Constraints on Dissipative Processes Allan Solomon 1,2 and Sonia Schirmer 3 1 Dept. of Physics & Astronomy. Open University, UK email: [email protected].

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