Introduction of the F=1 spinor BEC

Introduction of the F=1 spinor BEC

郭西川

國立彰化師範大學物理系

F=1 spinor BEC :23Na, 87Rb,...

Spinor BEC

electron spin

F I Shyperfine spin

nuclear spin

BECmagnetic trapping (one-component, scalar)

optical trapping (multi-component, vector)

2

22 2 2 4 †1 1

2 2 2n s

K H N

d V g gm

Ψ

r Ψ r Ψ Ψ Ψ Ψ FΨ

Grand canonical energy-functional for the spinor BEC (Ho, Ohmi)

2 20 2 0 224 4

, 3 3n s

a a a ag g

m m

(gs<0: ferromagnetic; gs>0: antiferromagnetic)

N is a fixed number

0

Ψ

order parameter:

1 3 U SO const Ψ

global phase

rotational symmetry in spin space

x x y y z zF F F F e e espin operators:

0 1 0 0 0 1 0 01 1

1 0 1 , 0 , 0 0 02 2

0 1 0 0 0 0 0 1x y z

i

F F i i F

i

†

* * *0 0

* * * * * *0 0 0 0 0 0

* *

0 1 0 0 0 1 0 01 1

ˆ ˆ ˆ1 0 1 0 0 0 02 2

0 1 0 0 0 0 0 1

1ˆ ˆ

2 2

ˆ

i

i i

i

i

Ψ FΨ

x y z

x y

z

2 2 2* 20 0

2 2† *0 0

2 2 2* 20 0

2

Ψ FΨ FΨ

Ground state structure of spinor BEC

2

0

0

, = ,0

(total particle density)

(total particle number)

n

n n n n

nd N N N N

r

0 0n n

Ψ r r χ

Define the normalized spinor byχ

† 1 χ χ

Such that all spinors are degenerate with the transformation

, , , ie U χ χ

, , yz ziFiF iFU e e e where

Euler angles

Define †F χ Fχ

2 2 22 2 2

2 2 2 n s

K

nd n V n g g

m m

Ψ

r χ r F

then the free energy can be expressed as

The free energy of K is minimized by:

sin0

1. 0, 0 1 2 cos2

0 sin

i

ii

si

ee

g e U

e

F χ

22

2

2 22

2

cos1

2. 0, 1 0 2 cos sin

0 sin

i

iis

i

e

g e U e

e

F χ

2 1U S

1 3U SO

2 2

* 20 02 n s si V g n g n n n g

t m

r

2 2

*00 0 02

2 n s si V g n g n n gt m

r

2 2

* 20 02 n s si V g n g n n n g

t m

r

Coupled GPE for spinor BEC

Time-dependent coupled GPE

2 2

* 20 02 n s sV g n g n n n g

m

r

2 2

*0 0 0 02

2 n s sV g n g n n gm

r

2 2

* 20 02 n s sV g n g n n n g

m

r

Time-independent coupled GPE

Modification of GPE – conservation of magnetization

The spin-exchange interaction also preserves the magnetization M, so we have two constraints for the GPE

0

01 0 1

N N N N const

M N N N N N const

222 2 2 4 †

2 2

22 2 2 2 2

0

24 †

1 1

2 2 2

2

1 1

2 2

n s

n s

K H N BM

d V g gm

B

d V B Bm

g g

Ψ

r Ψ r Ψ Ψ Ψ Ψ FΨ

r Ψ r Ψ

Ψ Ψ FΨ

N and M are both fixed numbers

The conservation of particle number and magnetization is equivalent to introduce the Lagrange multipliers and B in the free energy

2 2

* 20 02 n s sV g n g n n n g B

m

r

2 2

*0 0 0 02

2 n s sV g n g n n gm

r

2 2

* 20 02 n s sV g n g n n n g B

m

r

The corresponding GPE’s are modified to

Some interesting results

2 20

2

0+ -

0

21 12 4 ,

=

N NN M N MN N NN

N N N

For 0, M M N

Ferromagnetic:

anti-ferromagnetic:0 =0 N

3 identical decoupled equations

2 coupled-mode equations

Using the inequality rΦΦFΦ g2† ˆ

Case 1:

the ground state can be constructed by minimizing the spin-dependent part of the Hamiltonian

2†† ˆ

2

1ˆ ΦFΦΦΦ szL gF

22

0

22† ˆ ΦΦΦ zFChoose

However, since22† ˆ

ΦΦ zF

0

0

0

0

ig

fg

e

Φ (ferromagnetic state)

0sg

the spin configuration of

Field-induced phase segregation of the condensate configuration

0Φ

i

i

i

Fii

e

e

e

ee z00

ˆ

Let

the Hamiltonian is invariant under the gauge transformation

zLeB (Larmor frequency)

Case 2:

