topological excitations in spinor bose-einstein condensates...novel vortex chiral spin vortex 1/2...
TRANSCRIPT
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
The University of TokyoYuki Kawaguchi
Topological Excitations in Spinor Bose-Einstein Condensates
Muneto NittaMichikazu KobayashiMasahito 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 excitationsin 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 µm
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 ofthe 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 static
and dynamic properties of the system
Internal degrees of freedom analogy with anisotropic superconductors and QCD
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Internal Degrees of Freedom hyperfine spin
87Rb, 23Na, 7Li, 41K F=1, 2
85Rb F=2, 3133Cs F=3, 452Cr S=3, I=0
6Li F=1/2,3/240K F=7/2,9/2171Yb S=0, I=1/2173Yb 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 SystemsBEC-BCS crossover/ Unitarity gas(I=1, S=1/2)
Color Superconductor
3 internal statesSU(3) symmetry
173Yb: I=5/2SU(6) symmetry
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Physics in Cold Atomic Systems
Spinor BECspontaneous spin vortex creationin quantum phase transition
Sadler, et al. (Berkeley),Nature 443, 312 (2006)
Kibble Mechanisma scenario of defect formation after Phase transition
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
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 phasesdependes on the interaction parameters
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Ferromagnetic BEC Polar BEC Superfluid
3He A phase
Full Symmetry
Remaining Symmetry
Order Parameter
Characteristic Symmetry
spin-gauge (Berry phase)
discretespin-gauge
・orbital-gauge・discrete spin-gauge
Novel Vortex chiral spin vortex 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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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
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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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Topological ExcitationsInternal degrees of freedom
various kinds of topological excitations
vortex (line defect) Leonhardt and Volovik, JETP Lett. 72, 46 (2000) Zhou, PRL 87, 080401 (2001) Mäkelä, 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 numberKnot is classified with a linking number
order parametermanifold
vortex line
(nonsingular line structure)
mapping
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Linking Number = Hopf Charge Order Parameter: 3D unit vector
Boundary condition:
preimagelink
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Spin-1 Polar BECOrder Parameter
order parameter manifold
Invariant under
U(1) and Z2 contributeonly to vortex
e.g. 23Na BEC
KNOT
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Simplest Knot in Polar BEC ( Charge 1 )boundary condition rotate around the position vector as
link
spin matrix
torus: nz=0color: arg(nx+iny)
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
How to Probe
Cross Section of the density
Double rings
Slice the BEC
Stern-Gerlach experiment
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
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't matter as long as the zero point of the magnetic field is located in the condensate
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
YK, Nitta & Ueda, PRL 100, 180403 (2008)
Dynamical Creation & Destruction of Knots
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Dynamical Creation & Destruction of Knots
enter from periphery
The num. of knots %as n winds in time
The num. of rings %
YK, Nitta & UedaPRL 100, 180403 (2008)
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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Stability of Knots Energetical stability
unstable against shrinkagewithout 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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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Spin Current The dominant decay mechanism is related to
the spin current given by
Equation of continuity
n texture local magnetizationpolar state will be destroyedtoplogical 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 a linking number.
We can experimentally create a knot in a polar BEC and observe its dynamics.
Strictly speaking, the knot created in the quadratic field is unknot. 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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New Frontiers in QCD 2010 - QCD Phase Diagram - / March 10, 2010
Collision of Two Conventional VorticesWhen 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 VortexQuantum 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 triadT: tetrahedral groupnon-Abelian
Invariant under− π rotation around (1,0,0) (0,1,0) (0,0,1)- 2π/3 rotation around (1,1,1) (1,-1,-1) (-1,1,1) (-1,-1,1)accompanied with a phase transformation of -2π/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− π rotation around (1,0,0)
(0,1,0) (0,0,1)- independent from overall
phase
1/3 vortex- 2π/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. π 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
babab-1
b
b
a
ab-1
bab-1
or
a
b
a-1ba
a b
a-1baa
baa
a-1ba
b
a
b-1a
a
Abelian: equivalent
Abelian: reconnection or passingnon-Abelian: rung
passing
1
1
1
1
2 1 1
1 10
phase vortex
doubly quantizedvortex reconnection
rungrung
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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
Summary -non-Abelian vortices-
Future: Network structure in Quantum Turbulence
Unlike Abelian vortices, non-Abelian vortices do not reconnect themselves or pass through each other, but create a rung vortex between them. We have demonstrated this dynamics from a microscopic Hamiltonian.
Turbulence of Abelian vortices↓
Cascade process
Turbulence of non-Abelian vortices↓
Networking structures of vorticesNon-cascade processNew turbulence!
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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 BECKnot is a new type of topological excitation classified with a linking number.We can experimentally create a knot in a polar BEC and observe 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 vortex between them. We have demonstrated this dynamics from a microscopic Hamiltonian.
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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 BECKnot is a new type of topological excitation classified with a linking number.We can experimentally create a knot in a polar BEC and observe 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 vortex between them. We have demonstrated this dynamics from a microscopic Hamiltonian.
Topological Excitations in �Spinor Bose-Einstein CondensatesOutlineCold Atomic Systems Features of Cold Atomic SystemsInternal Degrees of FreedomPhysics in Cold Atomic SystemsPhysics in Cold Atomic SystemsSpinor BECSpin-1 Spinor BEC vs. Superfluid 3He AKnots in a Spin-1 Polar BECKnots in PhysicsTopological ExcitationsLinking Number = Hopf ChargeSpin-1 Polar BECSimplest Knot in Polar BEC ( Charge 1 )How to ProbeHow to CreateDynamical Creation & Destruction of KnotsDynamical Creation & Destruction of KnotsStability of KnotsSpin CurrentSummary - Knots -Collision Dynamics of Non-Abelian Vortices in a Spin-2 Cyclic BECCollision of Two Conventional VorticesFractional VortexSpin-2 cyclic PhaseVortices in the Cyclic PhaseY-Junction Crossing of Vortices Collision of Two VorticesNumerical ResultsSummary -non-Abelian vortices-Concluding RemarksConcluding Remarks