Download 量子状態操作と乱れ

10 Aug. 2010
Kyoto Yukawa
Reduction of the system dynamics
from the total system including the
environments
The University of Tokyo
Seiji Miyashita
Projection operator method
For the master equation
Real part
Phonon Bottleneck phenomena
in V15
Plateau induced by thermal effect
sample
Heat flow
Heat bath
Chiorescu, W. Wernsdorfer,
A. Mueller, H. Boegge, B. Barbara,
Phys. Rev. Lett. 84 (2000) 3454.
Field sweeping with thermal bath
Fast sweeping
vAD  v
LZS
Slow sweeping
vAD
vTH  v  vAD
Magnetic
Foehn Effect
K. Saito & SM.
JPSJ (2001) 3385.
Fe2
Fe-rings
Y. Shapira, et al PRB59 (1999) 1046
dM
dH
dM
dH
Y. Ajiro & Y. Inagaki
Y. Narumi & K. Kindo
H. Nakano & SM, JPSJ 70(2001) 2151
Fast Magnetization Tunneling in
Tetranicke(II) SMM
[Ni(hmp)(dmb)Cl]4

d

 iH ,    z  X , R   X , R 
dt
V=0.002, ..... , 0.28T/s
En-Che Yang,et al: Inorg. Chem. 45 (2006) 529
v=0.0512, ...., 0.0002

Boson system
Spin-boson system
from QMEnote (SM and T. Mori)
Relation between the equation of motion
and its steady solution
Equation of motion up to the second order

1
  L   O 4 , L   H s ,    2 (  )
t
i
(a situational solution)
L st   0
 
we may add any traceless W

  
L  st  2W  O 4
if T rW  0
 
 
The diagonal elements are arbitrary in the order of O 2
0
Master equation leads the system to the equilibrium of the system O 
 
2
The off-diagonal elements aredetermined in the order of O 
T. Mori and SM: JPSJ 77 (2008) 124005 (1-9).
Complex admittance
C. Uchiyama, M. Aihara, M. Saeki and S. Miyashita: PRE 80 (2009) 021128 (1-18).
M. Saeki, C. Uchiyama, T. Mori and S. Miyashita: PRE 81, (2010) 031131 (1-33)
10 Aug. 2010
Kyoto Yukawa
Study on the line shapes of
the response function
--Origins of the Width-The University of Tokyo
Seiji Miyashita
ESR line shape in strongly interacting spin systems
Temperature-dependence of the shift and width in lowdimensional quantum spin systems
Spin trimer: 3CuCl2 ・2Dioxane
g B H  
F F AF
(S=1/2)x3
paramagnetic
EPR
S=3/2
correlated state
Y. Ajiro, et al: JPSJ 63 (1994) 859.
Shift and width of the line shape
• Intrinsic width due to assembly of the delta-functions

Dmn    e  Em  e  En

x
mM n
2
, (  En  Em )
Z
• Quantum broadening due to quantum fluctuation of the field
H JC
0

2









g
b


b




b
b

 i
i
N
i 1
N
z
i
i 1
• Transmission spectrum (input-output modes)
C E 
T   E /   
AE 
• Broadening width due to the interaction with the thermal bath
2
i
1
 A 0  2  
     TrS B

i  iLS   
Microscopic expression of the line shape from
the Hamiltonian of the system
R. Kubo & K.Tomita JPSJ (1954) 888.
R. Kubo: JPSJ 12 (1957) 570.
Kubo formula
H  2 J ijS i  S j  H 0  Siz  H1 cost  S ix
ij 
i
i

1
 xx " ( )  (1  e  )  M x (0) M x (t ) eit dt
2

Isotropic models (Paramagnetic Resonance)
1
g
2
       E /  
R  H ,  
Perturbation

H perturbation  2
[ J ( Six S xj  Siy S jy )  J z Siz S zj ]
ij 
D

 mn
 S S

 m n  3S m  rmn S n  rmn    D 
5
 r3

rmn
 mn

S  S
i
ij 
j 
Expression of the admittance

1
 xx " ( )  (1  e  )  M x (0) M x (t ) eit dt
2

Eigenvalue and eigenvectors of the Hamiltonian
H | m  Em | m 
 "     D(mn ) (  ( En  Em ))
mn

