Entanglement and Coherence in Spin-1 Bose Gas --

Entanglement and Coherence in Spin-1 Bose Gas --

an exact solution for quantum dynamics of a many-body system

Jason Ho

The Ohio State University

International Conference in Recent Progress in Quantum Mechanics and Its Applications

Chinese University of Hong Kong December 14, 2005

Recent Progress in Quantum Mechanics and Its Applications

(1) Phase coherence, entanglement,

(2) Novel condensed matter systems/phenomena

Use BEC to illustrate these progress and application

Hallmark of BEC : phase coherence

Off-diagonal long range order -- coherence at different location

€

λα : Eigenvalues

€

να (r r ) : Eigenfunctions

€

λα

€

α

€

0

€

1

€

2

€

3

€

λ0~N

€

λ1,λ 2,...~O(1)

Existence of a single macroscopic eigenvalue in

€

<ψ+(r r )ψ (

r r ' ) >

€

<ψ+(r r )ψ (

r r ' ) >= λα να

* (r r )να (

r r ')

α∑

Penrose-Onsager characterization of Bose condensation

€

λα

€

α

€

0

€

1

€

2

€

3

€

λ0~N

€

λ1,λ 2,...~O(1)

€

< ψ+(r r )ψ (

r r ' ) >=< ψ

+(r r ) >< ψ (

r r ' ) >

+ …

Off-Diagonal Oder Phase Coherence

CN Yang, RMP, ORLRO paper

Other possibilities ?

€

λα

€

α

€

0

€

1

€

2

€

3

€

λ0~N

€

λ1,λ 2,...~O(1)

€

λα

€

α

€

0

€

1

€

2

€

3 €

λ0

,λ1,λ

2,...~ O(1)

Fragmented condensate

Strongly correlated

Nozieres, 1992

€

λα

€

α

€

0

€

1

€

2

€

3

€

λ0~N€

λα

€

α

€

0

€

1

€

2

€

3

€

λα

€

α

€

0

€

1

€

2

€

3

coherent Fragmented

Strongly correlated

Possible type of ground states:

Single condensate state of spin-F Bosons

€

<ψμ+ (

r r )ψ ν (

r r ') >= λα fμ

*(α ) (r r ) fν

(α ) (r r ' )

α∑

Single particle density matrix contains a single eigenvalue of order N

scalar Spin-1/2 Spin-1

€

<ψμ (r r ) >=

Ψ1

Ψ2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

<ψμ (r r ) >=

Ψ1

Ψ2

Ψ3

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

How to change phase coherence

in a controlled way?

Simple Example : Pseudo-spin 1/2 system

t

1 2

Hilbert space:

t

1 2 BEC --> Fragmentation Coherence --> squeezing --> entanglement

t

1 2 BEC --> Fragmentation Coherence --> squeezing --> entanglement

t

1 2

,

BEC --> Fragmentation Coherence --> squeezing --> entanglement

t

1 2

,


t

1 2

,


Schrodinger Cat State: huge number fluctuation => very delicate

Coherent State : very robust

If there is a robust mechanism that can Convert the coherent state into a Schrodinger Cat state from time to time, then can realize these cat state at these specific time.

Answer : Use Bosonic enhancement

Consider a term like this :

Spin-1 Bose gas


Spin-1 Bose gas


Spin-1 Bose gas


Spin-1 Bose gas


Spin-1 Bose gas

Dynamical Evolution of Spin-1 Bose Gas

Effect of spin degeneracy on BEC

Spin-1 Bose Gas

N spin-1 Boson in a single level

A deep harmonic trap

€

aμ

+

€

μ=1,0,−1

Spin-1 Bose Gas

Spin dynamics of spin-1 Bose gas


€

H = cr S

2

Hilbert space

€

H = cr S

2


Spin-1 Bose Gas



€

Ax = (−a1 + a−1) / 2

€

Ay = (a1 + a−1) / 2i

€

Az = a0

Under spin rotation, rotates like a 3D Cartesean vector .

€

aμ → (e−ir θ ⋅

r S a)μ

€

rA i → R(

r θ )ij

r A j

€

R(r θ ) : 3D rotation

€

aμ

+

€

μ=1,0,−1

Exact ground state :

€

| S = 0 >= ΘN / 2 | 0 >

€

< aμ

+aν >=N3

1 0 0

0 1 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

€

N0 = N1 = N−1 = N /3

€

H = cr S 2 C>0

€

Θ=2a1

+a−1

+ − a0

+2

€

ΔN12 ~ N 2

=

Spin-1 Bose Gas

Spin dynamics of spin-1 Bose gas


00

00

0

0

00

000

0

00

000

0

00 0

0

00 00

0000 0

0

00

Chapman et.al. PRL 04, PRL 05

Watch change with time

Small fluctuation

Large fluctuationsOscillatory behavior in

Data in Chapman et.al. PRL 04

0.1% magnetization Large fluctuation

Oscillatory

Hilbert space :

is a sum of two theta functions !

