Problem 1: Exponential Switching

Consider the following perturbation to a two-level system:

Vfi (t)=x(1-e-at) e[t]

Here x is the strength of the perturbation and a is the rate at which this strength is applied. In the plot below, the steepest curve is for the largest value of a.

"Exponential Switching

Figure 1

[a] Show that, in the limit of large time and infinitely slow exponential turn on, the transition probability as the following limit:

x²

w₂2

Use this expression is a useful asymptote that you will apply in parts [b] and [c] below.

[b] Consider an exponential switching perturbation for which x = 0.1 w₁ and a = 0.1 w₁i. Plot the transition probability, using time steps of 1/ wf, and show the

asymptote derived in [a].

Plt → arbitrarily large, a +0 =

(1)

(2)

[c] Repeat the analysis of [b] but now with a much slower rate of turn-on: a = 0.001 wf. Plot the transition probability, using time steps of 1/ w, and show the

asymptote derived in [a]. What is the difference between your results the two cases? Be sure o discuss the overall probability as well as the undulat in the

transition probability with time.

[d] Assume that the time-evolving state is a linear combination of the initial and final states. Substitute this into the Schrodinger equation, in the interaction picture,

to obtain an ODE for M₁. Numerically solve this ODE to obtain p[t], and compare the result with your perturbation approximations of [b] and [c]. Comment on the

accuracy of your perturbation approximation.