Dmn    e
 Em
e
 En

x
2
mM n
Z
,
(  En  Em )
M x (0) M x (t )  m 2e i0t t /
 "   

  0 2   
  0  R
width

shift
Nagata-Tazuke effect
K.Nagata and Y.Tazuke, JPSJ 32(1972)337.
(J. Kanamori & M.Tachiki : JPSJ 48 (1962) 50)
One-dimensional Heisenberg antiferromagnet
H0  c  axis
H0 // c  axis
Demonstration of the Nagata-Tazuke effects
 "     D(mn ) (  ( En  Em ))
N=8
mn
SM, T. Yoshino, A. Ogasahara: JPSJ 68 (1999) 655.
R.E. Dietz, et al. PRL 26 (1971) 1186.
T.T. Cheung, et al. PRB 17 (1978) 1266
Line shape of a spin chane with a staggered
DM interaction
Diz   1i D
S. El Shawish, O. Cepas, and SM: PRB81, 224421 (2010).
Line shape of a spin chain with a staggered
DM interaction
Diz   1i D
T 
cf.
D0
S xx  
  h/ 0
all T
S. El Shawish, O. Cepas, and SM: PRB81, 224421 (2010).

Models
Staggered DM model
XXZ model
Equivalence
Difference
Consideration on the line shape
F (t )  ie iHt [ H , S x ]e iHt
relaxation time
moments of (t )
Memory function (short time)
Memory function (long time)
Double peak structure

" ( )
it
S   
,

(

)

i

(
t
)
e
dt ' ( )  i" ( )
2
2

0
(  ' ( ))  (" ( ))
xx

( )  i  (t )e it dt ' ( )  i" ( )
0

(0)  i  (t ) dt 
0
i
C
 ' (0)  0, " (0) 
1
C
,

t
d
1
d
d
( ) | 0    t(t ) dt  t

 ' ( 0) 
,
" (0)  0,
0
d
C
d
 C d
t2

t
d2
1
d2
d2
2
2
( ) | 0  i  t (t )dt   2i t

 ' ( 0) 
,
" (0)  2
,
0
C
 C d 2
C
d 2
d 2


d
d
d


d2
4
" ( ) (  ' ( ))1 
' ( )   " ( )
" ( ) 

"
(

)
d
d
d
d




d 2
S xx   

2
2
2
2
2
2
d
(  ' ( ))  (" ( ))
(  ' ( ))  (" ( ))
2



2" ( )
(  ' ( ))
8" ( )
(  ' ( ))
 2 t
 (" ( )) 2
2
2
2
 C  2 C
 (" ( )) 2
3

 1  t

C


2

3

2
2
2
2


 1  d ' ( )   (  ' ( )) d ' ( )  " ( ) d " ( )   d " ( )  

d
d 2
d 2

 d
 



d
d


 (  ' ( ))1 
' ( )   " ( )
" ( ) 
d
d




2
2

 2
  2 t 2   2 3 1  t   2 3  t  1  t
C
C 
C

 2 
 C 
C



 C
2
 t2  
t 
d2
 
 1 
S xx   0

2
2
C 
C
d

2
2 t2

 

C2

 




2




Estimated line shape
in infinite chain
Exact short range + spin diffusion long time tail
with various cut-off times (tau_0,tau_c)
Shift and width of the line shape
• Intrinsic width due to assembly of the delta-functions

Dmn    e  Em  e  En

x
mM n
2
, (  En  Em )
Z
• Quantum broadening due to quantum fluctuation of the field
H JC
0

