Decoherence by Quantum Phase Transition
Transcript of Decoherence by Quantum Phase Transition
Chang-Pu Sun
(孙昌璞)
Institute of Theoretical Physics, Chinese Academy of Sciences
[email protected]://www.itp.ac.cn/~suncp
Decoherence
by Quantum Phase Transition
Lecture 4
Outlines
2. Quantum Measurement: Model of Decoherence
1. What is quantum decoherence
?
3. Environment induced Decoherence
and Quantum Chaos
5.
Emergence of Decoherence as a Phenomenon in
4. Quantum Phase Transition
Quan, Song , Liu, Zanardi, Sun, PRL, 2006
( ) ( )txtxdtd ,Hψ,ψi =
Wave Mechanics
Schrödinger Equation
Quantum Mechanics
Wawe function
( ) 2|ψ x, t |
Probability finding Particle
Matrix Mechanics
, Matrix mn mnP Q⎡ ⎤⎢ ⎥→ ⎢ ⎥⎢ ⎥⎣ ⎦
x x ..
x x ..
.. .. ..
[ ], 0Q P ≠
Observable spectrum relates two “orbits”
Q PΔ Δ ∼
Quantum Measurement Postulate
nncn n →=∑ψ
( )nA n a n=
naMeasuring A, once you obtain , then
nnc nψ =∑
Wave Function Collapse (WFC) !!!
5’th Solvay
conference (1927)
Quantum Theory –
“Quantum Wave Mechanics”
-
takes flight
“Electrons and photons”
Particle beam
probe“two slit ” experiment
Wave-Particle Duality
10 10 CC +=ψ
Quantum Decoherence
Wave Packet Collapse (WPC)
Decoherence
in Quantum Measurement
nncn n →=∑ψ nnc
n n2∑→ψψ
Single Particle Picture Ensemble Picture
Ensemble Explanation For Quantum Measurement
1 1 2 2( ) ( ) ( )n nnx c x c x c xψ φ φ φ= = +∑
2*m n M nn n
c c n m c n nρ ψ ψ ρ= = → =∑ ∑
Vanishing of Diagonal Elements due to QM
interference:
2 2 2 21 1 2 2 1 2 1 2( , ) | | | ( ) | | | | ( ) | * * ( ) ( ) . .x x c x c x c c x x c cρ φ φ φ φ= + + +
Quantum Probability to Classical Probability
Bohr’s Complementarity
~x pΔ Δi
Particle wave
Matter possesses particleMatter possesses particle--wave duality, but the particle and wave duality, but the particle and wave natures are wave natures are repulsive with each other in a same experimentrepulsive with each other in a same experiment
Uncertainty Principle p k p= + Δ
xΔ
Vanishing interference due to Perturbation of Momentum (PM) ?
Nature, 395,33(1998)
PM is Not Unique, Quantum Entanglement
From Conventional Copenhagen to Modern Point of View for Quantum Measurement
Modern Approach : Zurek, Zeh, Joos, Sun, et al
1.
Need not time to complete QM with WFC for the System (S)2.
Measuring Apparatus (D) must be Classical
van Neuman
Entanglement by interaction between S and D
Decoherence :Classical Correlation from Entanglement By Environment
Copenhagen
Von-Neumann Model for QM
( )x xφ φ=
IH SP−
=D S interaction measured system
, detectorSX P↔
↔
[ ][ ] [ ]
i
, , 0
X
X S P S
=
= =
,P
X
S s s s
x x x
=
=
( , ) ( ) ( )sx t x t c x st sψ ψ φ= = +∑(0) ( ) sx c sψ φ= ∑
( )x tφ + ( 2 )x tφ + ( 3 )x tφ + ( 3 )x tφ +
( ) ( )
( )
iSPts
isPs
t e c s
c s e
ψ φ
φ
−
−
= ⊗
= ⊗
∑
∑(0) ( )sc sψ φ= ⊗∑
Simple Mathematics
( , ) ( )
|
( )
isPts
s
x t x t
c x e s
c x st s
ψ ψ
φ
φ
−
=
=
= +
∑
∑
isPte x x st= +
Ideal Measurement =Schmidt decomposation
a t sΔ Δ( )x tφ + ( 2 )x tφ +
aΔ
t sΔ
Schmidt decomposition
sk n n nc s k bψ χ φ= ⊗ = ⊗∑ ∑
m n mn m nχ χ δ φ φ= =Overlaps :
a t sΔ Δ∼( )x tφ + ( 2 )x tφ +
aΔ
t sΔ
Stern-Gerlach
Experiment
( ) ( )0 0 1 Dψ α β= + ⊗
( )2
2 zpH g B zM
σ= +
2
2exp4xx Dσ
⎡ ⎤−⎢ ⎥⎣ ⎦
∼
( ) ( ) 1 00 1 0t D Dψ ψ α β→ = ⊗ + ⊗
( ) ( )2
1 or 0 exp2px D x iB z i DM
⎧ ⎫= ± −⎨ ⎬
⎩ ⎭
Dynamic Schmidt Decomposation
Reduced Density Matrix
( ) B z kz∼
( ) ( )2 2
( |)
1 1 0 0 * 1 0 .
