chang-pu sun ( 孙昌璞 ) institute of theoretical physics, chinese academy of sciences, beijing,...

Chang-Pu Sun ( 孙昌璞 )

Institute of Theoretical Physics,

Chinese Academy of Sciences, Beijing , China

[email protected]://www.itp.ac.cn/~suncp

and Quantum Phase Transition of Hybrid Systems

From Quantum Information to Quantum Thermodynamics

1.Quantum Information and Thermodynamics of Demon

From Classical Heat Engine (HE) to Quantum HE

2. Entanglement and Quantum Statistical Mechanics

Equilibrium Canonical State with Quantum Coherence

3. Quantum Phase Transition (QPT) of hybrid System

for Coupled Resonator Waveguide for Photonic Band

Outline

信息量子化时代的开始

Rolf Landauer (1927–99) Peter Shor

擦出一个比特信息要消耗能量 ln 2kT

Landauer 原理 , 1964 大数因子化量子算法 1993

( )P n计算步数

Charles H. Bennett

量子密码学 1984

量子离物传态 1993

计算的物理极限与量子计算

Landauer 原理预言了计算的物理极限的存在

摩尔定律（ Learn More ， 1965 ）的终结计算机 CPU （中心处理器）的运行速度每十八个月就会增加一倍

麦克斯韦妖“ PK” 热力学第二定律

通过一个热力学循环不可能从一个单一热源提取能量做功，而不对外界产生影响。

高温低温

James Clerk Maxwell, Theory of Heat , 1871

热力学第二定律开尔文表述：Maxwell’s demon

希拉德表述 : 单分子热机Leo Szilard, 1932

循环原理

2ln2/2/

KTdVV

KTPdVW

V

V

V

V

在宏观层面上破坏热力学第二定律是不可能的，因为要区分大量分子的个体速度是非常困难的。这个佯谬的提出只是表明热力学第二定律的原则上只能描述大量粒子组成的宏观物体，是一个统计性的原理，不能简单地应用到有限粒子系统。

过去错误的观念认为 , 确定系统在哪一个态上是物理上的一个测量过程，这种测量是一种不可逆过程，因此需要消耗能量。

Landauer 信息擦除原理保护热力学第二定律

k

n

kk PPS

1

ln

信息擦除需要耗能的物理过程

分子开始以 50% 的几率分别处于 A 和 B 区域，信息量 S=ln2 ，或称为 1 个比特的信息 ;

信息擦除后分子在确定的左态。于是，体系的由 S=ln2 变为 0

香农信息

(a)

(b)

(c)

(d)

(e)

(f)

麦克斯韦妖作为热机整体的循环过程

妖必须是热机的一部分，参与热力学循环必须要擦除自己信息，需要额外的能量 (Bennett ,1979)

麦克斯韦妖参入的量子热力学循环

ST

DT

S

D

S

D1D0

S

0D

1S

System’s bath

Demon’s bath

Quan et al Phys. Rev. Lett. 96, (2006)

模型超导电路实现

d s1 /

没有违背热力学第二定律的现象

CNOT and CEV operations

|,0,00,0||1,01,0|

|0,10,1||1,11,1|)1(0,0

,1,0,

0,1,

1,1,

DSDS

DSDS

pp

pp

.|0,00,0||1,01,0|

|1,11,1||0,10,1|)2(0,0

,1,0,

0,1,

1,1,

DSDS

DSDS

pp

pp

Measurement do not lead to entropy increase

3 pS ,D1,1 |1, 01, 0| pS ,D

1,0 1, 1 1, 1

pS ,D0,1 0, 1 0, 1 pS ,D

0,0 |0, 00, 0|.

#

CNOT

CEV Controlled Evolution

1 D

S

W

Q

in

.1/1/csc10,1

,1,0,

0,1,

1,1,

2 DSDSDSDS pppp

正功条件与热机效率

)/( DSDS TT

Similar to a simple quantum Otto engine

When we choose the CEV to be a special case-CNOT (among all CEVs the CNOT is the optimum operation to extract work) and the temperature of the demon bath very low (the demon can be restored to a zero-entropy standard state" to acquire information about the system in the most efficient way), we regain the efficiency of a simple Otto engine

1 D

S

It is the neglecting of erasure of the demon that leads to the perpetual machine of the second kind. Thus, even there exist the Maxwell’s demon, no violation of the second law at all !

