Jason Ho- Spinor-BEC and Multi-component Quantum Gases

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Spinor BEC andMulti-component Quantum Gases

Jason HoThe Ohio State University

Center for Advanced StudyTsinghua University

December 13, 2005



What have we learned ?

What novel things awaiting for us?

What can we do with multi-component

Bose gases?

What is new?



Spin´ 1/2 : Two different spin states of Rb

Spin-1 : |F=1 > hyperfine state of Rb-87 and Na-23

How do the ground states of these differ from scalar Bose gas and

from each other?

Spin-2 : |F=2 > hyperfine state of Rb-87

Examples of multi-component Bose gas

Do we expect any other ground states beside Bose condensed states?



What else besides Bose condensed states?

What are the reasons for un-condensed

ground states in Bose systems?

Why do they occur in multi-component systems?



Conventional Bose condensate :

all Bosons condenses into a single state.

Vanishing inthermodynamic

limit

3D 1D

No condensation



What happens when there are several degenerate

state for the Bosons to condensed in?

G: Number of degenerate states N: Number of Bosons



What happens when there are several degenerate

state for the Bosons to condense in?


Pseudo-spin 1/2 Bose gas: G =2



Spin-1 Bose gas : G=3, G<<N





Bose gas in optical lattice: G ~N





Bose gas in optical lattice: G ~N

Fast Rotating Bose gas: G>>N




PE : Eigenvalues

RE (&

r ) : EigenfunctionsPE

E0 1 2 3

P0

~ N

P1

,P2

,...~O(1)

Existence of a single macroscopic eigenvalue in

](&r )](

&r ' ) "

](&r )](&r ' ) "! PERE

* (&r )RE (&r ')

E§

Penrose-Onsager characterization of Bose condensation



PE

E0 1 2 3

P0

~ N

P1

,P2

,...~O(1)

]

(&r )](

&r ' ) "! ]

(&r ) " ](

&r ' ) "

+ «

Off-Diagonal Oder Phase Coherence



Other possibilities ?

PE

E

0

1

2

3

P0

~ N

P1

,P2

,...~O(1)

PE

E0 1 2 3

P0

,P1,P

2,...~O(1)

Fragmented condensate

Strongly correlated



PE

E

0

1

2

3

P0~ N

PE

E

0

1

2

3

PE

E0 1 2 3

By varying the external environment,

one can go from one regime to another



Single condensate state of spin-F Bosons

]Q

(&r )]R (

&r ') "! PE f Q

*(E ) (&r ) f R

(E ) (&r ' )

E§

Single particle density matrix contains a single eigenvalue of order N

scalar Spin-1/2 Spin-1

]Q (&

r ) "! =1

=2

¨ª©

¸ º¹

]Q (&r ) "! =

1

=2

=3

¨

ª

©©©

¸

º

¹¹¹



H ! ´ !2

2 M ] ] V (

&r )]]

g

2](r )](r )](r )] (r )

«

-¬»

½¼

H ! ´ !2

2 M ]Q

]Q V (r )]Q]Q

1

2 g QR]Q

]R]R]Q

«

-¬»

½¼

Q !o,q

H s

!!

2

2 M ]

Q]

QV (

&r )]

Q]

Q K

& B ]

Q & F QR

]R

«

-

¬

¬

»

½

¼

¼

1

2c

0]Q]

E]

E]Q c

2]Q]

E & F QR

& F EF

]F]R

«-¬

»½¼´́

L]Q ( Ö B

& F )

2«-¬

»½¼

QR

]R

´+

Single component

Two component

Spin-1

T.L. Ho, PR L (98), K . Machida et.al J. Phil Mag (98)



Internal conversion in Spin-1 and Spin-2 Bose gas

Absent in spin 1/2 Bose gas!



Can there be other possibilities ?

PE

E

0

1

2

3

P0

~ N

P1

,P2

,...~O(1)

PE

E0 1 2 3

P0

,P1,P

2,...~O(1)




Bosons in double well:



t

1 2Bose gas in double well <==> A simple model of

two component Bose gas







A Quick Overview

What we have now and what we look for.



