Transcript No Slide Title

Constraints on Dissipative Processes
Allan Solomon1,2
and
Sonia Schirmer3
1 Dept. of Physics & Astronomy. Open University, UK
email: [email protected]
2. LPTMC, University of Paris VI, France
3. DAMTP, Cambridge University , UK
email: [email protected]
[email protected]
DGMTP XXIII, Nankai Institute, Tianjin: 25 August 2005
Abstract
A state in quantum mechanics is defined as a positive
operator of norm 1. For finite systems, this may be
thought of as a positive matrix of trace 1. This constraint
of positivity imposes severe restrictions on the allowed
evolution of such a state. From the mathematical
viewpoint, we describe the two forms of standard
dynamical equations - global (Kraus) and local (Lindblad)
- and show how each of these gives rise to a semi-group
description of the evolution. We then look at specific
examples from atomic systems, involving 3-level
systems for simplicity, and show how these
mathematical constraints give rise to non-intuitive
physical phenomena, reminiscent of Bohm-Aharonov
effects.
Contents
Pure States
Mixed States
N-level Systems
Hamiltonian Dynamics
Dissipative Dynamics
Semi-Groups
Dissipation and Semi-Groups
Dissipation - General Theory
Two-level Example
Relaxation Parameters
Bohm-Aharonov Effects
Three-levels systems
Finite Systems
(1) Pure States
E.g. 2-level  
 
 
States
|  |2  |  | 21
qubit
Ignore overall phase; depends on 2 real
parameters Represent by point on Sphere
N-level
 1
 

 
 

 N







N
2
|

|
 i 1  i  C
1
(2) Mixed States
States
Pure state can be represented by operator
projecting onto y
r | y  y
For example (N=2) as matrix
 r is Hermitian
 Trace r = 1
 eigenvalues  0
  [ *  *]  *  *

 


*

*
 


This is taken as definition of a STATE (mixed or
pure)
(For pure state only one non-zero eigenvalue,
=1)
r is the Density Matrix
N - level systems
Density Matrix r is N x N matrix, elements rij
Notation: [i,j] = index from 1 to N2;
[i,j]=(i-1)N+j
Define Complex N2-vector V(r)
V[i,j] (r) = rij
Ex: N=2:
 r11
r
 21
r12 
r 22 
Vr
 r11 
r 
  12 
 r 21 
r 
 22 
Dissipative Dynamics (Non-Hamiltonian)
Ex 1: How to cool a system, & change a mixed state
to a pure state
 t 1 / 4  (1 e gt )3/4

 

3 / 4
0

1 / 4
0

0
 t   1
 
gt 
3 / 4e 
0
0
0

0
g is a Population Relaxation Coefficient
Ex 2: How to change pure state to a mixed state



1/ 4
3/4
3 / 4


 

3/ 4 

t
1/ 4
3 / 4 e  Gt
3 / 4 e  Gt ) 
3/ 4
G is a Dephasing Coefficient
1 / 4
   0


t 


3 / 4
0
Ex 3: Can we do both together ?
 γ 21 t
 γ12 t
e
ρ

(
1

e
)ρ 22

11
r (t )  

e G t ρ 21


e  γ12 t ρ 22  (1 e  γ 21 t )ρ11 
e G t ρ12
Is this a STATE?
(i)Hermiticity?
(ii) Trace  = 1?
(iii) Positivity?
Det r  (e
 ( γ 21  γ12 )t
ρ11ρ 22  e  2G t ρ12ρ 21)  ..
Constraint relations between G and g’s.
G  1/ 2( γ12  γ 21)
Hamiltonian Dynamics
(Non-dissipative)
[Schroedinger Equation]
Global Form: r(t) = U(t) r(0) U(t)†
Local Form: i  t r(t) =[H, r(t) ]
We may now add dissipative terms to this equation.
Dissipation Dynamics - General
Global Form* KRAUS Formalism
r   wi r
†
wi