,12

121

gS

gD

n

n

498.0gDn 492.0gSn 75.0

JpQW SS2510)1( in

s 10

sJWP /10/ 20

J.Q. You and F. Nori, Phys. Today 58, No. 11, 42 (2005).

超导量子热机的效率

||2min aE

Lht

The total time needed

to carry out a particular

algorithm consisting of

L elementary gates is

about .minLt

麦克斯韦妖型量子控制导致的算法物理极限

Xue, Yu, and Sun ， Phys. Rev. A 73, 013403 (2006)

Entanglement and Quantum Statistical Mechanics

H b b j

jaj aj j

jb aj h. c, #

|nnj |n j 1

N

|nj

Illustration with exactly solvable Model

Ennj n j

jnj

S. Popescu et al , Nature Physics 2, 754 (2006)S. Goldstein et al., Phy. Rev. Lett. 96, 050403 (2006)

Dong, Yang, Liu , quant-ph/0702027

{ } 1

{ }[ ] 11

[ ]1

1

( ) | ( , ) |

1| | ( , )

( , )

1| | 1

( , )

( , )| | ( ) | |

( , )

j

j j E n

j E n

N

S E j jm j

N

j jn m n jN

n mN

Nn

n nN

Tr m E m

n n m nD E

n nD E

D E nn n P n n

D E

From Microcanonical DM to Canonical DM

( ) nnP e

SE n SE dSEdE

n

SE n

DNE, DN 1E, SE lnDNE,

In thermodynamic limit

1

( ) ( )

( , )

( , )N

N

S E n S E n

D E n

D E n

e e

dSEdE

DNE, eSE

Generalized Quantum Thermolization

Almost all the pure states of the “Universe ” can give the Canonic Equilibrium State by tracing over the Environment

,

,[ , ] 11

( , )| | |

( , )j E

Nj

E jn n jN

C n nn n

D E

, ,1

( , )(| |) | |

( , )N

S B E En N

D E nTr n n

D E

Based on The Law of Large Numbers

Proof : with the Law of large numbers

| n n jEn ,

Cn, nj j 1

N

|nj

,|

1

| |

( , )E

n

n

N

n

D E

|| n||2 n jEn ,

|Cn, nj|2

1DN 1E,

n,n jE,

|Cn, nj|2 1,

1DN 1E,

n

|| n ||2 1

|| n ||2 n jEn ,

1 DNE n ,

S TrB| E, E, |

1DN 1E,

n

|| n ||2|nn|

Role of Interaction

H. Dong, S. Yang, X.F. Liu, , C.P. Sun , 2007

. ,j j j j jj j

H b b a a b b g a h c

Y.B Gao , C.P. Sun , PRE, 2007

2( { }) ( )j j jj

e n n n n n

|nnjn |n j 1

N

|njn

j

gj2

4 j

!

1| ( ) ( ) | 0 ( ) |n

j jn j j jn j

j

n n D a D nn

Deformation of Energy Shells due to Interaction

VE VE : |nnj |E n n2 j 1

N

jnj E

Universality of Model

A. O. Caldiera , A. J. Leggett, Ann. Phys. (NY) 149, 374(1983).

In the weak coupling limit, any heat bath could be universally modeled as a collection of harmonic oscillators with the linear couplings to the surrounded system according

,H.c.)(||,

agnnH jnnj

I M-level system

jj

N

jnj nnnE

1

}{,

2nnn j|gj |2/4 j

;

2

1

1

!1)(

jNj

Nn

N N

En

Micro State Numbers

)exp(][

)]([1

1

1

nnNn

Mn

Nn

n E

kEP

Gibbs distribution ?

Emergence Of “Temperature”

.

1|

22

nEn E

N

E

ESn

Quantum Thermolization with Interaction

S n

Pn |nn| nm

Fnm|nm|, #

, , .j

j

n n

nm j j m mF C n n C m m D

Off-Diagonal Elements

Novel Thermodynamics: von Noueman Entropy is not Thermodynamic Entropy

In general

pF

FpS )1/(1

ep

12FF

2)

