Constraints on N-Level Constraints on N-Level Systems*Systems*

Allan Solomon1,2 and Sonia Schirmer3

1 Dept. of Physics & Astronomy. Open University, UK email: a.i.solomon@open.ac.uk

2. LPTL, University of Paris VI, France

3. DAMTP, Cambridge University , UK email: sgs29@cam.ac.uk

*S. G. Schirmer and A. I. Solomon, PHYSICAL REVIEW A 70, 022107 (2004)Cozumel, December 2004

AbstractAbstract

Although dissipation may occasionally be Although dissipation may occasionally be exploited to induce changes in a system exploited to induce changes in a system otherwise impossible under Hamiltonian otherwise impossible under Hamiltonian evolution, more often it presents an evolution, more often it presents an unwelcome intrusion into the quantum unwelcome intrusion into the quantum control process. Generally, when we try to control process. Generally, when we try to control a system, say by a laser, then not control a system, say by a laser, then not only do we obligatorily have dissipation but only do we obligatorily have dissipation but positivity imposes certain constraints on the positivity imposes certain constraints on the dissipation. In this talk we describe these dissipation. In this talk we describe these constraints for N-level systems which for N>2 constraints for N-level systems which for N>2 are quite non-intuitive. We exemplify the are quite non-intuitive. We exemplify the relations obtained by discussing the effects relations obtained by discussing the effects in certain specific N-level atomic systems, in certain specific N-level atomic systems, such as lambda- and V-type systems. such as lambda- and V-type systems.

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.

PureStatesPureStatesFinite SystemsFinite Systems(1) Pure States (1) Pure States

N-levelN-level

2-level2-level qubit1|||| 22

Ci

N

i

N

1||1

2

1

General StatesGeneral States(2) Mixed States or Density Matrices(2) Mixed States or Density MatricesPurePure state can be represented by operator projecting onto For example (N=2) ]T 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

|

*****]*[

Hamiltonian Dynamics - Liouville FormHamiltonian Dynamics - Liouville Form

(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.

Dissipative Dissipative TermsTerms

Orthonormal basis:Orthonormal basis:

Population Relaxation Equations (

Phase Relaxation Equations

},...,2,1:{| Nnn

N - level systemsN - level systemsDensity 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 | -vector | > = V = VV(i,j)

() = ij

Ex: N=2:

22211211

|22211211

V

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 |(t)>

Non-zero elements of LD areare (m,n)=m+(n-1)N

)(],[)/( DLHi

Liouville Operator for a Three-Level Liouville Operator for a Three-Level SystemSystem

Three-state AtomsThree-state Atoms

1 3

2

1232

-system3

2

1

12

13

Ladder system

3

221 23

V-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/

13122/

2211 )1()(

tt

tt

eeee

t

And add another level all new =0.

Population evolutionPopulation evolutionInitial state T]1,1,1[

31

eigenvalues of eigenvalues of for this Three-level for this Three-level SystemSystem

Three-Level System:PurePhase Three-Level System:PurePhase DecoherenceDecoherence

333231

2322212/

13122/

11

)(

t

t

ee

t

Time (units of 1/)

Eigenvalues

Liouville Matrix for Pure Dephasing Liouville Matrix for Pure Dephasing ExampleExample

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

LD =

CLEARLY NEGATIVE EIGENVALUES!Similarly for previous example.

Dissipation Dynamics - GeneralDissipation Dynamics - General

Global Form*

†ii ww

iii Iww †

Maintains Positivity and Trace Properties

†U U Analogue of Global Evolution

*K.Kraus, Ann.Phys.64, 311(1971)

Dissipation Dynamics - GeneralDissipation Dynamics - GeneralLocal Form*

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)

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)

define

†† ,,],[ jjjj VVVVHi

121

0 0

0V

122

00 0

V

32 00 0

V

:= 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.

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

d

ij

p

ijij

kj

N

kki

p

ij

jkikij

ji

aaN

k

,)(

)|||(|

121

2],[

2],[2

1

1

V-matrices

s

s

Three Level Three Level SystemsSystems

3-Level Ladder system3-Level Ladder system

,1 1 ( )1 e

( ) ,1 2 t

,2 2 e

( ) t ,1 2 e

( ) 1 t

,1 3

e( ) t

,2 1 e( )

,1 2 t

,2 2 e( )

2 t

,2 3

e( )

1 t

,3 1 e( )

2 t

,3 2

,3 3

We assume population transition from level 2 to level 1 only.We want to derive relations on the dephasing Exponents , 1, 2.

Three-level Ladder SystemThree-level Ladder System3

2

1

12

13

Ladder system

‘isolate’ level 3

23=0 13=0

RESULT 23 > ½ 12If 13 =0 then 21 =23

Example: 3-level ladder Example: 3-level ladder systemsystem

Eigenvalue plot

12=1, 12=23=5/4

PureDephasing InequalitiesPureDephasing InequalitiesAmong some non-intuitive results between the pure dephasing rates are:

where a,b,c are any permutation of (1,2), (1,3),(2,3)

One conclusion: Pure dephasing in a 3-level system always affects more than one transition

Four-Level Four-Level SystemsSystems

Constraints on Four-Level SystemsConstraints on Four-Level Systems

ConclusionConclusionss

Equations for dissipative effects may Equations for dissipative effects may force relationships on dephasing of force relationships on dephasing of levels even when they are not levels even when they are not involved in the population involved in the population transitions.transitions.

These effects do not occur for all initial These effects do not occur for all initial configurations,e.g. for thermal states configurations,e.g. for thermal states or initial pure states where the or initial pure states where the ‘isolated’ state is not populated. ‘isolated’ state is not populated.