†
wi wi
I
i
Maintains Positivity and Trace Properties
Analogue of Global Evolution
*K.Kraus, Ann.Phys.64, 311(1971)
r U r U
†
Dissipation Dynamics - General
Local Form* Lindblad Equations
1
†
†
r  (i / ) [ H , r ]  {[Vi r ,Vi ]  [Vi , rVi ]}
2
Maintains Positivity and Trace Properties
Analogue of Schroedinger Equation
*V.Gorini, A.Kossakowski and ECG Sudarshan, Rep.Math.Phys.13, 149 (1976)
G. Lindblad, Comm.Math.Phys.48,119 (1976)
Dissipation and Semigroups
I. Sets of Bounded Operators
Def: Norm of an operator A:
||A|| = sup {|| A y || / || y ||, y  H}
Def: Bounded operator
The operator A in H is a bounded operator if
||A|| < K for some real K.
Examples:
X y( x ) = x y( x) is NOT a bounded operator
on H; but exp (iX) IS a bounded operator.
B(H) is the set of bounded operators on H.
Dissipation and Semigroups
II. Bounded Sets of operators:
Consider S-(A) = {exp(-t) A; A bounded, t  0 }.
Clearly S-(A)  B(H).
There exists K such that ||X|| < K for all X  S (A)
S-(A) is a Bounded Set of operators
Clearly S+(A) = {exp(t) A; A bounded, t  0 } does not
have this (uniformly bounded) property.
Dissipation and Semigroups
III. Semigroups
Def: A semigroup G is a set of elements
which is closed under composition.
Note: The composition is associative, as for
groups.
G may or may not have an identity element I,
and some of its elements may or may not
have inverses.
Example: The set { exp(-t): t>0 } forms a semigroup.
Example: The set { exp(-t):  0 } forms a semigroup with
identity.
Dissipation and Semigroups
One-parameter semigroups
T(t1)*T(t2)=T(t1 + t1)
with identity, T(0)=I.
Important Example: If L is a (finite)
matrix with negative eigenvalues, and
T(t) = exp(Lt).
Then {T(t), t  0 } is a one-parameter
semigroup, with Identity, and is a
Bounded Set of Operators.
Dissipation Dynamics - Semi-Group
Global (Kraus) Form:
r   wi r
wi†
•Semi-Group G:

†
wi wi
SEMI - GROUP G
I
i
g={wi}
then
g ’={w ’i }
g g ’ G
•Identity {I}
•Some elements have inverses:
{U} where UU+=I
Dissipation Dynamics - Semi-Group
1
†
†

Local Form r  (i / ) [ H , r ]  {[Vi r ,Vi ]  [Vi , rVi ]}
2

Superoperator Form
Vr  ( LH  LD )Vr
Pure Hamiltonian (Formal)

Vr  LHVr Vr  exp( LH t ) Vr (0)
LH generates Group
Pure Dissipation (Formal)

Vr  LDVr Vr  exp(LDt ) Vr (0)
LD generates Semi-group
Example: Two-level System (a)

 
r  i [ H , r ]   V j r ,V j†  V j , r V j†
Hamiltonian Part:
1
H := w 
0
0
 0
  fx 


-1
 1

(fx and fy controls)
1
0
  fy 


0
I
I 


0 
Dissipation Part:V-matrices
 0
V1  
 g 21
with
0

0
0
V2  
0
g12 

0 
~ 1
G  G 2 (g 12  g 21 )
 2G 0 
V3  

0
 0
Example: Two-level System (b)
(1) In Liouville form (4-vector Vr)
Vr := [ r1, 1, r1, 2, r2, 1, r2, 2 ]
.
Vr  ( LH  LD ) Vr
Where LH has pure imaginary eigenvalues
and LD real negative eigenvalues.
0
g2, 1 0
g 1, 2 




 0

0
0
G




LD :=
 0

0
0
G




 g

0
0
g
1, 2 
 2, 1
4X4 Matrix Form
2-Level Dissipation
Matrix
G


0


0


0
g 2, 1


 0

LD := 
 0


 g
 2, 1
0
0
G
0
0
G
0
0
g 1, 2 


0 

0 

g 1, 2



0
0
G

 2-Level Dissipation Matrix

0 g2, 1  g1, 2 g2, 1  g1, 2 (Bloch Form)


