Quantum response in dissipative environments

University of Tokyo

S. Miyashita

5 Nov. 2007Linear Response 50Equilibrium & NE response

collaborators: Akira Ogasahara, Keiji Saito, Chikako Uchiyama, and Mizuhiko Saeki

ESR line shape in strongly interacting spin systemsTemperature-dependence of the shift and width in low-dimensional quantum spin systems

Y. Ajiro, et al: JPSJ 63 (1994) 859.

Spin trimer: 3CuCl2 ・ 2Dioxane

F F AF

Microscopic expression of the line shape from Hamiltonian

Kubo Formula

dttMM tixxxx

e )()0()e1(2

1)("

R. Kubo: JPSJ 12 (1957) 570

R. Kubo & K.Tomita JPSJ (1954) 888

//)( )0()( iHtiHttiL MeeMetM mEmH m ||

))(()(" mnmn

mn EED

)( ,ee

2

mn

x

EEmn EE

Z

nMmD nm

Pure quantum dynamics

Shift from the PMR

i

xi

i

zi

jiij

ij

StHSH

JH

cos

2

10

SS

g2

1 ,R HParamagnetic Resonance

Isotropic models

ijji

mn mn

mnnmnm

mn

nm

zj

ziz

yj

yi

xj

ij

xi

rrD

SSJSSSSJH

SSD

rSrSSS53

onperturbati

3

])([2

Perturbation

Shift from the PMR

i

xi

i

zi

jiij

ij

StHSH

JH

cos

2

10

SS

g2

1 ,R HParamagnetic Resonance

Isotropic models

ijji

mn mn

mnnmnm

mn

nm

zj

ziz

yj

yi

xj

ij

xi

rrD

SSJSSSSJH

SSD

rSrSSS53

onperturbati

3

])([2

Perturbation

Studies on the line shape• F. Bloch: PR 70 (1946) 460. Nuclear Induction (Bloch equation)• J. H. Van Vleck: PR 74 (1948) 1168.

Dipolar broadening, and exchange narrowing• N. Bloembergen, E. M. Purcell and R. V. Pound: PR 73 (1948) 679.

Relaxation Effects in Nuclear Magnetic Resonance Absorption.• I. Solomon: PR 99 (1955) 559.

Relaxation processes in a system of two spins• F. Bloch: PR 105 (1957) 1206. General theory of relaxation

• A. Abragam: The principles of Nuclear Magnetism,

Oxford Univ. Press (1978)

Expression of the admittance

dttMM tixxxx

e )()0()e1(2

1)("

//)( )0()( iHtiHttiL MeeMetM

mEmH m ||

))(()(" mnmn

mn EED

)( ,ee

2

mn

x

EEmn EE

Z

nMmD nm

Pure quantum dynamics

Eigenvalue and eigenvectors of the Hamiltonian

Shift & Width

Peak position

Peak width )(

,ee

2

mn

x

EEmn

EEZ

nMmD nm

/2 0)()0( ttixx emtMM

2

0

"

R0

Nagata-Tazuke Dependence

(J. Kanamori & M.TachikiJPSJ 48 (1962) 50)

K. Nagata and Y. Tazuke: JPSJ 32 (1972) 337

1D Heisenberg model withDipole-dipole interaction

1D Heisenberg modelDipole-dipole interaction

Paramagnetic resonance

90

0

,0

HD ＝

D

N=4

Constant H

Frequency sweep abd Field sweep

00

00

" of smany value

"given :,"0

HH

HH

xx

Hxxxx

Line shape as an ensemble

of delta-function

)(

,ee

2

mn

x

EEmn

EEZ

nMmD nm

N=8

Shift

1D Heisenberg AF

Temperature Dependence

Angle Dependence

SM, T. Yoshino, A. OgasaharaJPSJ 68 (1999) 655

2/

0

Width

Magic AngleR.E. Dietz, et al. PRL 26 (1971) 1186.T.T. Cheung, et al. PRB 17 (1978) 1266

