1859-32

Summer School on Novel Quantum Phases and Non-EquilibriumPhenomena in Cold Atomic Gases

Masahito Ueda

27 August - 7 September, 2007

Tokyo Institute of Technology

Fundamentals of spinor Bose-Einstein Condensates

Masahito UEDA

Tokyo Institute of Technology,

ERATO, JST

Fundamentals

of Spinor Bose-Einstein Condensates

Collaborators

Yuki Kawaguchi (Tokyo Institute of Technology)

Hiroki Saito (University of Electro-Communications)

Contents

Comparing atomic-gas BEC with superfluid helium

Where to look for phenomena unique to atomic-gas BEC?

Basics of spinor BEC physics

ground-state phase diagram

many-body effects

topological excitations

Some more recent topics

spin vortex

quench dynamics

Atomic gases and helium both exhibit BEC and superfluidity,

but at the same time they show striking complementarity in

kinetics,

magnetism, and

symmetry breaking.

How different are they in these respects?

Atomic-gas BEC vs. Superfluid Helium

local equilibrium achieved

physics can be understood by conservation laws energy, continuity equation, etc. and hydrodynamics, even thought the system is strongly interacting.

local equilibrium not achieved

nonequilibrium relaxation and kineticsessential for understanding BEC phase transition and vortex nucleation

Magnetism nuclear spin (3He)

Kinetics

Symmetrybreaking

bulk He: thermodynamic limit achieved

spontaneous symmetry breakingof the relative gauge (phase)

emergence of the mean field

mesoscopic not thermodynamic limit

symmetry breaking may or may not occur

dynamics of symmetry breaking can be observed due to long collision time

Atomic-gas BEC

free free col

collisiontime

alkalis: electronic spin e 2000 N

control the spin texture locally

new quantum phases such as cyclic phase emerge in high-spin systems

3 1

col ~ 10 s ~

Superfluid helium

collectivemode

12 1

col ~ 10 s

Local Manipulation of Spin Textures

A. E. Leanhardt, et al., Phys. Rev. Lett. 90, 140403 (2003)

total density coreless vortex

density profiles of individual spin components

spin-1 23Na condensatetopological phase imprinting usingquadrupole magnetic field

zr, , z B z + B r cos 2 r -sin 2 ˆˆˆB

zB 0

2

2

r z cos 1, -12

sin1, 0

2

sin 1, + 12

i

i

re

re

r

2, ,

0 0,

2

(skyrmion)

(meron)

The local spin texture has thus been manipulated by external magnetic field.

Basics of spinor BEC physics

Internal Degrees of Freedom: Hyperfine Spin F

Alkali atoms have electronic and nuclear spins.

electronic spin

nuclear spin

Hyperfine interaction

12

S

1

23 39 87

85

133

12

32

52

72

H

Na, K, Rb

Rb

Cs

I 1,0

1,2

2,3

3,4

F

F

F

F

V AS I

1 1hf B2 2

~ a few GHz ~ 0.1K for BECF=I F=I

E V V k T

hyperfine spin

Therefore the hyperfinr spin F=S+I is a good quantum number.

Several spin states are available in BECs.

Novel quantum phases emerge for high-spin BECs.

Spinor BEC in an Optical Trap

In a magnetic trap, the spin of each atom is fixed by local magnetic field,

so that the internal degrees of freedom are frozen.

The order parameter is scalar.

In an optical trap, the atomic states for a given are degenerate

with respect to magnetic quantum number .

The order parameter is a spherical tensor of rank F

A rich variety of order-parameter manifolds

are available, depending on the value of F.

F

, 1 , ......,F

m F F F

, 1, ...,mm F F F

D. Stamper-Kurn, et.al.,

Phys. Rev. Lett. 80, 2027 (1998)

2

2

SO 3

U 1 S

Z

These symmetries of the order paramters are reflected in the nature of

Goldstone modes, spin textures, and topological excitations.