2









g
b


b




b
b

 i
i
N
i 1
N
z
i
i 1
• Transmission spectrum (input-output modes)
C E 
T   E /   
AE 
• Broadening width due to the interaction with the thermal bath
2
i
1
 A 0  2  
     TrS B

i  iLS   
Coupling between spin system and cavity
phonon system
Spin system N
m
H spin   H
Siz , H spin m   m m , H spin G   G G , spin
 m  G

i 1
Cavity photon system
Hcavity  cavitybb
Coupling
N
H couple   g

( Sib  Sib  ),
i 1
Transmission
H spin  H cavity  H couple k
g 0
trans
 Ek k
m
spin
H
trans  Ek  EG
cavity
H
Coupling between spin system and cavity
phonon system
Spin system N
m
H spin   H
Siz , H spin m   m m , H spin G   G G , spin
 m  G

i 1
Cavity photon system
Hcavity  cavitybb
Coupling
N
H couple   g

( Sib  Sib  ),
i 1
Transmission
H spin  H cavity  H couple k
 Ek k
g 0
m
spin
H
trans  Ek  EG
Enhancement of Rabi-oscillation
and the vacuum-field Rabi splitting
Enhancement of Rabi-oscillation
Y. Kaluzny, P. G. , M. Gross, J. M. Raimond and S. Haroche, PRL 51, 1175 (1983)
The vacuum-field Rabi splitting in the transmission spectrum
G. S. Agarwal:, PRL 53, 1732 (1984).
Splitting of PMR of DPPH
The vacuum-field
Rabi-splitting
G. S. Agarwal:, PRL 53, 1732 (1984).
DPPH
I. Chiorescu, N. Groll, S. Bertaina, T. Mori and SM: PRB (2010) in press. (1004.3605)
N-diamond
arXiv 1006.0251
arXiv 1006.0242
Rubby S=3/2 Cr3+
Multi-photon effect
N=nmax
Super-radiance?
(
...
,
0)
(
...
,
1)
(
...
,
N)
nmax: number of cavity
photons in the ground
state of spin system
At N=nmax, a wide distribution of
the Rabi frequences
Eigenvalues and the transmission spectrum
The vacuum-field
Rabi-splitting
G. S. Agarwal:, PRL 53, 1732 (1984).
Photon emission spectrum

 
d
1
  H ,     a  a  a  a  2a  a   ' aa   aa  2aa 
dt
i



   a  a,  ,  a (0)  a,  eq T 
 ' '     T ra   a  
i

  0.1,  '  0.0,   0  1.0, g  0.2
Shift and width of the line shape
• Intrinsic width due to assembly of the delta-functions

Dmn    e  Em  e  En

x
mM n
2
, (  En  Em )
Z
• Quantum broadening due to quantum fluctuation of the field
H JC
0

2









g
b


b




b
b

 i
i
N
i 1
N
z
i
i 1
• Transmission spectrum (input-output modes)
C E 
T   E /   
AE 
• Broadening width due to the interaction with the thermal bath
2
i
1
 A 0  2  
     TrS B

i  iLS   
Line shape of the transmission

g1

g2
c
vac
Thermal bath method
Transmission in a steady state
Aeikx it
Ce ikx it
Beikxit
j-1 j
j+1
j=0
i
d
 H 
dt
H  E
Input-output formulation
Shift and width of the line shape
• Intrinsic width due to assembly of the delta-functions

Dmn    e  Em  e  En

x
mM n
2
, (  En  Em )
Z
• Quantum broadening due to quantum fluctuation of the field
H JC
0

2









g
b


b




b
b

 i
i
N
i 1
N
z
i
i 1
• Transmission spectrum (input-output modes)
C E 
T   E /   
AE 
• Broadening width due to the interaction with the thermal bath
2
i
1
 A 0  2  
     TrS B

i  iLS   
C. Uchiyama, M. Aihara, M. Saeki and S. Miyashita: PRE 80 (2009) 021128.
Summary
• Explicit expression of the the spectrum line shape:
Line shale of a ring Heisenberg model with DM interaction

Dmn    e
  Em
e
  En
 mM
x
2
n
/ Z , (  En  Em )
• Quantum broadening due to quantum fluctuation of the field
Coupling of spin system and cavity photons
H JC
0

2









g
b


b




b
b

 i
i
N
i 1
N
z
i
i 1
• Transmission spectrum (steady flow method) vs
Broadening width due to the interaction with the thermal bath
C E 
T   E /   
AE 
i
1
 A 0  2  
     TrS B

i  iLS   
2
Thank you very much