Condensed

Fast Rotating : Great varietyof vortex lattices

Optical lattice: new states

Coherent Spin dynamics

From highly correlated toGeneralized Schrodinger cats

New quantum Hall states

Very rich correlated structures

Quantum Spin dynamics

Uncondensated

Textures New type of excitations

Low dimension: New solitons

New 1D and 2D correlated states

Spin 1/2

Spin-1

Spin-2

Coreless vortex

Spinor BEC



Condensed

Fast Rotating : Great varietyof vortex lattices

Optical lattice: new states

Coherent Spin dynamics

From highly correlated toGeneralized Schrodinger cats

New quantum Hall states

Very rich correlated structures

Quantum Spin dynamics

Uncondensated

Textures New type of excitations

Low dimension: New solitons

New 1D and 2D correlated states

Spin 1/2

Spin-1

Spin-2

Coreless vortex

Cornell

Ketterle

ChapmanSengstock

Ho

Machida

Pu, Law

Bigelow

Ueda

Li You Ho,

Ueda

Ho+Mueller,

Cornell, Ueda

Cirac+Zoller, Ho+Mueller,Shouten+Read

Bloch, Fei Zhou, Yip, Delmer

Chapman

Sengstock

Stamper-

Kurn

Li You, Bigelow,

Bong, Sengstock

Bigelo

w

(Diener +Ho)



Spin-1 and Spin-2 Bose Gases

Mean field description



Statistical´ Ferromagnet, spin-gauge symmetry



Na-23

Rb-87

c2

" 0

c2

0

antiferromagnetic

ferromagnetic

]Q "! (0,1,0) Polar phase

]Q "! 1,0,0 Quantum ferromagnet

Found at MITScience 1998

Bong,Sengstock, et.al. PRL 04, Chapman et.al PRL 04



Spin 1

Ferromagnetic Antiferromagnetic

R

1

0

0

¨

ª

©©©

¸

º

¹¹¹

R

0

1

0

¨

ª

©©©

¸

º

¹¹¹

Vector order parameter

Spin-Gauge Symmetry

Nematic order parameter

Analogous phases seen in3

He

Similar to 2-component case



A x

! (a1

a1) / 2

A y

! (a1

a1) / 2i

A z

! a0

Define

Under spin rotation,





Spin textures

Long sample

Short sample



Single Skyrmion

components densityTexture

Ferromagnet

Low rotation speed



Skyrmion pair Ferromagnet

Faster rotation speed




Skyrmion Lattice

Ferromagnet

Faster rotation speed




AntiferromagnetAngular momentum carrying object:

-disclination or 1/2 vortex

ze| z|2

0

e

| z|2

¨

ª

©©©

¸

º

¹¹¹

Topological Singularity

Nematic order vanishes at core -- replaced with Ferromagnetic



Single DisclinationOrder: pink=nematic

Green=ferromagnetic

Components

Density

Ferromagnetic

order



Four DisclinationsOrder: pink=nematic

Green=ferromagnetic

Components

DensityCores aligned

antiferromagnetically



Disclination LatticeOrder: pink=nematic

Green=ferromagnetic

Components

Density



Fast Rotating 2 componentMueller and Ho, PRL 88, 180403 (2002)

Triangle Skew Square StretchLocked

0 0.172 0.373 0.926

How lattices intermesh:

=Interaction between components

Interaction within components!

g 12

g 11 g 22





Fast Rotating Spin-1/2 and Spin-1 Bose gas



L=2

L=4 L=12

Difference between

BEC and

quantum Hall state

BEC QH

N=2

N=4

Difficulty in stabilizing correlated states :

expansive in kinetic energy

Advantage: Zero potential energy

1 2

Vortex core 2

1



New Quantum Hall systems:

Quantum Hall states with large spins

Bilayer (or Multilayer ) quantum Hall systems

Composite QH systems

Spin-1 Bose gas

scalar

scalar



Density -- 5 particles

Total Density Component Densities

T.L. Ho and E. Mueller, PR L89, 050401 (2002)



Optical Lattice:

A whole host of new states



Effect of spin degeneracy on BEC

Only the lowest harmonic state is occupied

=> a zero dimensional problem

Spin-1 Bose Gas


A deep

harmonic

trap

aQ

Q!