Elliptic Functions are mathematicians' fairy land.

Those who once glazed upon them are forever captured.

Relate to those at

Relate to those at

time

N=1000

Generality of the quantum Carpet

•Periodic generation of Cat state and coherent state.•Self similar in “space-time”.•Quantum dynamics (periodic revival) can be revealed at short times.

**Fluctuation in Chapman’s experiment is physical.**Specific predictions to reveal quantum dynamics

Printing and deprinting phases on the quantum carpet

New Era in ultra-cold atoms

Strongly correlated cold atoms

Engineering of quantum states

N=100

N=1000


Spin-1 Bose Gas



€

N = aμ

+aμμ∑

€

rS = aμ

+ r S μν aν

μν∑

€

rS μν

: spin-1 matrix

€

rS = −i

r A

+×

r A

€

N =r A

+⋅

r A €

Ax = (−a1 + a−1) / 2

€

Ay = (a1 + a−1) / 2i

€

Az = a0

€

aμ

+

€

μ=1,0,−1

Spin-1 Bose Gas



€

H = cr S

2

€

rS = aμ

+ r S μν aν

μν∑

€

c > 0: Ferromagnetic ,

€

c < 0 : Antiferromagnetic

€

H = cN −c(r A + )2

r A 2

What is the ground state for c>0 ?

To find the optimal spinor condensate ,

€

|ψ >=( ζ μ

μ∑ aμ

+ )+N

N!| vac >

€

ψ | H |ψ = cr S

2

+ cNwe minimize .

For c>0, diamagnetic case,

€

ζ =0

1

0

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

€

|ψ >=a0

+N

N!| vac >=

(ˆ z ⋅r A + )N

N!| vac >

€

rS = 0sincewe have

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

; and

€

|ψ >=( ˆ n ⋅

r A + )N

N!| vac > real.

€

ˆ n , the “polar family”

Conventional condensate :

€

| c >=a0

+N | 0 >N!

€

< aμ

+ aν >=N

2

1

0

1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟1 0 1( )

€

N0 = 0,

€

N±1 = N /2

€

H = cr S 2 C>0

€

< r

S >= 0

€

ΔN12 ~ N

Average the coherrent state over all directions

Relation between singlet state and coherent state

x

y

z

Because

€

ΔN12 ~ 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

Spin textures

Long sample

Short sample

Single Skyrmion

components density Texture

Ferromagnet

Low rotation speed

Skyrmion pair

Ferromagnet

Faster rotation speed


Skyrmion Lattice

Ferromagnet

Faster rotation speed


AntiferromagnetAngular momentum carrying object:

p-disclination or 1/2 vortex

€

ze−|z|2

0

e−|z|2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Topological Singularity

Nematic order vanishes at core -- replaced with Ferromagnetic

Single Disclination

Order: pink=nematicGreen=ferromagnetic

Components

Density

Ferromagneticorder

Four Disclinations


Components

DensityCores alignedantiferromagnetically

Disclination Lattice


Components

Density

A Quick Overview

What we have now and what we look for.

“Statistical” Ferromagnet, spin-gauge symmetry

Na-23

Rb-87€

c2 > 0

€

c2 < 0

antiferromagnetic

ferromagnetic

€

<ψμ >=(0,1,0) Polar phase

€

<ψμ >= 1,0,0( ) Quantum ferromagnet

Found at MIT Science 1998

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

Spin 1• Ferromagnetic • Antiferromagnetic

€

R

1

0

0

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

€

R

0

1

0

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Vector order parameter

Spin-Gauge SymmetryNematic order parameter

Analogous phases seen in 3He

Similar to 2-component case

€

Ax = (−a1 + a−1) / 2

€

Ay = (a1 + a−1) / 2i

€

Az = a0

Define

Under spin rotation,

Optical Lattice:A whole host of new states

Cr Condensate : Tilman Pfau, PRL 2005.

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

a

Triangle Skew Square StretchLocked

0 0.172 0.373 0.926

How lattices intermesh:

a=Interaction between components

Interaction within components

€

=g12

g11g22

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

L=2

L=4 L=12

Difference betweenBEC 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, PRL89, 050401 (2002)

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

What have we learned ?

What novel things awaiting for us?

What can we do with multi-component Bose gases?

What is new?

What have we learned ?

What novel things awaiting for us?

What can we do with multi-component Bose gases?

What is new?

Periodic changes from coherent state to Schrodinger Cat state

Can introduce phase difference between difference component of the Cat, and retrieve those information later.

