman photon-association to detect the percentage of Mott
region. (D. J. Heinzen’s group)
• The Josephson junction array system in the magnetic field.
From the electrical field distribution, it is possible to determine the
vortex charge.
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Summary
• The vortex core is a more strongly coupled region compared to
the bulk area.
• Near the SF-MI transition, the vortex core is nearly Mott
insulating and the core particle density approaches the nearest
commensurate value.
• As t/U increases, the vortex evolves from the strong coupling
configuration to weak coupling one.
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