constraints on dissipative processes allan solomon 1,2 and sonia schirmer 3 1 dept. of physics &...
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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
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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.
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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
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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
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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
|
**
***]*[
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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
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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
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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
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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.
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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)
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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)
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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.
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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
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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.
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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.
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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
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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
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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
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:= 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.
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:= 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
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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
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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
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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!
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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.
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Dissipative Dissipative TermsTerms
Orthonormal basis:Orthonormal basis:
Population Relaxation Equations (
Phase Relaxation Equations
knknHi ],[
kknk
nknnnk
knnnnn Hi
],[
},...2,1:{| Nnn
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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
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Liouville Operator for a Three-Level Liouville Operator for a Three-Level SystemSystem
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Three-state AtomsThree-state Atoms
1
3
2
1
12
13
3
2
12
32
V-system
Ladder system
3
221
23
-system
1
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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.
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““Eigenvalues” of a Three-level Eigenvalues” of a Three-level SystemSystem
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Phase Decoherence in Three-Level Phase Decoherence in Three-Level SystemSystem
333231
2322212/1312
2/11
)(
te
te
t
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““Eigenvalues” of a Three-level Eigenvalues” of a Three-level SystemSystem
Pure DephasingPure Dephasing
Time (units of 1/)
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Three Level Three Level SystemsSystems
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Four-Level Four-Level SystemsSystems
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Constraints on Four-Level SystemsConstraints on Four-Level Systems