1,0,1



Spin-1 Bose Gas

Spin dynamics of spin-1 Bose gas

A deep

harmonic

trap

H ! c&S

2

Hilbert space




Spin-1 Bose Gas


A deep

harmonic

trap

A x

! (a1

a1) / 2

A y

! (a1

a1) / 2i

A z

! a0

Under spin rotation, rotateslike a 3D Cartesean vector .

aQ p (ei

&

&S

a)Q

& A

ip R(

&)

ij

& A

j

R(&) : 3D rotation

aQ

Q!

1,0,1




Spin-1 Bose Gas


A deep

harmonic

trap

N !a

Q

a

QQ§

&

S

!aQ

&

S QRaRQR

§

: spin-1 matrix

&S ! i

& A

v

& A N !

& A

& A

A x

! (a1

a1) / 2

A y

! (a1

a1) / 2i

A z

! a0

aQ

Q!

1,0,1



Spin-1 Bose Gas


A deep

harmonic

trap

H ! c&S

2

&

S ! aQ &

S QR aRQR§

c " 0: Ferromagnetic ,

c 0 : Antiferromagnetic

H ! c N c(

&

A

)2&

A

2

What is the ground state for c>0 ?



To find the optimal

spinor condensate ,

| ] "!( ^Q

Q§ aQ

) N

N !

| vac "

] | H | ] ! c&S

2

c N we minimize .

For c>0, diamagnetic case,

^ !

0

10

¨

ª

©

©©

¸

º

¹

¹¹

| ] "!

a0

N

N !| vac "!

( Ö z & A

) N

N !| vac "

&

S ! 0sincewe have

Since H is rotationally invariant, the optimal states are given by

; and

|] "!

( Ö n & A

) N

N !| vac " real.

Ö n ,the ³polar family´



Conventional condensate :

| c "! a0 N | 0 " N !

aQ aR "! N

2

1

0

1

¨

ª©©©

¸

º¹¹¹

1 0 1

N 0 ! 0,

N s1 ! N /2

H ! c&S

2 C>0

&

S "! 0

( N 12 ~ N



Exact ground state :

| S ! 0 "! 5 N /2 | 0 "

aQaR "! N

3

1 0 0

0 1 0

0 0 1

¨

ª©©©

¸

º¹¹¹

N 0 ! N 1 ! N 1 ! N / 3

H ! c&S

2 C>0

5 ! 2a1

a1

a0

2

( N 1

2~ N

2

=



Average the coherrent state over

all directions

Relation between singlet state and coherent state

x

y

z

Because

( N 1

2~ N

2

The system is

easily damaged



Transformation of singlet into coherent states as a function of

External field and field gradient:

If the total spin is non-zero

Bosonic enhancement






Bosonic enhancement











With field gradient



S/N

=/N^2







Dynamical Evolution of

Spin-1 Bose Gas



Spin-1 Bose Gas

Spin dynamics of spin-1 Bose gas

A deep

harmonic

trap

00

00

0

0

00

000

0

0 00

00

000 0

0

00 00

0000

0

0

0

0

Chapman et.al.

cond-mat/0309164

Watch how change with time

D i Ch l d /0309164



Small fluctuation

Large fluctuationsOscillatory behavior in

t=0 starts here,

According to Chapman

Data in Chapman et.al. cond-mat/0309164



To a large extent, can be explained by

time-dependent GP equation

What happen if one evolves the state properly by quantum mechanics?



0.1% magnetizationLarge fluctuation

Oscillatory



For simplicity, consider zero field







is a sum of two theta functions !





Relate to those at



N=100

N=1000





time









Cr Condensate :

Tilman Pfau, PR L 2005.