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 new quantum phenomena, forces on to re-examine BEC with greater depth

Work done with Dr. Roberto DienerProf. 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

€

H = ∫ h2

2M∇ψ + ∇ψ + V (

r r )ψ +ψ +

g

2ψ +(r)ψ +(r)ψ (r)ψ (r)

⎡

⎣ ⎢ ⎤

⎦ ⎥

€

H = ∫ h2

2M∇ψ μ

+∇ψ μ + V (r)ψ μ

+ψ μ + 1

2gμνψ μ

+ψ ν

+ψ νψ μ

⎡

⎣ ⎢ ⎤

⎦ ⎥

€

μ =↑,↓

€

Hs

=h2

2M∇ψ

μ+∇ψ

μ+ V (

r r )ψ

μ+ψ

μ− γ

r B ⋅ψ

μ+ r

F μν

ψν

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

+1

2c0ψ

μ+ψ

α+ψ

αψ

μ+ c

2ψ

μ+ψ

α+ r

F μν

⋅r F αβ

ψβ

ψν

⎡ ⎣ ⎢

⎤ ⎦ ⎥

€

∫

€

∫

€

ηψμ+ ( ˆ B ⋅

r F )2

⎡ ⎣ ⎢

⎤ ⎦ ⎥μν

ψν

€

∫+

Single component

Two component

Spin-1

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

f = 1

€

Na23, K 39 Rb87

€

V (1,2) = δ(r r 1 −

r r 2) c01+ c2

r F 1 ⋅

r F 2( )

€

c0 = (2g2 + g0) /3,

€

c2 = (g2 − g0) /3,

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

Na-23

Rb-87

€

c2 > 0

€

c2 < 0

antiferromagnetic

ferromagnetic

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

€

1= P0

+ P2

€

2r F 1 ⋅

r F 2 = ∑F=0

2 f F(F +1) ˆ P f − 2 f ( f +1)ˆ 1

€

2r F 1⋅

r F

2= (

r F 1

+r F

2)2 −

r F 12 −

r F

22

€

rF 1

2 =r F 2

2 = f ( f +1)

€

2r F 1 ⋅

r F 2 = 6 ˆ P 2 + 2 ˆ P 1 − 4( ˆ P 2 + ˆ P 1 + ˆ P 0)

€

rF 1 ⋅

r F 2 = 2 ˆ P 2 − 2 ˆ P 0

€

1= P0

+ P1

+ P2

€

V (1,2) = δ(r r 1 −

r r 2) g0P0 + g2P2( ) = δ(

r r 1 −

r r 2) c01+ c2

r F 1 ⋅

r F 2( )

€

c0 = (2g2 + g0) /3,

€

c2 = (g2 − g0) /3,

f = 1

€

Na23, K 39 Rb87

f = 2

€

Rb83 Rb85

€

Cs133,Cs135,Cs137 f = 3

€

V (1,2) = δ(r r 1 −

r r 2) c01+ c2

r F 1 ⋅

r F 2( )

€

V (1,2) = δ(r r 1 −

r r 2) c01+ c2

r F 1 ⋅

r F 2 + c4 (

r F 1 ⋅

r F 2)

2( )

€

V (1,2) = δ(r r 1 −

r r 2) c01+ c2

r F 1 ⋅

r F 2 + c4 (

r F 1 ⋅

r F 2)

2 + c6(r F 1 ⋅

r F 2)

3( )

86.2days

30.2yrs3Million yrs

Fragmented vs Coherent Condensates

Can there be other possibilities ?

€

λα

€

α

€

0

€

1

€

2

€

3

€

λ0~N

€

λ1,λ 2,...~O(1)

€

λα

€

α

€

0

€

1

€

2

€

3 €

λ0

,λ1,λ

2,...~ O(1)

Fragmented condensate

Various states of light: Coherent (Gaussian) State

Squeezed State

Cat State

Bosons in double well:

t

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

Zero

The Ground States of

Zero Dimensional Spin-1 Bose Gas

S

singlet

Squeezed

coherent

Field gradient

Quantum carpet for a particle in a box

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

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

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

Our approach: Quantum evolution of the initial state

Note: According to mean field, if all bosons are initially in the state, should remain constant, .

within the single mode approximation.

Key findings:

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 studies of “quantum carpets” in the last decade in atomic and molecular physics. It is a concise summary of all numerical results in the last decade.

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

More details of our approach:

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

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

Schrodinger Cat running on a Quantum Carpet:

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, with magnetization, etc.

Phase impriting

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.







M=10, N=1000























non-zero q, q=0.1



Initial state : N_{o}=N, M=0









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

Entanglement and Coherence in Spin-1 Bose Gas --