SM, T. Yoshino, A. OgasaharaJPSJ 68 (1999) 655

parallelmagic angleperpendicular

Zigzag Chain

A. Ogasahara and S. MiyashitaJ. Phys. Soc. Jpn. Suppl. B 72,44-52 (2003).

Spiral structure

Dipole-dipole interaction

DM interaction parallel perpendicular

1 2

3

r=1

a

Spriral structure Dipole-dipole interaction

r

r=0.1 parallel r=0.2

r=0.5 r=0.5 modified

Spiral structure 　DM interaction d(x,z)

(0,0) r=0.5 parallel

(0,0) r=0.5 perpendicular

DM parallel d(1,0)

(1,0) r=0.5 parallel (1,0) r=0.5 perpendicular

D

DM perpendicular d(0,1)

(0,1) r=0.5 parallel (0,1) r=0.5 perpendicular

d(0,5)

(0,5) r=0.5 parallel (0,5) r=0.5 perpendicular

Response in dissipative dynamics

dttMM tixxxx

e )()0()e1(2

1)("

pure quantum dynamics

//)( iHtxiHtx eMetM

quantum dynamics with dissipation Relaxation effects:I. Solomon: PR 99 (1955) 559. Relaxation processes in a system of two spinsF. Bloch: PR 105 (1957) 1206. General theory of relaxation

Y. Hamano and F. Shibata: JPSJ 51 (1982) 1727,2721,2728.M. Saeki: Prog. Theor. Phys. 67 (1982) 1313. : relaxation method Prog. Theor. Phys. 115 (2006) 1. : TCLE method

POSTERpresentation

Dissipative dynamicsQuantum Master equation method

dttMM tixxxx

e )()0()e1(2

1)("

Quantum master equation

//BTr)( tHHHixtHHHix BISBIS eMetM quantum dynamics with dissipation

RXRXH

i

dt

d,,,

2

F. Bloch: PR 105 (1957) 1206.S. Nakajima: PTP 20 (1958) 987, R. Zwanzig: J. Chem. Phys. 33 (1960) 1338.A. G. Redfield: Adv. Magn. Reson. 1 (1965) 1.H. Mori: PTP 33 (1965) 423. M. Tokuyama and H. Mori: PTP 55 (1976) 411.N. Hashitsume, F. Shibata and M. Shingu: J. Stat. Phys. 17 (1977) 155 & 171.T. Arimitsu and H. Umezawa: PTP 77 (1987) 32.

Time evolution of the density matrixin dissipative system

K. Saito, S. Takesue and SM. Phys. Rev. B61 (2000) 2397.

RXRXH

i

dt

d,,,

2

II

mXkEEnEE

mRk mkmk , )(

kkk

k kkk

bbH

XbbH

HHHH

B

I

BI0

,

,

density spectral the:0

1e)(e)(

)( operators sreservoie' theoffunction n correlatio time

)()()()(e

)()(e)()()(e

,1

02

2

2

2

IDI

DDtdt

t

XtsXXsXt

sXtXtsXXdds

Hidt

d

ti

tti

　　

　 　 　 　 　 　 　 　 　

　 　

Independent phonon bath

Quantum dynamics of magnetization

Molecular magnets

V6 Cu3 Ni4

V15Mn12 Fe8

Phonon-bottleneck effect

)2/3(V15 S

2/1zM

I. Chiorescu, W. Wernsdorfer, A. Mueller, H. Boegge, B. Barbara,Phys. Rev. Lett. 84 (2000) 3454.K. Saito & SM. JPSJ (2001) 3385.

Plateau in the magnetization process due to thermal contact with the bath

Field sweeping with thermal bath

Fast sweeping Slow sweeping

vv AD ADTH vvv ADv

K. Saito & SM. JPSJ (2001) 3385.

MagneticFoehn EffectLZS

Fe-rings

H. Nakano & SM, JPSJ 70(2001) 2151

Y. Ajiro & Y. Inagaki

Y. Narumi & K. Kindo

Fe2 Y. Shapira, et al PRB59 (1999) 1046

dH

dM

dH

dM

Fast sweep region?

