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Constraints on Dissipative ProcessesConstraints on Dissipative Processes Constraints on Dissipative ProcessesConstraints 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
AbstractAbstractAbstractAbstractA state in quantum mechanics is defined as a positive A state in quantum mechanics is defined as a positive operator of norm 1. For operator of norm 1. For finitefinite systems, this may be systems, this may be thought of as a positive matrix of trace 1. This thought of as a positive matrix of trace 1. This constraint of positivity imposes severe restrictions on constraint of positivity imposes severe restrictions on the allowed evolution of such a state. From the the allowed evolution of such a state. From the mathematical viewpoint, we describe the two forms of mathematical viewpoint, we describe the two forms of standard dynamical equations - global (Kraus) and local standard dynamical equations - global (Kraus) and local (Lindblad) - and show how each of these gives rise to a (Lindblad) - and show how each of these gives rise to a semi-group description of the evolution. We then look at semi-group description of the evolution. We then look at specific examples from atomic systems, involving 3-level specific examples from atomic systems, involving 3-level systems for simplicity, and show how these systems for simplicity, and show how these mathematical constraints give rise to non-intuitive mathematical constraints give rise to non-intuitive physical phenomena, reminiscent of Bohm-Aharonov physical phenomena, reminiscent of Bohm-Aharonov effects.effects.
ContentsContentsContentsContentsPure StatesPure StatesMixed StatesMixed StatesN-level SystemsN-level SystemsHamiltonian DynamicsHamiltonian DynamicsDissipative DynamicsDissipative DynamicsSemi-GroupsSemi-GroupsDissipation and Semi-GroupsDissipation and Semi-GroupsDissipation - General TheoryDissipation - General TheoryTwo-level ExampleTwo-level ExampleRelaxation ParametersRelaxation ParametersBohm-Aharonov EffectsBohm-Aharonov EffectsThree-levels systemsThree-levels systems
StatesStatesStatesStatesFinite SystemsFinite Systems(1) Pure States(1) Pure States
Ignore overall phase; depends on 22 real parameters Represent by point on SphereSphere
N-levelN-level
E.g. 2-levelE.g. 2-level qubit1|||| 22
Ci
N
i
N
1||1
2
1
StatesStatesStatesStates(2) Mixed States(2) Mixed States
PurePure state can be represented by operator
projecting onto
For example (N=2) as matrix
is Hermitian Trace = 1 eigenvalues 0
This is taken as definition of a STATESTATE (mixedmixed or purepure)
(For purepure state only one non-zero eigenvalue, =1)
is the Density Matrixis the Density Matrix
This is taken as definition of a STATESTATE (mixedmixed or purepure)
(For purepure state only one non-zero eigenvalue, =1)
is the Density Matrixis the Density Matrix
|
**
***]*[
N - level systemsN - level systemsN - level systemsN - level systems
Density MatrixDensity Matrix is N x N matrix, elements ij
Notation: Notation: [i,j] = index from 1 to N2; [i,j]=(i-1)N+j
Define Complex NDefine Complex N22-vector V-vector V(()) V[i,j]
() = ij
Ex: N=2:
22
21
12
11
2221
1211
V
Dissipative Dynamics (Non-Hamiltonian)
Ex 1: How to cool a system, & change a mixed state to a pure state
Ex 2: How to change pure state to a mixed state
is a Population Relaxation Coefficient
00
01
4/30
0)3/41(4/1
4/30
04/1
t
tt
e
et
is a Dephasing Coefficient
4/30
04/1
4/34/3
)4/34/1
4/34/3
4/34/1
t
tt
e
et
Ex 3: Can we do both together ?
1121
2212
21
122212
1121
ρ)tγ1(ρtγρt
ρtρ)tγ1(ρtγ
)(eee
eeet
Is this a STATE? (i)Hermiticity? (ii) Trace = 1?
(iii) Positivity?
..)ρρρρ( Det 2112t2
2211t)γγ( 1221 ee
Constraint relations between and ’s.
