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Constraints on N-Level Systems*

Constraints on N-Level Systems*. Allan Solomon 1 , 2 and Sonia Schirmer 3. 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.

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Constraints on N-Level Systems*

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  1. Constraints on N-Level Systems* Allan Solomon1,2and 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

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

  3. 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.

  4. qubit PureStates Finite Systems (1) Pure States 2-level N-level

  5. General States Purestate y can be represented by operator projecting ontoy (2) Mixed States or Density Matrices For example (N=2) y= [a,b]Tas matrix r • 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. Hamiltonian Dynamics - Liouville Form(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.

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

  8. 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:

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

  10. Liouville Operator for a Three-Level System

  11. 2 32 12 1 3 L-system 3 23 3 1 13 2 21 23 12 2 Ladder system 1 V-system Three-state Atoms

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

  13. Population evolution Initial state

  14. eigenvalues of r for this Three-level System

  15. Three-Level System:PurePhase Decoherence Eigenvalues Time (units of 1/G)

  16. Liouville Matrix for Pure Dephasing Example LD = CLEARLY NEGATIVE EIGENVALUES! Similarly for previous example.

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

  18. Dissipation Dynamics - General Local Form* 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)

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

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

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

  22. Three Level Systems

  23. 3-Level Ladder system We assume population transition from level 2 to level 1 only. We want to derive relations on the dephasing Exponents G, G1, G2.

  24. Three-level Ladder System 3 23 13 2 12 1 Ladder system ‘isolate’ level 3 23=0 13=0 RESULT G23 > ½ g12 If G13 =0 then G21 =G23

  25. Example: 3-level ladder system Eigenvalue plot g12=1, G12=G23=5/4

  26. PureDephasing Inequalities Among 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

  27. Four-Level Systems

  28. Constraints on Four-Level Systems

  29. Conclusions • Equations for dissipative effects may force relationships on dephasing of levels even when they are not involved in the population transitions. These effects do not occur for all initial configurations,e.g. for thermal states or initial pure states where the ‘isolated’ state is not populated.

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