0
0
G 0
0
0
0



0
0
0
2-Level Dissipation
Matrix (Bloch Form,
Spin System)

0


0


0
G
0
0
2 g
0
0

0


0


0
g2, 1  g1, 2
Solution to Relaxation/Dephasing
Problem
Choose Eij a basis of Elementary Matrices, i,,j = 1…N
V-matrices
Gs
V[ i , j ]  a[ i , j ] Eij
Gij  12
N
2
2
(|
a
|

|
a
|
)
 [ k ,i ]
[k , j]
k 1,  i ,  j
~
G ij  12 (| a[ i ,i ] |2  | a[ j , j ] |2 )
gs
g ij  | a[ i , j ] |
2
Solution to Relaxation/Dephasing Problem (contd)
Determine V-matrices in terms of
physical dissipation parameters
V[ i , j ]  a[ i , j ] Eij 
x[ i , j ] Eij
( N2 x’s may be chosen real,positive)
g ij  x[i,j]
(i  j )
~
1
G ij  2 ( x[ i ,i ]  x[ j , j ] )
N(N-1)
N(N-1)/2 G s
~ 1 N
Gij  G ij  2  (g ki  g kj )
k 1
gs
Solution to Relaxation/Dephasing Problem (contd)
V[ i , j ]  a[ i , j ] Eij 
x[ i , j ] Eij
Problem: Determine N2 x’s in terms of the N(N-1)
relaxation coefficients g and the
~
N(N-1)/2 pure dephasing parameters Γ
x[ i , j ]  g ij
(i  j )
~
1
G ij  2 ( x[ i ,i ]  x[ j , j ] )
N(N-1)
gs
N(N-1)/2 G s
There are (N2-3N)/2 conditions on
the relaxation parameters; they are
not independent!
Bohm-Aharanov–type Effects
“ Changes in a system A, which is
apparently physically isolated from a
system B, nevertheless produce phase
changes in the system B.”
We shall show how changes in A – a
subset of energy levels of an N-level
atomic system, produce phase changes in
energy levels belonging to a different
subset B , and quantify these effects.
Dissipative Terms
Orthonormal basis:
{| n : n  1,2,...N }
Population Relaxation Equations (g
r nn  i [ H , r ]nn   g kn r nn   g nk r kk
k n
k n
Phase Relaxation Equations
r  i [ H , r ]   G r
kn kn
Quantum Liouville Equation (Phenomological)
Incorporating these terms into a
dissipation superoperator LD
r  i [ H , r ]  LD ( r )
Writing r(t as a N2 column vector V

Vr  ( LH  LD )Vr
Non-zero elements of
LD are
(m,n)=m+(n-1)N
Liouville Operator for a Three-Level System
Three-state Atoms
2
g32
g12
1
3
g23
V-system
3
1
g21
2
3
g12
2
1
g23
Ladder system
L-system
g13
Decay in a Three-Level System
 γ 21 t
 γ12 t
e
ρ

(
1

e
)ρ 22

11
r (t )  

e G t ρ 21


 γ12 t
 γ 21 t
e
ρ 22  (1 e
)ρ11 
e G t ρ12
Two-level case
In above choose g21=0 and G=1/2 g12 which
satisfies 2-level constraint
G  1 / 2( γ12  γ 21)
And add another level all new g=0 .
tg / 2 r
 r  (1  etg ) r
e
22
12
 11
r (t )  
etg / 2 r 21
etg r 22

r31
r 32

r13 

r23 

r33 

“Eigenvalues” of a Three-level System
Phase Decoherence in Three-Level System

t
G
/
2

r11
e
r12

r (t )  etG / 2 r 21
r22

r31
r 32

r13 

r23 

r33 

“Eigenvalues” of a Three-level System
Pure Dephasing
Time (units of 1/G)
Three Level Systems
Four-Level Systems
Constraints on Four-Level Systems