s DTr t t
F h c
ρ ψ ψ
α β αβ
=
+ + +
{ }2 2 2 41 0 exp 2 0tF D D a f t tξ →∞= = − − ⎯⎯⎯→
Deacying Decoherence Factor
SH C P= Ultra-Relativistic particle + N Spin Array
Model Hamiltonian
Xσ
HEPP-Coleman Model
HI ∑j1
N12 Vx − j ct1 302j,
(Hepp,1975, Sun ,1993)
UIt j1
N
e−iFx−jct2j 12 130
Fx − j ct −
tVx − j ct ′dt ′ 1
c −x−jct
Vydy
Ut|↑ 0 ⊗ |↓ ⊗ ⊗ |↓ |↑ 0|↑ ⊗. . .⊗|↑,Ut|↓ 0 ⊗ |↓ ⊗ ⊗ |↓ |↓ 0 ⊗ |↓ ⊗ ⊗ |↓
Ujt|↓ |uj − sin Fx − jcos Fx − j
−
Vxdx
2
Exact Solution
ΔSz 12 ∑
j1
N
⟨t|3j|t 12 N ||2 ∑
j1
N
sin2Fx − j
sin2Fx − j ≃ F2x − j ΔSz 2NV2t2
N → V N g→
FN, t j1
N
⟨↓ |uj j1
N
cos Fx − j
|cos Fx − j|≤ 1
Decoherence
due to the macroscopicness
Principle Difficulty :
Uncertainty inEntanglement
Entanglement Quantum Mesurement ≠
12
112
1 0 ,
D D D± −
± = ±⎡ ⎤⎣ ⎦= ⎡ ⎤⎣ ⎦∓
11 02
12
1, 0,
, ,
D D
D D
η
+ −
= −⎡ ⎤⎣ ⎦= + − −⎡ ⎤⎣ ⎦
QM Needs Classical Correlation
2 21 1 0 01, 1, 0, 0,D D D Dα β+
n n n nn nnC n D B S A⊗ = ⊗∑ ∑
nDn ⊗ nn AS ⊗or
Zurek
Triple Model
( )1
2
ES2 z
H
pH g B z HM
σ= + +
Interaction with Screen
Zurek: Environment Induced Decoherence
( ) ( )0 0 1 D Eψ α β= + ⊗ ⊗
( ) 2 2SD 1 1 0 01, 1, 0, 0,ETr D D D Dρ ψ ψ α β= = +
Step 1: ( ) ( )1 01 0 1 0D E D D Eα β α β+ ⊗ ⊗ → ⊗ + ⊗ ⊗
( )1 0 1 1 0 01 0 1 0D D E D E D Eα β α β⊗ + ⊗ ⊗ → ⊗ ⊗ + ⊗ ⊗0 1 0E E =
Step 2:
More About Environment induced decoherence
ESSE HHHH ++=
( ) ( ) ( ),0 0 0 ;S D Eψ φ ϕ= ⊗Initial state: ( )0SD g eC g C eφ = +
State evolution: ( ) ( ) ( )teCtgCt eegg ϕϕψ ⊗+⊗=
( ) ( ) ( ) ,0exp EtiHt ϕϕ αα −= ( )eg,=α
C.P. Sun, Phys. Rev. A (1993).
K. Hepp, Hev. Phys. Acta, 45, 237 (1972).J.S. Bell, Hev. Phys. Acta, 48, 93 (1975).
( ) ( ) ( )( ) eeCggCttTrt egES22
+== ψψρ
( ) ( ) ( ) ( )g e e gge g eC C e gt t gt Ct C eϕ ϕ ϕ ϕ∗ ∗+ +
Reduced density matrix of the system:
( ) ( )1
N
e g j jj
t t e gϕ ϕ=
=∏
Decoherence due to Factorization :
0 5 10 15 200.0
0.2
0.4
0.6
0.8
1.0
D(T
,t)
ωZt
α2=10 α2=1 α2=0.1
C.P. Sun , PRA, 1993
Decoherence
due to Quantum Chaos
Classical Chaos: Butterfly Effect:
Slightly different initial condition leads
to exponential divergence of trajectories
Same Dynamics
Slightly differentinitial condition
Largely differentFinal State
Zurek,Nature,412,712(2001);
PRL, 89,170405 (2002)
No Butterfly Effect in Quantum Mechanics? Quantum Chaotic Environment :
Unitary Transformation
†0 1 0 1
0 1
( ) ( ) (0) | ( ) ( ) (0)
(0) (0)
E t E t E U t U t E
E E
=
=
Peres, Conception of Quantum Chaos ,1995
Same initial state evolve according to two slightly different Hamiltonian
U1
U2†
0 1( ) ( ) | ( ) ( ) 0E t E t E U t U t E= =
1( )E t
0( )E t
E
1(0)E
1(0)E1( )E t
1( )E t
Quantum phase transition (QPT)
Ising
model in a transverse field
( )0 1 1x z zj j j
jJ gH H H σ σ σ += + = − +∑
Quasi-
classical dynamics
Quasi-
classical dynamics
12
Gg
→→→→ →→ → → → →= − −←← →
2gG ↑ = − −↑↑↑ ↑↑↑↑ ↑ ↑↑ ↑↑↓↑:1<<g
:1>>g
Motivation (uncertainty relation)
In decoherence
process, e.g., vanishing of interference paptern, the randomness of relative phase has its source in uncertainty relation
In QPT, the quantum fluctuation is also due to the uncertainty relation
Is there any intriguing relation between them???