Dipolar coupling in fluids

Ferrofluids

~ 2-20 nm

Application:

rotary seals in disk drives

dampers for audio speakers



Cond-mat/ Nov 2005









Signiture of Quantum dynamics:

(1) Complete revival

(2) Periodic variation between cat states and coherent states

(3) Entire spacetime structure is control by scaling

(4) Printing of phase can be deprint at later time

(5) Can be detected by population histogram at time p/q






Bose gases?

What is new?






Bose gases?

What is new?



Periodic changes from coherent state toSchrodinger Cat state

Can introduce phase difference between difference component

of the Cat, and retrieve those information later.



Question: What really new things BEC



Question: What really new things BEC

has brought us?

Ans: Bose gas with internal degrees

of freedom --Pseudo-spin 1/2 Bose gas,

Spin-1 and Spin-2 Bose gas

New ground states, a whole host of newquantum phenomena, forces on to re-

examine BEC with greater depth



Work done with

Dr. Roberto Diener Prof. S.K. Yip

Work supported by NSF and NASA

Tin-Lun Ho and Sung Kit Yip,Physical Review Letters 84,

4031 (2000)

R. Diener and Tin-Lun Ho, to be published

´ !2« »

Single component



H ! ´ !2

2 M ] ] V (

&r )]]

g

2](r )](r )](r )] (r )

«

-¬»

½¼

H ! ´ !2

2 M ]Q

]Q V (r )]Q]Q

1

2 g QR]Q

]R]R]Q

«

-¬»

½¼

Q !o,q

H s

!!

2

2 M ]

Q]

QV (

&r )]

Q]

Q K

& B ]

Q & F QR

]R

«

-¬¬

»

½¼¼

1

2c

0]Q]

E]

E]Q c

2]Q]

E & F QR

& F EF

]F]R

«-¬

»½¼´́

L]Q ( Ö B

& F )

2«-¬

»½¼QR

]R

´

+

Two component

Spin-1

T.L. Ho, PR L (98), K . Machida et.al J. Phil Mag (98)

T.L. Ho, PR L 87, 81, 742 (1998)



f = 1

N a23

, K 39

Rb87

V (1,2) ! H(&

r 1

&

r 2) c

01 c

2

&

F 1

&

F 2

c0

! (2 g 2 g 0) /3,

c2

! ( g 2 g 0) /3,

Na-23

Rb-87

c2

" 0

c2 0

antiferromagnetic

ferromagnetic

It is useful to rewrite P in terms of spin operators



It is useful to rewrite P in terms of spin operators.

1! P 0 P

2

2&

F 1

&

F 2

! § F ! 0

2 f F ( F 1) Ö P

f 2 f ( f 1)Ö 1

2&

F 1

&

F 2

! (&

F 1

&

F 2)2

&

F 12

&

F 22

&

F 1

2!

&

F 2

2! f ( f 1)

2&

F 1

&

F 2

! 6 Ö P 2 2 Ö P

1 4( Ö P

2 Ö P

1 Ö P

0)

&

F 1

&

F 2

! 2 Ö P 2 2 Ö P

0

1! P 0 P

1 P 2

V (1,2) ! H(&

r 1

&

r 2) g

0 P

0 g

2 P

2 ! H(&

r 1

&

r 2) c

01 c

2

&

F 1

&

F 2

c0

! (2 g 2 g

0) /3, c

2! ( g

2 g

0) /3,

23 39 87



f = 1

N a23

, K 39

Rb87

f = 2

Rb83

Rb85

C s133,C s135,C s137f = 3

V (1,2) ! H(&

r 1

&

r 2) c

01 c

2

&

F 1

&

F 2

V (1,2) ! H(&

r 1

&

r 2) c

01 c

2

&

F 1

&

F 2 c

4(&

F 1

&

F 2)2

V (1,2) ! H(&

r 1

&

r 2) c

01 c

2

&

F 1

&

F 2 c

4(&

F 1

&

F 2)2 c

6(&

F 1

&

F 2)3

86.2days

30.2yrs3Million yrs







Fragmented vs Coherent Condensates





PE

E

0

1

2

3

P0

~ N

P1

,P2

,...~O(1)