)21γ12γ(2/1
Hamiltonian Dynamics Hamiltonian Dynamics
(Non-dissipative)(Non-dissipative)
Hamiltonian Dynamics Hamiltonian Dynamics
(Non-dissipative)(Non-dissipative) [[Schroedinger Equation]Schroedinger Equation]
Global Form: (t) = U(t) (0) U(t)†
Local Form: i t (t) =[H, (t) ]
We may now add dissipative terms to this equation.
Dissipation Dynamics - GeneralDissipation Dynamics - GeneralDissipation Dynamics - GeneralDissipation Dynamics - General
Global Form* KRAUS Formalism
†ii ww
iii Iww †
Maintains Positivity and Trace Properties
†U U Analogue of Global Evolution
*K.Kraus, Ann.Phys.64, 311(1971)
Dissipation Dynamics - Dissipation Dynamics - GeneralGeneral
Dissipation Dynamics - Dissipation Dynamics - GeneralGeneral
Local Form* Lindblad Equations
Maintains Positivity and Trace Properties
]},[],{[21
],[)/( ††iiii VVVVHi
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
Dissipation and Semigroups
I. Sets of Bounded Operators
B(H) is the set of boundedbounded operators on H.
Def: Norm of an operator AA:
||AA|| = sup {|| AA || / || ||, H}
Def: Bounded operator The operator AA in H is a bounded operator if ||AA|| < K for some real K.
Examples: X ( x ) = x( x) is NOT a bounded operator on H; but exp (iX) IS a bounded operator.
Dissipation and Semigroups
II. Bounded Sets of operators:
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)
Clearly S+(A) = {exp(t) A; A bounded, t 0 } does notnot have this (uniformly bounded) property.
S-(A) is a Bounded SetBounded Set of operators
Dissipation and Semigroups
III. Semigroups
Dissipation and Semigroups
III. Semigroups
Example: The set { exp(-t): t>0 } forms a semigroup.
Example: The set { exp(-t): 0 } forms a semigroup with identity.
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.
Dissipation and SemigroupsDissipation and Semigroups
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.
One-parameter semigroups
T(t1)*T(t2)=T(t1 + t1)
with identity, T(0)=I.
Dissipation Dynamics - Semi-GroupDissipation Dynamics - Semi-GroupDissipation Dynamics - Semi-GroupDissipation Dynamics - Semi-Group
Global (Kraus) Form: SEMI - GROUP G
†ii ww
iii Iww †
� Semi-Group G: g={wi} g ’={w ’i }
then g g ’ G
� Identity {I}
� Some elements have inverses:
{U} where UU+=I
Dissipation Dynamics - Semi-GroupDissipation Dynamics - Semi-GroupDissipation Dynamics - Semi-GroupDissipation Dynamics - Semi-Group
Local Form ]},[],{[2
1],[)/( ††
iiii VVVVHi
Superoperator Form VLLV DH )(
Pure Hamiltonian (Formal)
)0()exp( VtLVVLV HH
Pure Dissipation (Formal)
)0()exp( VtLVVLV DD
LH generates Group
LD generates Semi-group
Example: Two-level System (a)
Dissipation Part:V-matrices
)(~
21122
1
:= H w
1 0
0 -1fx
0 1
1 0fy
0 I
I 0
Hamiltonian Part: (fx and fy controls)
with
†† ,,],[ jjjj VVVVHi
1
21
0 0
0V
122
0
0 0V
3
2 0
0 0V
:= LD
,2 1 0 0
,1 2
0 0 0
0 0 0
,2 1 0 0
,1 2
Example: Two-level System (b)
(1) In Liouville form (4-vector V)
:= V [ ], , ,,1 1
,1 2
,2 1
,2 2
VLLV DH )(.
Where LH has pure imaginary eigenvalues and LD real negative eigenvalues.