Generalized Hepp-Coleman model:
( )1+− += ∑ x z zj j j
j
J eH g e σ σ σ
e
g
z zjg j j 1H J += − σ σ∑
( )e e g eH H H V= λ = +
xje jV J .= −λ σ∑
Ising
model in a transverse field
Exact solution of Loschmidt
echo
( ) ( )kke e k kH A A 1/ 2+λ = ε −∑
( )( )k k 2e e ( ) 2J 1 2 cos kaε = ε λ = + λ − λ
[ ] [ ] [ ]( )ikal
x k kk s e l e l
l s l
eA u iv ,N
−+ −
<
= σ σ − σ∑ ∏
K=2n/Na
ke ( )ε λ
c 1λ = λ =
Large N
Finite N
Ground states
g gG ......= ↓↓ ↓
( ) ( )k k k kg ek 0
G icos sin A A G+ +−
>
⎡ ⎤= α + α⎣ ⎦∏
( ) ( ) ( )k k k k kB cos A isin A ,+
± ±= α − α ∓
k gB G 0=
k eA G 0=
z zjg j j 1H J += − σ σ∑z z x
j je j j 1 jH J J+= − σ σ − λ σ∑ ∑
( ) [ ] gt exp iH t Gα αϕ = −
( ) ( ) ( )2 2g eL( , t) | D t | | t | t |λ = = ⟨ϕ ϕ
Decoherence
&
Loschmidt
echo
( ) ( )s eg g e[ t ] c c D t∗ρ =
( ) ( ) ( )g eD t t | t= ⟨ϕ ϕ
Reduced Density matrix
Loschmidt echo
( ) ( )2 2 kk e
k 0
L( , t) [1 sin 2 sin t ].>
λ = − α ε∏
Loschmidt
echo
via Local density of state
( ) tL t e γ−∼
2( ) ( ) = | | | ( ),i ts s
s
L L t e dt G E Eωω δ ω−= ⟨ ⟩ −∑∫
2 2
1( )( )
L ωω γ−Ω +
∼
A heuristic analysis
cK
c kk 0
L ( , t) F L( , t),>
λ ≡ > λ∏ cKk 0c kS( , t) ln L | ln F |>λ = ≡ −∑
( )2
2c2
E(K )S( , t) sin 2J[1 ]t(1 )λ
λ ≈ − − λ− λ
( )2cL ( , t) exp tλ ≈ −γ
2 2c c c cE(K ) 4 N (N 1)(2N 1) /(6N )= π + +
2c4J E(K )γ =
[ ] ( )22 2ksin 2 k a /(1 )α ≈ λ − λ
ke 2J |1 |ε ≈ − λ
Numerical results: far from the critical point
Our result: Unpublished ,
but even posted in Arxive before the PRL paper
( )2cL ( , t) exp tλ ≈ −γ
Short time behavior
Universality of Loschmidt
Echo
transverse field Ising model
Cucchietti, Fernandez-Vidal, Paz, quantph/0604136
Boson Habburd model
( )2cL ( , t) exp ut F(t)λ ≈ −
Coupling independent u
Numerical results : near the critical point
Large N
Small N
⊗⊗ ⊗ ⊗ ⊗
Implementation :Superconducting Quantum Network
Circuit QED for Charge Qubit array
( ) ( ) ( 1)0 [ ] ( )x z zH h B α α α
α α
λ λ σ σ σ += ≡ +∑ ∑
: / 0.05mNEC C CΣ ≈
( )†x a aφ η= +
( )1/ 2( / ) /S d l Lη ω=
( )† † † ( )F xH a a g a a aa α
α
ω σ= − +∑
Model Hamiltonian
0 0
| ,k kkk k
G O O−> >
= − ≡∏ ∏
Pseudo-Spin Representation:
† †
† †
† †
1,
,
.
zk k k k k
xk k k k k
yk k k k k
S
S i
S
γ γ γ γ
γ γ γ γ
γ γ γ γ
− −
− −
− −
= + −
= +
= −( )
0k
kn nH H>= ∑
( ) ( cos2 sin 2 )kn nk zk nk xk nkH S Sε α α= +
0
k km niH t iH t
mn kk
D e e−
>
= − −∏
Summary and conclusion1.
We establish the connection between the QPT and the decoherence
for the first time.
2. We obtain the decohence
factor (Loschmidt
echo) quantitatively through the exact calculation
3. Both the analysis and the numerical result confirm our assumption that the quantum critical behavior of the environment strongly enhances its ability of inducing decoherence.
4. A possible physical implementation is proposed based on the superconducting Circuit QED
Recent Objectives and Their Status
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Theory of Quantum Measurement in Practice
?
We have worked More
We have initialed some
We are ready to work
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?