F=1 FM BEC

F=1 AFM BEC

(continuous spin-gauge symmetry)

(discreate spin-gauge symmetry)

Mean Field Theory of Spin-1 BECsT.Ohmi and K.Machida, J. Phys. Soc. Jpn. 67, 1822 (1998)

T.-L.Ho, Phys. Rev. Lett. 81, 742 (1998)

20 2 2 0

22 2 2 2 2

trap2 6 6

g g g g

m mM

m m

E d U n nr F

2 *

,

2

1,

4 ( 0, 2)

m m mn n

m m n

i i

nn

g a im

F F

2 0

ˆˆ ˆ

2

2

2

2

1

1 0

0

ˆ , ,

cos1

1ˆ , , 0 sin2

0

:

In general,

sin

yz ziFiF iF

i

ii

i

n

U e e e

e

e nU n

e

g

e

g

F

a

1

0

1

1

0

2 0

1

1 0

0

:

In general,

order-parameter ma

0 or 12

1 0

sin0nˆn , , 1 2 cos2

n

i

0

n

si

i

i i

i

nn

e

e

e

g

e U

g

F

b antiferromagnetic or polar

2

2

1fold

U S

Z

Hartree spin-exchange

Gauge angle and rotation angle appear as - .

This continuous spin-gauge symmetry allows the

system to have coreless vortex

texture-induced supercurrent .

order –parameter manifold SO(3)

The order parameter is invariant under simultaneous

- & + .

This discrete spin-gauge symmetry allows the

system to hold a 1/2 vortex.

F. Zhou, Phys. Rev. Lett. 87, 80401 (2001)

x

y

z

kinetic one-body

potential

gauge rotation

spin-gauge symmetry

local spin density

Euler angles , ,

total spin 2: scattering length

total spin 0: scattering length

collide

2a

0a

2 0

†11!

ˆBEC vacN

N

a a

a

2† † † † † 20 1 1 1 0 1

1ˆ ˆˆ ˆ ˆ2 BEC ~ vac symmetry U 1 33

NN

S a a a S n n n

21

1 0

1 2

1 =0 m

m

m

U Sa Y n

Z

1 1 2 0 1

2 0a a

forbidden by Bose symmetry

Many-Body Gound State of a Spin-1 BEC

Law, et al., Phys. Rev. Lett. 81, 5257 (1988)

Koashi and MU, Phys. Rev. Lett. 84,1066 (2000)

Ho and Yip, Phys. Rev. Lett. 84, 4031 (2000)

cf. mean field:

Spin-singlet correlation is favored. Bose antiferromagnet with no Neel order

How can this discrepancy be

reconciled ?

Bose ferromagnet

In fact, mean-field theory breaks down at zero magnetic field, but its

validity is quickly restored as the magnetic field increases.

Suppose that all bosons form spin-singlet pairs and all magnetic

sublevels are equally populated.

2† † † 2

0 1 1 1 0 1ˆ ˆ ˆBEC ~ 2 vac

3

NN

a a a n n n

As the magnetic field increases, singlet pairs are broken one by one

via spin flip: .

Connection between Many-Body Theory and Mean-Field Theory

Each time a spin-singlet pair is broken, the m=0 component is exponentially

suppressed due to inverse bosonic enhancement effect .

The m =0 component virtually disappears at m ~ N½,

beyond which mean field theory is restored.

M. Koashi & MU, Phys. Rev. Lett. 84 1066 (2000)

2

† 2

2† † † † † †21 0 1 1

ˆ vac

1ˆ ˆˆ ˆ ˆ ˆvac 23

N

Nmm

S

a S S a a a

†ˆ 1 1a n n n

1U

spin flip of m singlet pairs

1

3

N

0n

N

manybody

fragmented

condensate

Mean field regime

Magnetization

g B

5

4

3

2

1

g B

When m pairs are broken, the many-body state becomes

Connection between Many-Body Theory and Mean-Field Theory

2

2

1U S

Z

The fractional dependence on N

indicates that the manybody effect

can appear in the mesoscopic

regime.