PE

E0 1 2 3

P0

,P1,P

2,...~O(1)


Various states of light:



Coherent

(Gaussian)

State

Squeezed

State

Cat State

Bosons in double well:



1 2Bose gas in double well <==> A simple model of

B



t

1 2two component Bose gas







Zero



The Ground States of

Zero Dimensional Spin-1 Bose Gas

S



S

singlet

Squeezed

coherent

Field gradient



Quantum carpet for a particle in a box

M V Berry Quantum fractals in boxesJ. Phys. A 29 6617-6629 (1996)

C Leichtle, I S Averbukh and W P Schleich 1996 Multilevel quantum beats: an analytical approachPhys. Rev. A 54 5299-312 (1996)

O Friesch, I Marzoli and W P SchleichQuantum carpets woven by Wigner functions

N ew J. Phys. 2, 4.1-4.11 (2000)



Many simple questions have

led us to remarkable surprises Nature has for us

0.2% magnetization



0.2% magnetization



g

Note: According to mean field, if all bosons are



Our approach: Quantum evolution of the initial state

g ,

initially in the state, should

remain constant, .

within the single mode approximation.

Key findings:



y g

1. Find all the features observed in expt.

2. Obtain an analytic solution for the time evolution

of the wavefunction.

3. The exact solution reveals many additional

revival features at longer times

4. Our exact solution is also applicable to the studiesof ³quantum carpets´ in the last decade in atomic

and molecular physics. It is a concise summary

of all numerical results in the last decade.

More details of our approach:



A. We note that single mode approximation is valid in this case.

Hence

B. Different spin component of a quantum state will dephase

over time . Beyond this time, mean field approach is

no longer valid.

C. The general quantum state for is

Schrodinger Cat running on a Quantum Carpet:



The large fluctuation is due to dynamical fragmentation of the

condensate -- a periodic transformation between Schrodinger

Cat state and a coherent state.

Schrodinger Cat state occurs at ct = coherent state

occurs at ct =

Phase imprinting at pi/8 (using quadratic Zeeman effect) will affect

subsequent time evolution, and will change the coherent state

structure.

Deprinting the phase at 3\pi/8 can restore the original time evolution

Reference:



(a) Roberto Diener and Tin-Lun Ho, to be published.

( b) M.-S. Chang, C.D. Hamley, M. D. Barrett, J.A. Sauer,

K.M. Fortier, W. Zhang, L. You, M.S. Chapman,

cond-mat/0309164.

(c) Schmaljohann, M. Erhard, J. Kronjäger, M. Kottke,

S. van Staa, L. Cacciapuoti, J. J. Arlt, K. Bongs, and K. Sengstock

Phys. Rev. Lett. 92, 040402 (2004)





















N o as a function of time without magnetization



N_o as a function of time, without magnetization,

with magnetization, etc.

Phase impriting

M=10, N=1000



QuickTime and a

TIFF

(Uncompressed) decompressor are needed to see this picture.

QuickTime and a

TIFF (Uncompressed) decompressor are needed to see this picture.

QuickTime and aTIFF (Uncompressed) decompressor

are needed to see this picture.















QuickTime and a

TIFF (Uncompressed) decompressor are needed to see this picture.





non-zero q, q=0.1Initial state : N_{o}=N, M=0























We have only talked about 3/5 of the major development

last year



last year.

Two other major developments:

*Spin-1 and Spin-2 Bose gas:

Expt. (Chapman) (Sengstock )

=> Quantum dynamics of

Spinor BEC and Superfragmented condensates

=> Periodic generation of Schrodinger Cat state

Low dimensional quantum gases

T.L.Ho, PRL, 81, 742 (1998)

T.L. Ho and S.K. Yip,PRL 84, 4031(2000)

Documents

Jason Ho- Spinor-BEC and Multi-component Quantum Gases