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Constraints on Dissipative Processes

Constraints on Dissipative Processes. Allan Solomon 1 , 2 and Sonia Schirmer 3. 1 Dept. of Physics & Astronomy. Open University, UK email: a.i.solomon@open.ac.uk 2. LPTMC, University of Paris VI, France

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Constraints on Dissipative Processes

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  1. Constraints on Dissipative Processes Allan Solomon1,2and Sonia Schirmer3 1 Dept. of Physics & Astronomy. Open University, UK email: a.i.solomon@open.ac.uk 2. LPTMC, University of Paris VI, France 3. DAMTP, Cambridge University , UK email: a.I.solomon@open.ac.uk sgs29@cam.ac.uk DGMTP XXIII, Nankai Institute, Tianjin: 25 August 2005

  2. Abstract 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.

  3. Contents • 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

  4. qubit States Finite Systems (1) Pure States E.g. 2-level Ignore overall phase; depends on2real parameters Represent by point onSphere N-level

  5. States Purestate can be represented by operator projecting ontoy (2) Mixed States For example (N=2) as matrix • r is Hermitian • Trace r= 1 • eigenvalues³0 This is taken as definition of aSTATE (mixedorpure) (Forpurestate only one non-zero eigenvalue, =1) ris the Density Matrix

  6. N - level systems Density Matrix  is N x N matrix, elements ij Notation: [i,j] = index from1 to N2; [i,j]=(i-1)N+j Define Complex N2-vector V() V[i,j]() = ij Ex: N=2:

  7. Dissipative Dynamics (Non-Hamiltonian) gis a Population Relaxation Coefficient Gis a Dephasing Coefficient 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

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

  9. 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.

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

  11. 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)

  12. Dissipation and Semigroups I. Sets of Bounded Operators Def:Norm of an operator A: ||A|| = sup {|| Ay || / || y ||, yÎH} Def: Bounded operatorThe operator A in H is a bounded operator if ||A|| < K for some real K. Examples: Xy( x ) = x y( x)is NOT a bounded operator onH; butexp (iX)IS a bounded operator. B(H)is the set of bounded operators onH.

  13. Dissipation and Semigroups II. Bounded Sets of operators: Consider S-(A) = {exp(-t) A; A bounded, t ³ 0 }. Clearly S-(A)  B(H). There exists K such that ||X|| < K for all X Î S-(A) S-(A) is a Bounded Set of operators Clearly S+(A) = {exp(t) A; A bounded, t ³ 0 } does not have this (uniformly bounded) property.

  14. Dissipation and Semigroups III. Semigroups 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. Example: The set{ exp(-t):t>0 }forms a semigroup. Example: The set{exp(-t):³0 }forms a semigroup with identity.

  15. Dissipation and Semigroups One-parameter semigroups T(t1)*T(t2)=T(t1 + t1) with identity,T(0)=I. Important Example: IfLis a (finite) matrix with negative eigenvalues, andT(t) = exp(Lt). Then {T(t), t³0 }is a one-parameter semigroup, with Identity, and is a Bounded Set of Operators.

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

  17. Dissipation Dynamics - Semi-Group Local Form Superoperator Form Pure Hamiltonian (Formal) LHgenerates Group Pure Dissipation (Formal) LDgenerates Semi-group

  18. Example: Two-level System (a) Hamiltonian Part: (fx and fy controls) Dissipation Part:V-matrices with

  19. Example: Two-level System (b) (1) In Liouville form (4-vectorVr) Where LHhas pure imaginary eigenvalues and LD real negative eigenvalues.

  20. 4X4 Matrix Form 2-Level Dissipation Matrix 2-Level Dissipation Matrix (Bloch Form) 2-Level Dissipation Matrix (Bloch Form, Spin System)

  21. Solution to Relaxation/Dephasing Problem Choose Eij a basis of Elementary Matrices, i,,j = 1…N V-matrices G s g s

  22. Solution to Relaxation/Dephasing Problem (contd) Determine V-matrices in terms of physical dissipation parameters ( N2 x’s may be chosen real,positive) N(N-1)gs N(N-1)/2 s

  23. Solution to Relaxation/Dephasing Problem (contd) Problem: Determine N2 x’s in terms of the N(N-1) relaxation coefficients g and the N(N-1)/2 pure dephasing parameters N(N-1)gs N(N-1)/2 s There are (N2-3N)/2 conditions on the relaxation parameters; they are not independent!

  24. 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.

  25. Dissipative Terms • Orthonormal basis: Population Relaxation Equations (g³0) Phase Relaxation Equations

  26. Quantum Liouville Equation (Phenomological) • Incorporating these terms into a dissipation superoperator LD Writing r(t) as a N2 column vector V Non-zero elements of LDare (m,n)=m+(n-1)N

  27. Liouville Operator for a Three-Level System

  28. 2 32 12 3 V-system 3 23 13 2 12 1 Three-state Atoms 1 3 1 21 23 2 Ladder system L-system

  29. Decay in a Three-Level System Two-level case In above choose g21=0 and G=1/2 g12which satisfies 2-level constraint And add another level all new g=0.

  30. “Eigenvalues” of a Three-level System

  31. Phase Decoherence in Three-Level System

  32. “Eigenvalues” of a Three-level SystemPure Dephasing Time (units of 1/G)

  33. Three Level Systems

  34. Four-Level Systems

  35. Constraints on Four-Level Systems

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