Chang-Pu Sun

(孙昌璞)

Institute of Theoretical Physics, Chinese Academy of Sciences

suncp@itp.ac.cnhttp://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

Quantum Information Quantum Physics

Sources of Decoherence in Multi-Qubit System (esp. in Solid State System, 1/f)

Decoherence Models for Many Body System with Inner or External Environment

Protocols for Quantum Storage and Memory

Exotic Light Propagation in Coherence Medium

Quantum Information Transfer for Multi-Qubits

Correlation in Engineered FM (AF) Quantum Spin Chain

Quantum State Control for Quantum Open System

Fast Quantum Logic Gate with Noise Suppression

Non-Desolation Readout of Quantum Information

Theory of Quantum Measurement in Practice

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We have worked More

We have initialed some

We are ready to work

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