Spin-2 BEC

Koashi & MU, Phys. Rev. Lett. 84, 1066 (2000 )

Ciobanu, et al., Phys. Rev. A 61,033607 (2000)

MU & Koashi., Phys. Rev. A 65, 063602 (2002 )

Interaction Hamiltonian

forbidden by Bose symmetry

2 2 0 2 4 1 3

0 2 4a a a

2 2 †

0 1 2

1 ˆ ˆˆ ˆˆ: : : :2

V d c n c c S Sr F

2

2 4

0

2

4 2

1

2

0 2 4

2

4 34

74

77 10 34

7

a ac

Ma a

cM

a a ac

M

21

20

cc

2†S vacN

33 †

0A vacN

†

2 vacN

a

Cyclic phasespin-singlet trio BEC

Ferromagnetic phase

single boson BEC

Antiferromagnetic phase

spin-singlet pair BEC

†

†

ˆ ˆˆ

ˆ ˆ ˆ

1ˆ ˆ ˆ5

F f

m m

m

m mn n

mm

m m

m

n

S

particle density

spin density

spin-singlet pair amplitude

c2

c1The pairwise and trio-wise units

bring about some unique features

of spin-2 BEC.

“ Meissner Effect ” of the Antiferromagnetic Spin-2 BEC

MU & M. Koashi, Phys Rev. A 65, 063602 (2002)

This term counteracts B because c2 < 0.

spin 2 †1 2seff eff

2 ˆ ˆˆ ˆ ˆ: : , , , ;2 5

F z z

c cH S S pF p g B N N F F

V V

2eff

1 2 2

eff eff eff

1

' 1, 2 6 const.

2 ' 8 2 40z z

c c cg VE F l F B l l F

V c g V V

0 p B

2

eff

s

11 1 20

2

2 2

' 0

0 for AFM

z

c

V

l N N F

c c

c

# of spin-single pairs

Ground-state

mangetization

of AFM phase

Physically, the F =2 spin-singlet pair

condensate acts to screen the external

magnetic field until the magnetic field

reaches a critical value.

Bcritical ~1/Veff

mesoscopic Meissner effect

zF

Minimize the spin-dependent part of the Hamiltonian

with respect to magnetization Fz :

Zeeman term

eff

1

zF v

p c

2

eff8

c

V

appliedmagnetic field

# of trio singlets

Spinor BECs can hold various topological excitations.

Fractional Vortices

The order parameter invariant under

F=1 polar ( pair singlet ) F=2 cyclic ( trio singlet )

invariant under

1/2 vortex

cf. no 1/2 vortex for F=2 AF BEC

1/3 vortex

F. Zhou, Phys. Rev. Lett. 87, 080401 (2001)

Y. Zhang, et. al. cond-mat/0404138

H. Mäkelä, J. Phys. A: Math Gen. 39, 7423 (2006)

G. W. Semenoff and F. Zhou, Phys. Rev. Lett. 98, 100401 (2007)

0

0

cos

two-fold symmetry

ie

(spatial inversion)

+ (gauge transformation)

0

2 /3 rotation about n=(1,1,1)

three-fold symmetry

Fragmentation of BEC

What’s BEC?

The system exhibits BEC when the largest eigenvalue of the one-

particle reduced density matrix is an extensive rather than an intensive

quantity.O. Penrose & L. Onsager, Phys. Rev. 104, 576 (1956)

Is the number of extensive eigenvalues necessarily one?

If there is more than one such eigenvalue, the BEC is said to be

fragmented.

P. Nozieres and D. Saint James, J. Physique 43, 1133 (1982)

Three Conditions for Fragmented BEC

The system must have exact symmetry that allows degeneracy.

