Constraints on Dissipative Processes Allan Solomon 1,2 and Sonia Schirmer 3 1 Dept. of Physics & Astronomy. Open University, UK 2. LPTMC, University of Paris VI, France 3. DAMTP, Cambridge University, UK DGMTP XXIII, Nankai Institute, Tianjin: 25 August 2005

AbstractAbstract 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.

ContentsContents 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

StatesStates Finite Systems (1) Pure States 2 Sphere Ignore overall phase; depends on 2 real parameters Represent by point on Sphere N-level E.g. 2-level qubit

StatesStates (2) Mixed States Pure Pure state can be represented by operator projecting onto  For example (N=2) as matrix  is Hermitian  Trace  = 1  eigenvalues 0 STATEmixed pure This is taken as definition of a STATE (mixed or pure) pure (For pure state only one non-zero eigenvalue, =1) is the Density Matrix  is the Density Matrix

N - level systems Density Matrix Density Matrix  is N x N matrix, elements  ij Notation: Notation: [i,j] = index from 1 to N 2 ; [i,j]=(i-1)N+j Define Complex N 2 -vector V () V [i,j] () =  ij Ex: N=2:

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  is a Dephasing Coefficient

Ex 3: Can we do both together ? Is this a STATE? (i)Hermiticity? (ii) Trace  = 1? (iii) Positivity? Constraint relations between  and ’s.

Hamiltonian Dynamics (Non-dissipative) [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 - General Global Form* KRAUS Formalism Maintains Positivity and Trace Properties Analogue of Global Evolution *K.Kraus, Ann.Phys.64, 311(1971)

Dissipation Dynamics - General Local Form* Lindblad Equations 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 Dissipation and Semigroups I. Sets of Bounded Operators bounded B(H) is the set of bounded operators on H. A Def: Norm of an operator A : AA ||A|| = sup {|| A  || / ||  ||,   H } A Def: Bounded operator The operator A in H is a bounded operator if A ||A|| < 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) not Clearly S + (A) = {exp(t) A; A bounded, t  0 } does not have this (uniformly bounded) property. Bounded Set S - (A) is a Bounded 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 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(t 1 )*T(t 2 )=T(t 1 + t 1 ) with identity, T(0)=I.

Dissipation Dynamics - Semi-Group Global (Kraus) Form: SEMI - GROUP G  Semi-Group G: g={w i } g ’={w ’ i } then g g ’ G  Identity {I}  Some elements have inverses: {U} where UU + =I

Dissipation Dynamics - Semi-Group Local Form Superoperator Form Pure Hamiltonian (Formal) Pure Dissipation (Formal) L H generates Group L D generates Semi-group

Example: Two-level System (a) Dissipation Part:V-matrices Hamiltonian Part: (f x and f y controls) with

Example: Two-level System (b) (1) In Liouville form (4-vector V  ) Where L H has pure imaginary eigenvalues and L D real negative eigenvalues.

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 Problem Choose E ij a basis of Elementary Matrices, i,,j = 1…N V -matrices  s s

Solution to Relaxation/Dephasing Problem (contd) ( N 2 x’s may be chosen real,positive) Determine V-matrices in terms of physical dissipation parameters N(N-1)  s N(N-1)/2  s

Solution to Relaxation/Dephasing Problem (contd) N(N-1)  s N(N-1)/2  s Problem: Determine N 2 x’s in terms of the N(N-1) relaxation coefficients  and the N(N-1)/2 pure dephasing parameters There are (N 2 -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: Population Relaxation Equations (  Phase Relaxation Equations

Quantum Liouville Equation (Phenomological) Incorporating these terms into a dissipation superoperator L D Writing t as a N 2 column vector V  are Non-zero elements of L D are (m,n)=m+(n-1)N

Liouville Operator for a Three-Level System

Three-state Atoms    12   12  32 V-system Ladder system 3 2  21  23  -system 1

Decay in a Three-Level System Two-level case In above choose  21 =0 and  =1/2  12 which satisfies 2-level constraint And add another level all new  =0 .

“ Eigenvalues” of a Three-level System

Phase Decoherence in Three-Level System

“ Eigenvalues” of a Three-level System Pure Dephasing Time (units of 1/)

Three Level Systems

Four-Level Systems

Constraints on Four-Level Systems