2coth()( FpFP

Equivalent Thermal State with Effective Temperature

,

)(

)(

FP

FPS

Effective Temperature )exp()(

)()(

eff

FP

FPtr

)2

coth()2

(cosh4 2

2

F

eff

)(ln)( FPFPSVN

)2

coth(2

FESSVN

),1ln(1

ee

ES

von Neumann entropy

It is observed that due to the system-bath interaction the von Neumann entropy explicitly deviates from the thermodynamic entropy

von Noueman Entropy or Thermodynamic Entropy

Thermodynamic Entropy

新奇量子热机

11 2

2

T T

Scully 光子气体热机

比经典系统作正功的条件要苛刻 1 2T T

Quan ， Zhang, Sun, Phys. Rev. E 72, 056110 (2005)

Scullyet al , Science 299, 862 (2003)

环境偏离通常的热平衡态，这样的“热库”事先具有量子相干性，不是处在一个最大混合态上

什么是非平衡态有效温度？

2. NAMR + Cahrge Qubit1.NAMR + Spins

Nature 430, 329 (04) PRL.88,148301 (02).

Hybrid Systems as Quantum Coherent Devices GHz-Nano-Mechanical Resonator (NAMR) : Cavity QED Analog

Standard Quantum Limit (SQL):

PRL. 94, 030402 (05) disputed ! La Haye et al., Science， (04)

3. NAMR + Flux Qubit

Xue et al,(CAS & NTT) New J.Phs. 2007

Natural Atom,Ions

Superconducting Qbits

Nuclear Spin,Excitation

Quantum Dots

Q-EM Field Cavity QED Semi-Clas. Quant. Quant.

S-TLR T Circuit QED × ×

NAMR T NM-QED Semi-Clas T

Large JJ T Circuit QED × ×

Spin Wave × × SW.QED SW.QED

2-Level Artificial Atoms

Artificial Cavity QED Construct

Qu

an

tum

data

bu

s

T=Theoretical Protocols

probe

|a>

|c>

|b>

classical

光控周期半导体光子结构材料

g1

g2

1δ 2δ

Quantum Cascade Laser固态级联激光

g

e

g

e

Yanik,Fan,Phys.Rev. Lett.92, 083901 (2004)

cavity

available evanescent coupling

ωA : resonance frequency of Cavity A

ωB : resonance frequency of Cavity B

δ= ωA - ωB

δ>>βδ<<-β

Recent Experiment:

Stopping Light All Optically by On-Chip Setup

Xue, et al, Phys. Rev. Lett. 96, 123901 (2006)

Is Laser a quantum phase transition (QPT) ?

1.Greentree, Tahan, Cole, Hollenberg (GTCH), Nature Phys. 2, 856 (2006).

2. Hartmann, Brandao , Plenio (HBP), Nature Phys. 2, 849 (2006).3. Angelakis, Santos, Bose, (ASB) quant-ph/0606159. (2006).

0 . Zhou, Gao, Song, Sun (ZGSS) cond-mat/0608577, submit to

Different points of View

ZGSS: Laser Like Output

GTCH: Mott Phase Transition

HBP: Polariton QPT

ASB: Polariton QPT

Hybrid Setups Based on Coupled Resonator Optical Waveguide (CPOW) by Doping Artificial Atoms

g

e

g

e

(b)

(c)

(a)

a

1. Hu, Zhou, Shi, Sun:quant-ph/0610250 : Coupled cavity QED for coherent control of photon transmission (I): Green function approach for hybrid systems with two-level doping

Probe

b

a

c

b: Colored EIT

0

a: Photonic band k

2. Lan Zhou, Jing Lu, C. P. Sun quant-ph/0611159Coupled cavity QED for coherent control of photon transmission (II) : Slowing light in coupled resonator waveguide doped with $\Lambda $ Atoms

Optical resonances

ω0ω δω0+

ω δω0-

ω0

input output

ω0

ω δω0+

ω δω0-

ω0

input output

Photonic Band Structure of CROW

† †1ˆ ˆ ˆ ˆC j j j j

j j

H a a J a a hc k 2J cosk

Optical analog of tight binding Bloch Electron

k

What is the observables order parameter for QPT?

ZGSS: Laser –Atomic Coherence and Photon visibility in k-space

GTCH: the Average of photon operators, only a guess!!

HBP and ASB : the Average of Polariton operators

Correct, but not observable?

Fluctuation of t Total Excitation (ASB)

Hfree a i 1

N

aiai b

i 1

N

|eie|, #

Hint g i 1

N

ai|gie| H.c., # ,H.c.1

†

1

ii

N

ihop aatH

the cavity-mode-atom interaction the photon hopping between NN defects

2

1†

11

ziii

ii

i

SaaPP

Model with Conservation

Total Excitation is Conservative

Atomic entanglement characterized by concurrence

jj

N

jNNjj snssnnsn ||,..,;,..,|},{|

111In the basis by

])[(},{},{

},{},{

jjjjsnsn

snsn snsnjj

jj

jj

jj

)(,][ 2121)12(

2121

ssssssss

.