The interaction between the degenerate states must be attractive

to gain the Fock exchange energy

The system must be mesoscopic to avoid collapse or symmetry

breaking into a single BEC

classic examples

rotating BEC with attractive interaction in a harmonic trap

N. Wilkin, J. Gunn & R. Smith, Phys. Rev. Lett. 80, 2265 (1998)

antiferromagnetic BEC effective” attractive interaction (i.e., ) in

the spin-singlet channel

T. -L. Ho and S. K. Yip, Phys. Rev. Lett. 84, 4031 (2000)

M. Koashi and MU, Phys. Rev. Lett. 84, 1066 (2000)

2 00a a

Why is the fragmented BEC so difficult to observe in reality?

The fragmented BEC is very fragile against symmetry-breaking

perturbations.

Fragmented BEC fragile against symmetry-breaking perturbation

example 2. (antiferromagnetic BEC)

E. Mueller, et al., Phys. Rev. A 74, 33612 (2006)

2† † † 1 -1 1 -10y zx

-0 vac vac A = - A = A

4 2 2

N N ˆ ˆ ˆ ˆa a a adˆ ˆ ˆ ˆ ˆ ˆS , , ai

nA A n A

example 1.

matrix element

2† †

1 20

1 2

†

1 2

1vac

2 2 2 2 !

h.c.

Ni i

N

N N dˆ ˆe c e c

N

ˆ ˆ ˆH tc c

1 2 1 2 (extensive) for 2t N N t N N N N /

† †

1 2

1 2

1vac

2 2 2 !

i i N

N

N Nˆ ˆ( e c e c )

N

This fragmented BEC is fragile against magnetic field because of bosonic

stimulation.

symmetry-breaking per turbation

Topological Defect Formation in Quenched Spinor BEC

Quench=rapid change in external parameters such as

magnetic field

K. Murata, et al., Phys. Rev. A 75, 013607 (2007)

Topological Defect Formation in Quenched Ferromagnetic BEC

line

ar

Ze

em

an e

ne

rgy

quadratic Zeeman energy

m=0

m=1

H. Saito, Y. Kawaguchi, MU, Phys. Rev. A75, 013621 (2007)

The AM conservation prohibits a uniform

magnetization.

What happens to the BEC if we quench

the B-field from the polar to the broken-

axisymmetric phase?

Phase diagram of spin-1 FM BEC

Bogoliubov Spectrum of a Ferromagnetic BEC

Suppose that all atoms are prepared

in the m=0 state in a pancake-shaped trap.

H.Saito, Y.Kawaguchi, and M.U., Phys. Rev. Lett. 96, 065302 (2006)

Bogoliubov spectrum for l =0, 1 as a

function of the spin-exchange interaction g12D

2 0a a

1l unstable

In the region the modes with orbital angular

momentum l = 1 have imaginary parts; they

are therefore dynamically unstable and grow

exponentially.

Bogoliubov Spectrum of a Ferromagnetic BEC

2 0a a

1l unstable

Then the system has orbital AM

instability.

Prediction: the l = 1 modes will start

to grow and rotate spontaneously!

Suppose that the parameter of 87Rb BEC

is prepared at this point.

H.Saito, Y.Kawaguchi, and M.U., Phys. Rev. Lett. 96, 065302 (2006)

Bogoliubov spectrum for l =0, 1 as a

function of the spin-exchange interaction g12D

Chiral Symmetry in a Ferromagnetic BEC

The angular momentum conservation implies that the l =1 and l = -1 modes

must be created simultaneously by the same amount.

There are two possibilities:

spin orbital

degenerate1

1

m

m

1

1

l

l

1

1

m

m

1

1

l

l

spin orbital

* *

1 0 0 12

cos

sin

i

x y

x

y

F F iF e

F

F

1

0

1

1

i

i

e

e

spin texture spin texture

Chiral Symmetry in a Ferromagnetic BEC

The angular momentum conservation implies that the l =1 and l = -1 modes

must be created simultaneously by the same amount.

There are two possibilities:

spin orbital

degenerate1

1

m

m

1

1

l

l

1

1

m

m

1

1

l

l

spin orbital

These two possibilities are degenerate, this degeneracy being a statement

of the chiral symmetry.

spin texture spin texture

Chiral Symmetry Breaking in a Ferromagnetic BEC

The chirally symmetric state lies higher

in energy than the chiral-symmetry broken

states.

