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Constraints on N-Level Systems* Allan Solomon 1,2 and Sonia Schirmer 3 1 Dept. of Physics & Astronomy. Open University, UK 2. LPTL, University of Paris VI, France 3. DAMTP, Cambridge University, UK *S. G. Schirmer and A. I. Solomon, PHYSICAL REVIEW A 70, (2004) Cozumel, December 2004

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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.

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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.

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PureStatesPureStates Finite Systems (1) Pure States N-level 2-level qubit

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

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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.

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Dissipative Terms Orthonormal basis: Population Relaxation Equations ( Phase Relaxation Equations

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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 Define Complex N 2 -vector | > = V V (i,j) () = ij Ex: N=2:

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

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Liouville Operator for a Three-Level System

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Three-state Atoms 13 2 12 32 -system 12 13 Ladder system 3 2 21 23 V-system 1

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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 .

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Population evolution Initial state

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eigenvalues of for this Three-level System eigenvalues of for this Three-level System

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

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Liouville Matrix for Pure Dephasing Example L D = CLEARLY NEGATIVE EIGENVALUES! Similarly for previous example.

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

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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)

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

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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.

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Solution to Relaxation/Dephasing Problem Choose E ij a basis of Elementary Matrices, i,,j = 1…N V -matrices s s

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Three Level Systems

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

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Three-level Ladder System 12 13 Ladder system ‘isolate’ level 3 23 =0 13 =0 RESULT 23 > ½ 12 If 13 =0 then 21 = 23

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Example: 3-level ladder system Eigenvalue plot 12 =1, 12 = 23 =5/4

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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

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Four-Level Systems

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Constraints on Four-Level Systems

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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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