Entanglement and Coherence in Spin-1 Bose Gas --
description
Transcript of 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
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λα : Eigenvalues
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να (r r ) : Eigenfunctions
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λα
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α
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0
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1
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2
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3
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λ0~N
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λ1,λ 2,...~O(1)
Existence of a single macroscopic eigenvalue in
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<ψ+(r r )ψ (
r r ' ) >
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<ψ+(r r )ψ (
r r ' ) >= λα να
* (r r )να (
r r ')
α∑
Penrose-Onsager characterization of Bose condensation
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λα
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α
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0
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1
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2
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3
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λ0~N
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λ1,λ 2,...~O(1)
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< ψ+(r r )ψ (
r r ' ) >=< ψ
+(r r ) >< ψ (
r r ' ) >
+ …
Off-Diagonal Oder Phase Coherence
CN Yang, RMP, ORLRO paper
Other possibilities ?
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λα
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α
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0
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1
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2
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3
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λ0~N
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λ1,λ 2,...~O(1)
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λα
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α
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0
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1
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2
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3 €
λ0
,λ1,λ
2,...~ O(1)
Fragmented condensate
Strongly correlated
Nozieres, 1992
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λα
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α
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0
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1
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2
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3
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λ0~N€
λα
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α
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0
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1
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2
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3
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λα
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α
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0
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1
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2
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3
coherent Fragmented
Strongly correlated
Possible type of ground states:
Single condensate state of spin-F Bosons
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<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
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<ψμ (r r ) >=
Ψ1
Ψ2
⎛
⎝ ⎜
⎞
⎠ ⎟
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<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
,
BEC --> Fragmentation Coherence --> squeezing --> entanglement
t
1 2
,
BEC --> Fragmentation Coherence --> squeezing --> entanglement
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
Consider a term like this :
Spin-1 Bose gas
Consider a term like this :
Spin-1 Bose gas
Consider a term like this :
Spin-1 Bose gas
Consider a term like this :
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
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aμ
+
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μ=1,0,−1
Spin-1 Bose Gas
Spin dynamics of spin-1 Bose gas
A deep harmonic trap
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H = cr S
2
Hilbert space
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H = cr S
2
Effect of spin degeneracy on BEC
Spin-1 Bose Gas
Effect of spin degeneracy on BEC
A deep harmonic trap
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Ax = (−a1 + a−1) / 2
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Ay = (a1 + a−1) / 2i
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Az = a0
Under spin rotation, rotates like a 3D Cartesean vector .
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aμ → (e−ir θ ⋅
r S a)μ
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rA i → R(
r θ )ij
r A j
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R(r θ ) : 3D rotation
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aμ
+
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μ=1,0,−1
Exact ground state :
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| S = 0 >= ΘN / 2 | 0 >
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< aμ
+aν >=N3
1 0 0
0 1 0
0 0 1
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
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N0 = N1 = N−1 = N /3
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H = cr S 2 C>0
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Θ=2a1
+a−1
+ − a0
+2
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ΔN12 ~ N 2
=
Spin-1 Bose Gas
Spin dynamics of spin-1 Bose gas
A deep harmonic trap
00
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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
Effect of spin degeneracy on BEC
Spin-1 Bose Gas
Effect of spin degeneracy on BEC
A deep harmonic trap
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N = aμ
+aμμ∑
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rS = aμ
+ r S μν aν
μν∑
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rS μν
: spin-1 matrix
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rS = −i
r A
+×
r A
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N =r A
+⋅
r A €
Ax = (−a1 + a−1) / 2
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Ay = (a1 + a−1) / 2i
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Az = a0
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aμ
+
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μ=1,0,−1
Spin-1 Bose Gas
Effect of spin degeneracy on BEC
A deep harmonic trap
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H = cr S
2
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rS = aμ
+ r S μν aν
μν∑
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c > 0: Ferromagnetic ,
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c < 0 : Antiferromagnetic
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H = cN −c(r A + )2
r A 2
What is the ground state for c>0 ?
To find the optimal spinor condensate ,
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|ψ >=( ζ μ
μ∑ aμ
+ )+N
N!| vac >
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ψ | H |ψ = cr S
2
+ cNwe minimize .
For c>0, diamagnetic case,
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ζ =0
1
0
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
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|ψ >=a0
+N
N!| vac >=
(ˆ z ⋅r A + )N
N!| vac >
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rS = 0sincewe have
Since H is rotationally invariant, the optimal states are given by
; and
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|ψ >=( ˆ n ⋅
r A + )N
N!| vac > real.
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ˆ n , the “polar family”
Conventional condensate :
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| c >=a0
+N | 0 >N!
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< aμ
+ aν >=N
2
1
0
1
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟1 0 1( )
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N0 = 0,
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N±1 = N /2
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H = cr S 2 C>0
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< r
S >= 0
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ΔN12 ~ N
Average the coherrent state over all directions
Relation between singlet state and coherent state
x
y
z
Because
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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
Transformation of singlet into coherent states as a function of External field and field gradient:
If the total spin is non-zero
Bosonic enhancement
Transformation of singlet into coherent states as a function of External field and field gradient:
If the total spin is non-zero
Transformation of singlet into coherent states as a function of External field and field gradient:
If the total spin is non-zero
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
components density Texture
Skyrmion Lattice
Ferromagnet
Faster rotation speed
components density Texture
AntiferromagnetAngular momentum carrying object:
p-disclination or 1/2 vortex
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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
Order: pink=nematicGreen=ferromagnetic
Components
DensityCores alignedantiferromagnetically
Disclination Lattice
Order: pink=nematicGreen=ferromagnetic
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
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c2 < 0
antiferromagnetic
ferromagnetic
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<ψμ >=(0,1,0) Polar phase
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<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
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R
1
0
0
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
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R
0
1
0
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
Vector order parameter
Spin-Gauge SymmetryNematic order parameter
Analogous phases seen in 3He
Similar to 2-component case
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Ax = (−a1 + a−1) / 2
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Ay = (a1 + a−1) / 2i
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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
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=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
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Na23, K 39 Rb87
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V (1,2) = δ(r r 1 −
r r 2) c01+ c2
r F 1 ⋅
r F 2( )
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c0 = (2g2 + g0) /3,
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c2 = (g2 − g0) /3,
T.L. Ho, PRL 87, 81, 742 (1998)
Na-23
Rb-87
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c2 > 0
€
c2 < 0
antiferromagnetic
ferromagnetic
It is useful to rewrite P in terms of spin operators.
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1= P0
+ P2
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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
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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
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Rb83 Rb85
€
Cs133,Cs135,Cs137 f = 3
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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 ?
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λα
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α
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0
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1
€
2
€
3
€
λ0~N
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λ1,λ 2,...~O(1)
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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
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M=10, N=1000
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non-zero q, q=0.1
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Initial state : N_{o}=N, M=0
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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)