Lect11_QuantRes - Quiz#4 A system is described by the Hamiltonian H = h"0"1"1 4"2"2 9"3"3 j"k = jk 1 What are the eigenvalues of H!1=!2= 2 What is

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Quiz #4 A system is described by the Hamiltonian: 1. What are the eigenvalues of H ? ( ! 1 = ?, 2 = ?, …) 2. What is the density operator at temperature T ? (Hint: express " using the basis of eigenstates of H ) 3. For T = 4 h 0 /K , what is the probability to find the system in state | 2 # ? ( ) jk k j H = + + = 3 3 2 2 1 1 0 9 4 h
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Review The Rabi Model: By choosing a suitable zero for energy, any two- level Hamiltonian can be written as: $ is the ‘Rabi Frequency’ and % is the ‘detuning’ The eigenvalues and eigenvectors are: ‘Level repulsion’: when two levels are coupled, they always tend to ‘repel’ each other. ‘Avoided crossing’: when two degenerate levels are coupled, the coupling ‘lifts’ the degeneracy, creating a gap of energy E gap = h H = h 2 "# $ % $ # & ( ) * + " ± = ± 1 2 2 + $ 2 ± = ± 2 + $ 2 % # & ( ) * + 1 + $ 2 ± 2 + $ 2 % # ) * + 2 + $ 2 Adiabatic transfer ‘Adiabatic transfer’: slowly vary a parameter to take a system through and avoided crossing Start in | ! 1 # with large positive % , then vary % to large negative value 1. Slow: System goes smoothly from | 1 # to | 2 # while staying in | - # Transfer time condition: 2. Fast: system stays in physical state | 1 #
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This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.

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Lect11_QuantRes - Quiz#4 A system is described by the Hamiltonian H = h"0"1"1 4"2"2 9"3"3 j"k = jk 1 What are the eigenvalues of H!1=!2= 2 What is

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