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852_2010hw13_Solutions

# 852_2010hw13_Solutions - PHYS852 Quantum Mechanics II...

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PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 13: Solutions Topics covered: Hilbert-space Frame Transformations, Time-Dependent Perturbation Theory 1. The Hamiltonian for a driven two-level system is H = ~ ω 0 | 2 ih 2 | + ~ Ω cos( ωt ) ( | 1 ih 2 | + | 2 ih 1 | ) , (1) where ω 0 is the separation between the bare levels, and ω is the driving frequency. (a) Make a frame transformation generated by the operator G = ~ ω | 2 ih 2 | , and determine the equation of motion for the state-vector in the new frame, defined by | ψ G ( t ) i = U G ( t ) | ψ ( t ) i . First, we note that U GS ( t ) = e iGt/ ~ = | 1 ih 1 | + e iωt | 2 ih 2 | (2) (the | 1 ih 1 | term must be there in order to satisfy U GS (0) = I ). Following the general theory in the lecture notes, we have H G ( t ) = U GS ( t ) H S ( t ) U SG ( t ) - G = ( | 1 ih 1 | + e - iωt | 2 ih 2 | ) H ( | 1 ih 1 | + e iωt | 2 ih 2 | ) - ~ ω | 2 ih 2 | = ~ ( ω 0 - ω ) | 2 ih 2 | + ~ Ω cos( ωt ) ( | e iωt | 1 ih 2 | + e - iωt | 2 ih 1 | ) = ~ Δ | 2 ih 2 | + ~ Ω 2 ( | 1 ih 2 | + | 2 ih 1 | ) + ~ Ω 2 ( e i 2 ωt | 1 ih 2 | + e - i 2 ωt | 2 ih 1 | ) . (3) Hence the equation of motion for | ψ G ( t ) i is: d dt | ψ G ( t ) i = - i ~ H G ( t ) | ψ G ( t ) i (4) (b) Make the rotating wave approximation (RWA) by assuming that ω ω 0 , and dropping any terms that oscillate at or near 2 ω 0 . Write, in terms of the detuning Δ = ω 0 - ω , the effective time-independent Hamiltonian, H G , that then governs the time evolution of | ψ G ( t ) i . Dropping the terms rotating at ± 2 ω gives: H G,RWA = ~ Δ | 2 ih 2 | + ~ Ω 2 ( | 1 ih 2 | + | 2 ih 1 | ) , (5) In terms of the Rabi Hamiltonian, H Rabi = Δ S z + Ω S x , we have H G,RWA = H Rabi + ~ Δ 2 . (6) 1

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(c) Assume that the system begins at time t = 0 in the ground-state of H G , and calculate | ψ G ( t ) i . Is this a stationary state in the rotating frame? Now use | ψ S ( t ) i = U SG ( t ) | ψ G ( t ) i to see what this state looks like in the Schr¨odinger picture. Is it a stationary state in the Schr¨ odinger picture? The eigenstates of H G are those of H Rabi , and the eigenvalues of H G , are those of H Rabi plus the constant shift ~ Δ 2 . The ground state of H G is therefore | ω - G i = ( Ω + Ω 2 + Δ 2 ) | 1 i - Δ | 2 i q 2 ( Ω 2 + Δ 2 + Ω Ω 2 + Δ 2 ) , (7) which is a stationary state in the rotating frame (Recall that a stationary state evolves as | φ ( t ) i = e - iωt | φ (0) i ). In the Schr¨odinger picture, this state becomes | ω - S i = ( Ω + Ω 2 + Δ 2 ) | 1 i - Δ e - iωt | 2 i q (Ω + Ω 2 + Δ 2 ) 2 + Δ 2 , (8) which is not a stationary state. The point is the eigenstates of H G are not energy eigen- states, but the problem can be solved in the rotating frame, and then transformed at the end to get the solution in the Schr¨ odinger picture. Note that for time-dependent V , energy is not a conserved quantity. Up to a tiny correction due to the dropped terms in the RWA, H G,RWA is a constant of motion. This constant is often referred to as the ‘quasi energy’. 2
(d) Assuming the system begins in the ground state of H G , use second-order time-dependent perturbation theory to treat the fast-oscillating terms that were discarded in the RWA, and compute the probability to find the system in the excited state of H G , at time t > 0.

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852_2010hw13_Solutions - PHYS852 Quantum Mechanics II...

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