Yuki Kawaguchi- Topological Excitations in Spinor Bose-Einstein Condensates

8/3/2019 Yuki Kawaguchi- Topological Excitations in Spinor Bose-Einstein Condensates

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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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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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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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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


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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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


7/34New 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 Mechanism

a scenario of defect formation after Phase transition



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



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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Knots in a Spin-1 Polar BEC

YK, M. Nitta, and M. Ueda,Phys. Rev. Lett. 100, 180403 (2008)



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 !!!



Topological Excitations


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



Linking Number = Hopf Charge

Order Parameter: 3D unit vector

Boundary condition:

preimage

link



Spin-1 Polar BEC

Order Parameter

order parameter manifold

Invariant under

U(1) and Z2 contribute

only to vortex

e.g. 23Na BEC

KNOT



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)



How to Probe

Cross Section of the density

Double rings

Slice the BEC

Stern-Gerlach experiment



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



YK, Nitta & Ueda, PRL 100, 180403 (2008)

Dynamical Creation & Destruction of Knots



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)



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



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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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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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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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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(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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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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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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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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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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Numerical Results

commutable pair non-commutable pair

reconnection

passing through

rung


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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.


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.


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.

Documents

Yuki Kawaguchi- Topological Excitations in Spinor Bose-Einstein Condensates