000

00

00

000

2

1)(

ij

ijij

ijij

ij

ij

u

wz

zw

u

X. Wang , 2002

).,0(max2 ijijijij uuzC

zij |Si Sj | uij |1/2 Siz1/2 Sjz|

,

Photon visibility in hybrid system

V Vmax VminVmax Vmin

. #

.1

)( †)(

,lj

ljik

lj

aaeN

kV

Quantum Feature of Photon=Second Order Coherence

Functions

Fluctuation of t Total Excitation (ASB) =Variance of Polariton

N

Naaaa iiiii

1†† N

222 )()( PPP

Excitation Variance [dark lines in (a, c)],

Photon Visibility [dark lines in (b, d)],

Atomic Concurence (color maps in (a, d)

by

Exact Diagonalization 2 Particles-:(a,b),4 Particles- (c,d)..

Phase Diagram of the Hybrid System

References : Some of Our Recent Publications

1. H. T. Quan, Y. D. Wang, Y Liu, C. P. Sun, F. Nori Phys. Rev. Lett. 97, 180402 2. H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, C. P. Sun , Phys. Rev. Lett. 96, 140604 (2006) 3.. P. Zhang, H. H. Jen, C. P. Sun, L. You , Phys. Rev. Lett. 98, 030403 (2007) 4. L. F. Wei, Yu-xi Liu, C. P. Sun, F Nori, Phys. Rev. Lett. 97, 237201 (2006)5. Fei Xue, Ling Zhong, Yong Li, C. P. Sun , Phys. Rev. B 75, 033407 (2007)6. Y. B. Gao and C. P. Sun Phys. Rev. E 75, 011105 (2007) ?7. Yu-xi Liu, C. P. Sun, F Nori, Phys. Rev. A 74, 052321 (2006) 8. Lan Zhou, F. M. Hu, Jing Lu, C. P. Sun, Phys. Rev. A 74, 032102 (2006) 9. Nan Zhao, L. Zhong, Jia-Lin Zhu, C. P. Sun , Phys. Rev. B 74, 075307 (2006) 10. Yong Li, Z. D. Wang, C. P. Sun,Phys. Rev. A 74, 023815 (2006) 11. S. Yang, Z. Song, C. P. Sun , Phys. Rev. B 73, 195122 (2006) 12.Yong Li, Li Zheng, Yu-xi Liu, C. P. Sun , Phys. Rev. A 73, 043805 (2006)13. H. T. Quan, P. Zhang, C. P. Sun , Phys. Rev. E 73, 036122 (2006) 15. S. Yang, Z. Song, C. P. Sun , Phys. Rev. A 73, 022317 (2006) 16. C. P. Sun, L. F. Wei, Yu-xi Liu, F Nori , Phys. Rev. A 73, 022318 (2006)17.Fei Xue, S. X. Yu, C. P. Sun , Phys. Rev. A 73, 013403 (2006)

EJ

Vg Cg

CJ

(a)

(b)

(c)

Superconductor transmission line is cut into equal pieces.

Φx and the biased voltage controls the energy difference of this two-level system

Each dcSQUID acts as a two-level system.

Input

output

Hybrid Setup based on superconducting circuit

Zhou, Gao, Song, Sun cond-mat/0608577, submit to PRL

P k cos kak sin kbkQ k sin kak cos kbk

polariton operators

tan k 2g/ k A

dispersion relation k 12

k A k A 2 4g2

Spin-wave dressed photonic band

group velocity

Laser like coherent output

Laser-like equation

tak d0g2L , k a k 4d0g4

N L , k|L , k|2akakak

tak a k i kak igBk, tBk Bk i ABk igS0ak,

tS0 d0 S0 i 2gN

k

a kBk h. c.

L , k 1 i A k

η: Cavity damping rateγ: charge qubit decay-rateΓ: relaxation time to equilibriumd0: tinput rate for equilibrium inversion

Quantum

Physics

Information

Science

Quantum Physics of Information:

chang-pu sun ( 孙昌璞 ) institute of theoretical physics, chinese academy of sciences, beijing,...

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