Topological Excitations in Spinor Bose-Einstein Condensates...Novel Vortex chiral spin vortex 1/2...

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


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


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


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


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


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


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


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


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


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 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


Linking Number = Hopf Charge Order Parameter: 3D unit vector

Boundary condition:

preimagelink


Spin-1 Polar BECOrder Parameter

order parameter manifold

Invariant under

U(1) and Z2 contributeonly 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=0color: 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't matter as long as the zero point of the magnetic field is located 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 & UedaPRL 100, 180403 (2008)


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


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


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 -


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)


Collision of Two Conventional VorticesWhen two vortices collide, they RECONNECT

vortex line

Abelian non-Abelian

rung


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


(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?


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)


Y-Junction

cb

a

cba=1

e.g. π rotationaround (1,0,0)

base point

b

a

c

cba=1

(1,1,1)


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)


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


Numerical Results

commutable pair non-commutable pair

reconnection

passing through

rung


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!


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.


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

Topological Excitations in Spinor Bose-Einstein Condensates...Novel Vortex chiral spin vortex 1/2...

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