V=0.002, ..... , 0.28T/s

[Ni(hmp)(dmb)Cl]4

En-Che Yang,et al: Inorg. Chem. 45 (2006) 529

LZ transition + Thermal relaxation + MFE

v=0.0512, ...., 0.0002

RXRXzHi

dt

d,,,

Formulation of line-shape with dissipative dynamics

)0(e)(

)(

),(,),(,,1

cf.

TrTr

00

00

/0

/0

/0

/0

//

tAttA

tALttAt

LtRXtRXHit

etett

etAAetAAeeAtA

Lt

iHtiHt

iHtiHtiHtiHt

Eigenmode of time-evolution operator

)0( , ,)(

)0(e)( )( )(

)1,),(( , vector

)1,,( , ),(matrix

),(,),(,,1

21

2

cect

et

iL

ttLtt

Nkk

Njiji

tRXtRXHit

Mmti

mm

mti

m

mim

Lt

i

i

I. Knezevic and D. K. Ferry: Phys. Rev. E66(2003) 016131, Phys. Rev.A 69 (2004) 012104.S. Miyashita and K. Saito: Physica B 329-333 (2003) 1142.

Explicit form of the autocorrelation

ikMmti

M

mm

M

ikik

ki

M

ikik

iecA

ttAAAtA

)1(

0

)(

2

ikMm

M

mmik

i

M

ik

ikMm

M

mmik

i

iM

ik

ti

cAi

cAi

edteAtA

i

)1(

)1(0

1

1

2

2

021

0

,

)(

tAc

ecttA

M

mti

mm

i

Dynamical susceptibility

ikMm

M

mmmik

i

M

ikAA

ikMm

M

mmik

i

M

ik

ti

tiAA

dcA

dAidtetAA

dtetAAAtAi

)1(

)1(0

0

1

ReIm

1

2

2

Line shape

Atd

edtAt

M

mti

mm

i

021

0

,

)(

where

Condition for relaxation to the equilibrium distribution

KMS relation for correlation of the bath

e

eEE

eEE

km EkmEmk

0t tt 0

t

Zet SHS /eq0

0),(,),(,,1

eqeq2

eqeq tRXtRXH

it

Steady state

ZAeAtA SHS /eq0

Paramagnetic Resonance

ee

I

SSSX

SHH

zii

yii

i

xii

i

zi

1

20

1.02

01.02

xx

Exchange narrowing

1.02

01.02

xx

zii

yii

i

xii

i i

jizi

SSSX

SSJSHH

Dipole-dipole interaction

1.02

01.02

xx

zii

yii

i

xii

i i

jizi

SSSX

SSJSHH

DD

Resonance and dissipation

0001.02

001.02

Summary• Direct numerical estimation of the line shape

　　　 Ensemble of the delta-functions

　　　 Geometrical effects• Estimation of the width due to dissipative dynamics

Quantum Master equation method

Width due to the dissipative dynamics• Analysis of the themal bath:

　　　　　 Coupling to the system : X

　　　　　 Relaxation function: Φ

　　　 short-relaxation approximation?

Exchange narrowing

Motional narrowing? • Other related topics

Quantum narrowing effect in the spin-Peierls transition

Micro-wave heating in quantum system

Quantum narrowing effect

H. Onishi and SM: JPSJ 72(2003) 392

H J 1 ui1 ui Si Si+1i1

N

1

2mpi

2 k

2ui1 ui 2

i1

N

◆ effects of quantum lattice fluctuation becomes small when m small

uniform

dimerization

Spin-Peierls systems

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1 1.2

magnetic susceptibility

adiabaticm=10000m=100m=1uniform

/

N

T

N=64

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

magnetic excitation spectrum

m=10000m=100m=1uniform

q / 2

E(q

)

N=64

-1 0 1 2 3 4 5 6 70

10

20

30

40

50

lattice position i

ima

gin

ary

tim

e

m=1, T=0.02, N=64

Effect of AC field in complicated system-- Micro-wave heating --

M. Machida, K. Saito and SM: JPSJ 71(2002) 2427

Relation between the eigen state ofthe Hamiltonian and that of the Floquetoperator: (POSTER by Hijii)

/2

0expT

cos

dssHiF

BtAtH

Thank you very much