Therefore the chiral symmetry will be

dynamically broken and each spin component

will begin to rotate spontaneously.

chirallysymmetric

chiral-symmetry broken

Chiral Symmetry Breaking in a Ferromagnetic BEC

Time development of orbital AM of m = -1 component

chiral symmetry broken

AM 0

latencyperiodAM=0

orbital AM of the m =-1 component

The angular momentum remains zero

for a certain latency period and then acquires

a non-zero value due to the chiral symmetry

breaking.

Chiral Symmetry Breaking in a Ferromagnetic BEC

domain wall topological spin textures

The chirally symmetric state has a domain wall which

costs the ferromagnetic energy. The chiral-symmetry-

broken state circumvents this energy cost by developing

topological spin textures.

Spin

texture

dissipative

Time development of orbital AM m = -1 component

symmetric sym.-broken

chiral symmetry broken

AM 0

latencyperiodAM=0

orbital AM of the m =-1component

Prediction observed by the Berkeley group

snapshots of the transverse magnetization

of an elongated BEC

The system remained unmagnetized for a

certain latency period before magnetization

developed spontaneously.

L. E. Sadler, et al., Nature 443, 312 (2006)

spontaneousmagnetization

Initial conditions: all atoms in the m=0 state.

unmagnetized for a certain latency period

Prediction observed by the Berkeley group

unmagnetizedduring a certain latency period

vortex of the m= 1components with its core filled wit the m=0

component

This polar-core spin vortex corresponds to our chiral-symmetry broken state.

chiral-symmetry

broken states

spontaneousmagnetization

H. Saito, Y. Kawaguchi, and M.U., Phys. Rev. Lett. 96, 065302 (2006)

Initial conditions: all atoms in the m=0 state.L. E. Sadler, et al., Nature 443, 312 (2006)

Formation Dynamics of Spin Vortices

F

Farg

H.Saito, Y.Kawaguchi, and M.U., PRA 75, 013602 (2007)

Minute fluctuations (quantum, thermal, etc.) at the moment of defect nucleations

are amplified to yield observable magnetic domain structures.

Spin correlations give us information about the nature of initial noise.

line

ar

Ze

em

an e

ne

rgy

quadratic Zeeman energy

m=0

m=1

magnitude

phase

A New Testing Ground for Kibble-Zurek Mechanism

Basic idea: spontaneous symmetry breaking at causally disconnected

places generates topological defects (singularities) where the order

parameter does not connect smoothly.

Order parameter

Superfluid phase

magnetization

vortices

Topological defects

Cold Atoms

Detailed comparison between theory

and experiments are feasible.

Cosmology, Superfluid He

T. W. Kibble, J. Phys. A 9, 1387 (1976)

W. H. Zurek, Nature 317, 505 (1985)

q = 0

q = qc / 2

200 m

H. Saito, Y. Kawaguchi, and MU, to be published in PRA

(cond-mat/0704.1377)

line

ar

Ze

em

an e

ffe

ct

quadratic Zeeman effect

m=0

m=1

Spin Vortex Formation in a Quenched BEC

Winding Number and Scaling Law: Prediction

2

1 1

2 2w d F F F F

i Fr

spin winding number

winding number

R

KZ

BEC

Zurek, Nature 317, 505 (1985)

R=linear dimension of the system

vortex spacingR

Winding Number and Scaling Law: Numerical Results

1/ 2w R for large R1w R for small R

H. Saito, Y. Kawaguchi, and MU, to be published in PRA

(cond-mat/0704.1377)

1/ 2w RKZ prediction

--- A rich variety of spinor BEC ---

New Quantum Phases, Magnetism, and Symmetry Breaking

Spin-1 polar BEC: 1/2 vortex

Spin-2 cyclic BEC: 1/3 vortex

Spin-2 polar BEC: Meissner-like effect

Topological defects: spin vortex with broken

chiral symmetry

Kibble-Zurek mechanism

Summary