Choose 0ˆˆ †† ΦΦΦΦ yx FF

s

L

s

LzszszL gg

FgFgF2

ˆ2

1ˆ2

1ˆ22

†2

††

ΦΦΦΦΦΦ

so that we have

The minimum is achieved if sLz gF ΦΦ ˆ†

00**

0*

0*

0

We may assume that 0 0 and is real but since

* 0ˆ 22† ΦΦ zF

0ˆˆ †† ΦΦΦΦ yx FF

00

0sg

which is in contraction with the condition sLz gF ΦΦ ˆ†

and we must conclude that

ig

ig

pg

e

e

0Φ (polar state )

sLg gr

2

sLgg

g

rr

For

where

,sL grNote that when the condition sLz gF ΦΦ ˆ†

cannot be fulfilled and we must choose such that ΦΦ zF̂†

is as close to sL g as possible. This implies that

0

0

0

0

ig

fg

e

Φ

and thus the ground state is described by

,0

(ferromagnetic state )

ig

ig

pg

e

e

0Φ sLg gr

For

0

0

ig

fg

e

Φ sLg gr

gΦ

0sg the two different configurations coexist:

The phase boundary rb is determined by

sLbg gr

(ferro region)

(polar region)

2322

ferro

3

2

2

2

polar

3

2

1

2

1

2

1

2/1

1

2

1

gngextgsgLg

s

L

sgL

g

gVrdgM

rd

ggMrdNHG

r

The free energy is given by

,0/ G

regionpolar

region ferro

n

ext

sn

extL

g

g

V

gg

V

r

r

r

In the Thomas-Fermi limit, the minimization of the free energy, leads to

Example: 2202r

MVext r

2/5

2/52/3

20

2

15

8LL

s

n

n

s

sn g

g

g

g

MggN

bn

bsn

L

g

rrg

rM

rrgg

rM

2/

2/

220

220

r

The total particle number and the chemical potential is related by

2

0

20

/2

//2

MR

Mggr

L

sLnb

phase boundary

radius of atomic cloud

ti

ti

et

t

et

,0

,

,

,

,

r

r

r

r

r

t

t

et ti

,

,

,

0

,

r

r

r r

t,rΦ

collective excitations tt g ,, rΦrΦrΦ

tt g ,, rrr

hydrodynamic-like mode

tittt

tittt

ss

exp,,,

exp,,,

rrrr

rrrr

Furthermore, we let fluctuation of number density

fluctuation of spin density

Derivation of hydrodynamic equations

polar region

ferro region

rrrr

rrrr

sgsL

s

ns

sL

gn

M

g

Mg

g

MM

g

2

2

rrr

gsn

M

gg2

s ,

ΦFΦFΦΦrΦ

ˆˆˆ

2

1 †22snzLext ggFV

Mti

Substituting into the time-dependent GP equation

Upon linearization we obtain the hydrodynamic equations in different regions:

polar region

ferro region

(Stringari)

Example: 2202r

MVext r

0

0

22

222222

2222222

sL

ss

sL

rrRrR

rrRrR

Rr /

where nsLL ggMr / ,/2 ,/2 20

220

2

01412

2

22

22

2

DfffA ll

d

d

d

dll

d

d

d

d

sL

L

f

f

Rr

RrfDA ,

/0

0 ,

1/

/1 where

222

22

Now let ,,,, lmsslm YfYf rr

322

232

2

2-2

uuD

uu kkkk cclkk

lkklkcc

/1 22 Rrλ L

uuuu kkkk ccc

Denote u as the eigenvectors of A with eigenvalues and let

To solve the coupled equations, let

, 0

1

k

kk

k

kk b

ax uuf

322

2322

2 kk lkk

lkklku

DAu

and we obtain the recursion relation

Boundary condition requires that the series must terminate at some interger k=2n

0 2644 22 nlnnln uD

322/ 0

322 0

22/10

22

20

22

lnnlnggEb

lnnlnEa

nss

nln

n

nln

n

u

u

and the solution is

The dispersion relations do not depend on the magnetic field!

L-independence of solutions of polynomial-type

3

1,2

1

jijiijext xxMV r

Consider a general quadratic potential

where the 33 matrix ijis positively definite.

Consider a polynomial solution

r

r

r

r

k

k

nks Q

P

where Pk(r) and Qk(r) are homogeneous polynomials of degree k.

Note that if Pk(r) is a polynomial of x,y,z with degree k

rkP2 is a polynomial of degree k-2

rr kg P is a polynomial of degree k

Clearly, terms of even degree are decoupled from terms of odd degree. So we may assume

r

r

r

r

r

r

k

k

gsL

Lg

n

nkk

k

nk Q

P

M

g

M

MM

g

Q

P

2

2

2

.03131 rrrr nnnn QQPP

Collecting terms of degree n on both sides

rr

rr

r

r

ngn

s

ng

n

n

QMg

g

PM

Q

P

1

2 The obtained frequency does not depend on L

In memory of Prof. W.-J. Huang (黃文瑞 )

— A friend and a tutor

ΦFΦBΦFΦ

ΦΦrΦ

ˆˆ2

1

2

1

2

1

†2

†

4223

s

next

g

gVM

rdH

Gross-Pitaeviski Hamlitonian for the spinor BEC (Ho, Ohmi)

3

4,

3

24 2020 aa

Mg

aa

Mg sn

(gs<0: ferromagnetic; gs>0: antiferromagnetic)

ratio icgyromagnet: