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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
The University of Tokyo
Yuki Kawaguchi
Topological Excitations inSpinor Bose-Einstein Condensates
Muneto NittaMichikazu Kobayashi
Masahito Ueda
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Outline
Introduction
cold atomic systems
Internal degrees of freedom
Topological excitations
in spinor BECs (BECs with spin degrees of freedom)
Knot soliton in a spin-1 polar BEC Non-Abelian vortices in a spin-2 cyclic BEC
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Cold Atomic Systems
Atomic cloud trapped in vacuum
Number of atoms
105-106
Temperature 100nK
Cloud size a few-100 m
Both Fermionic and Bosonic atoms
Photo by I. Bloch's group
5 order of magnitude diluter than the air
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Features of Cold Atomic Systems
High-precision measurements
high tunability of experimental parameters:interaction strength, density, trap geometry, external field, etc.
direct observation of
the momentum distribution, spin structure, vortices, etc.
Extremely Dilute gas
long relaxation time ~ ms
real-time observation of non-linear dynamics
good agreement with the mean field theory
quantitative comparison with theory and experiment of staticand dynamic properties of the system
Internal degrees of freedom
analogy with anisotropic superconductors and QCD
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Internal Degrees of Freedom
hyperfine spin
87Rb, 23Na,
7Li, 41KF= 1, 2
85Rb F= 2, 3
13 3Cs F= 3, 4
52Cr S= 3, I= 0
6Li F= 1/ 2,3/ 2
40K F= 7/ 2,9/ 2
171Yb S= 0, I= 1/ 2
173Yb S= 0, I= 5/ 2
Boson Fermion
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Physics in Cold Atomic Systems
BEC-BCS crossover/ Unitarity gas(I=1, S=1/2)
Color Superconductor
3 internal statesSU(3) symmetry
173Yb: I=5/2
SU(6) symmetry
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Physics in Cold Atomic Systems
Spinor BECspontaneous spin vortex creationin quantum phase transition
Sadler, et al. (Berkeley),Nature 443, 312 (2006)
Kibble Mechanism
a scenario of defect formation after Phase transition
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Spinor BEC
Hamiltonian
Mean-field approximation:
Assume all atoms are in the same single-particle state
The multi-component order parameter
spin-1
spin-2
m: magnetic sublevel
spin is conserved in the scattering: SO(3)Symmetry of the Hamiltonian G=U(1) x SO(3)
Several phases
dependes on the interaction parameters
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FerromagneticBEC
Polar BEC Superfluid 3He A phase
FullSymmetry
Remaining
Symmetry
OrderParameter
CharacteristicSymmetry
spin-gauge(Berry phase)
discrete
spin-gauge
orbital-gauge
discrete spin-gauge
Novel Vortexchiral spinvortex
1/2 vortexMermin-Ho vortex
1/2 vortex
Spin-1 Spinor BEC vs. Superfluid 3He A
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Knots in a Spin-1 Polar BEC
YK, M. Nitta, and M. Ueda,Phys. Rev. Lett. 100, 180403 (2008)
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Knots in Physics
Faddeev and Niemi, Nature 387, 58 (1997)Low energy excitation in QCD
However,
experimental realization is highly nontrivial
Realizable by using Spinor BECs !!!
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Topological Excitations
Internal degrees of freedom
various kinds of topological excitations
vortex (line defect) Leonhardt and Volovik, JETP Lett. 72, 46 (2000)
Zhou, PRL 87, 080401 (2001)
Mkel, Zhang, and Suominen, J. Phys. A 36, 8555 (2003)
Barnett, Turner, and Demler, PRA 76, 013605 (2007)
monopole (point defect) Stoof, Vliegen, and Khawaja, PRL 87, 120407 (2001)
Roustekoski and Anglin, PRL 91, 190402 (2003)
skyrmion (nonsingular point structure)
Khawaja and Stoof, Nature 411, 918 (2001)
classified with a winding number
Knot is classified with a linking number
order parametermanifold
vortex line
(nonsingular line structure)
mapping
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Linking Number = Hopf Charge
Order Parameter: 3D unit vector
Boundary condition:
preimage
link
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Spin-1 Polar BEC
Order Parameter
order parameter manifold
Invariant under
U(1) and Z2 contribute
only to vortex
e.g. 23Na BEC
KNOT
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Simplest Knot in Polar BEC ( Charge 1 )
boundary condition
rotate around the position vector as
link
spin matrix
torus: nz=0
color: arg(nx+iny)
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How to Probe
Cross Section of the density
Double rings
Slice the BEC
Stern-Gerlach experiment
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How to Create
Linear Zeeman effect
n rotate around the local magnetic field
1. Prepare a n-polarized BEC in an optical trap
2. Suddenly apply a quadrupole field
3. n field develops as
4. Knot appears
* precise configuration of the magnetic field doesn'tmatter as long as the zero point of the magnetic field islocated in the condensate
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YK, Nitta & Ueda, PRL 100, 180403 (2008)
Dynamical Creation & Destruction of Knots
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Dynamical Creation & Destruction of Knots
enter from periphery
The num. of knots%
as n winds in time
The num. of rings %
YK, Nitta & Ueda
PRL 100, 180403 (2008)
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Stability of Knots
Energetical stability
unstable against shrinkage
without higher derivative term
(Faddeev term)
However,
the cold atomic system is isolated in a vacuum
total energy : conserved
kinetic energy density
volume
shrink
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Spin Current The dominant decay mechanism is related to
the spin current given by
Equation of continuity
n texture local magnetization
polar state will be destroyed
toplogical stability of knots is violated
spin expectation value
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Knot is a new type of topological excitation classified with alinking number.
