Quantum response in dissipative environments University of Tokyo S. Miyashita 5 Nov. 2007 Linear...
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Transcript of Quantum response in dissipative environments University of Tokyo S. Miyashita 5 Nov. 2007 Linear...
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