:= LD
,2 1 0 0
,1 2
0 0 0
0 0 0
,2 1 0 0
,1 2
0 0 0
0 0 0
0 0 ,2 1
,1 2
,2 1
,1 2
0 0 0 0
,2 1
,1 2
0 0 0
0 0 0
0 0 2 0
0 0 0 0
2-Level Dissipation Matrix
2-Level Dissipation Matrix (Bloch Form)
2-Level Dissipation Matrix (Bloch Form, Spin System)
4X4 Matrix Form
Solution to Relaxation/Dephasing Solution to Relaxation/Dephasing ProblemProblem
Solution to Relaxation/Dephasing Solution to Relaxation/Dephasing ProblemProblem
Choose Eij a basis of Elementary Matrices, i,,j = 1…N
ijjiji EaV ],[],[
2
],[ || jiij a
)|||(|~
)|||(|
2],[
2],[2
1
2],[
2
,,1],[2
1
jjiiij
jk
N
jikikij
aa
aa
V-matrices
s
s
Solution to Relaxation/Dephasing Problem (contd)Solution to Relaxation/Dephasing Problem (contd)Solution to Relaxation/Dephasing Problem (contd)Solution to Relaxation/Dephasing Problem (contd)
( N2 x’s may be chosen real,positive)
ijjiijjiji ExEaV ],[],[],[
)( ji[i,j]ij x
)(~
)(~
12
1
],[],[2
1
kj
N
kkiijij
jjiiij xx
Determine V-matrices in terms of physical dissipation parameters
N(N-1) s
N(N-1)/2 s
Solution to Relaxation/Dephasing Problem (contd)Solution to Relaxation/Dephasing Problem (contd)Solution to Relaxation/Dephasing Problem (contd)Solution to Relaxation/Dephasing Problem (contd)
ijjiijjiji ExEaV ],[],[],[
)(],[ jiijjix
)(~
],[],[2
1
jjiiij xx
N(N-1) s
N(N-1)/2 s
Problem: Determine N2 x’s in terms of the N(N-1) relaxation coefficients and theN(N-1)/2 pure dephasing parameters Γ
~
There are (N2-3N)/2 conditions on the relaxation parameters; they are not independent!
Bohm-Aharanov–type EffectsBohm-Aharanov–type Effects
“ “ Changes in a system A, which is Changes in a system A, which is apparently physically isolated from a apparently physically isolated from a system B, nevertheless produce phase system B, nevertheless produce phase changes in the system B.”changes in the system B.”
We shall show how changes in A – a subset We shall show how changes in A – a subset of energy levels of an N-level atomic of energy levels of an N-level atomic system, produce phase changes in energy system, produce phase changes in energy levels belonging to a different subset B , levels belonging to a different subset B , and quantify these effects.and quantify these effects.
Dissipative Dissipative TermsTerms
Orthonormal basis:Orthonormal basis:
Population Relaxation Equations (
Phase Relaxation Equations
knknHi ],[
kknk
nknnnk
knnnnn Hi
],[
},...2,1:{| Nnn
Quantum Liouville Equation Quantum Liouville Equation (Phenomological)(Phenomological)
Incorporating these terms into a Incorporating these terms into a dissipation superoperatordissipation superoperator L LDD
Writing t as a N2 column vector V
Non-zero elements of LD areare (m,n)=m+(n-1)N
VLLV DH )(
)(],[ DLHi
Liouville Operator for a Three-Level Liouville Operator for a Three-Level SystemSystem
Three-state AtomsThree-state Atoms
1
3
2
1
12
13
3
2
12
32
V-system
Ladder system
3
221
23
-system
1
Decay in a Three-Level Decay in a Three-Level SystemSystem
1121
2212
21
122212
1121
ρ)tγ1(ρtγρt
ρtρ)tγ1(ρtγ
)(eee
eeet
Two-level case
In above choose 21=0 and =1/212 which
satisfies 2-level constraint)21γ12γ(2/1
333231
2322212/1312
2/22)1(11
)(
tete
tete
t
And add another level all new =0.
““Eigenvalues” of a Three-level Eigenvalues” of a Three-level SystemSystem
Phase Decoherence in Three-Level Phase Decoherence in Three-Level SystemSystem
333231
2322212/1312
2/11
)(
te
te
t
““Eigenvalues” of a Three-level Eigenvalues” of a Three-level SystemSystem
Pure DephasingPure Dephasing
Time (units of 1/)
Three Level Three Level SystemsSystems
Four-Level Four-Level SystemsSystems
Constraints on Four-Level SystemsConstraints on Four-Level Systems