We can experimentally create a knot in a polar BEC andobserve its dynamics.
Strictly speaking, the knot created in the quadratic field isunknot. Is it possible to create true knot, such as trefoil ?
Summary - Knots -
/
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Collision Dynamics of Non-AbelianVortices in a Spin-2 Cyclic BEC
M. Kobayashi, YK, M. Nitta, and M. Ueda,Phys. Rev. Lett. 103, 115301 (2009)
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Collision of Two Conventional Vortices
When two vortices collide, they RECONNECT
vortex line
Abelian non-Abelian
rung
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Fractional Vortex
Quantum number of a vortex
= the circulation around the vortex in a unit of
Z2 symmetry
integer vortex
Invariant under
Half-quantum vortex
Spin-1 Polar Phase
Abelian
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
(1,1,1)
Spin-2 cyclic Phase
Shape of the order parameter in spin space
Cyclic Phase
headless triad
T: tetrahedral group
non-Abelian
Invariant under rotation around (1,0,0) (0,1,0) (0,0,1)- 2/3 rotation around (1,1,1) (1,-1,-1) (-1,1,1) (-1,-1,1)accompanied with a phase transformation of -2/3
87Rb?
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Vortices in the Cyclic Phase
1/2 vortex
rotation around (1,0,0)(0,1,0) (0,0,1)
- independent from overallphase
1/3 vortex
- 2/3 rotation around(1,1,1) (1,-1,-1) ...- coupled with overall phase
Vortices can be characterized with a rotation operator
They cannot commute with each other
(1,1,1)
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Y-Junction
cb
a
cba=1
e.g. rotationaround (1,0,0)
base point
b
a
c
cba=1
(1,1,1)
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Crossing of Vortices
ba
b
=bab-1?a'
When another vortex crosses between the base point and the vortex,it looks as if the kind of the vortex has changed.
base point
(1,1,1)
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Collision of Two Vortices
or
a
b
b
bab-1 b
ba
ba
bab-1
b
b
a
ab-1
bab-1
or
a
b
a-1ba
a b
a-1baa
ba
a
a-1ba
b
a
b-1a
a
Abelian: equivalent
Abelian: reconnection or passingnon-Abelian: rung
passing
1
1
1
1
2 1 1
1 1
0
phase vortex
doubly quantized
vortex reconnection
rung
rung
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Numerical Results
commutable pair non-commutable pair
reconnection
passing through
rung
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Concluding Remarks
Introduction of the cold atomic gases
Knots in a spin-1 Polar BEC
Knot is a new type of topological excitation classified with a
linking number.
We can experimentally create a knot in a polar BEC andobserve its dynamics.
non-Abelian vortices in a spin-2 cyclic BECUnlike Abelian vortices, non-Abelian vortices do not reconnect
themselves or pass through each other, but create a rung vortexbetween them.
We have demonstrated this dynamics from a microscopicHamiltonian.
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Concluding Remarks
Introduction of the cold atomic gases
Knots in a spin-1 Polar BEC
Knot is a new type of topological excitation classified with a
linking number.
We can experimentally create a knot in a polar BEC andobserve its dynamics.
non-Abelian vortices in a spin-2 cyclic BECUnlike Abelian vortices, non-Abelian vortices do not reconnect
themselves or pass through each other, but create a rung vortexbetween them.
We have demonstrated this dynamics from a microscopicHamiltonian.