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PHAS0069

Advanced Quantum Theory Problem Sheet 3

2022


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1. Time-dependent perturbation theory. A spin-1 particle is held in a strong magnetic field in the z-direction. Immediately prior to time t = -t0 , a measure- ment of its spin indicates that it is in the state |s = 1, ms  = 1). At t = -t0 the experiment is perturbed by a weak magnetic field in the x-direction which ramps up to a maximum and then decays back down to zero at time t = t0 .

(a) The resulting Hamiltonian is H = Ω z  + λ(t)x  where λ(t) = λ0 (1 - |t|/t0 ) for |t| < t0 and λ(t) = 0 for |t| 2 t0 and |λ0 | < Ω . Using pertur- bation theory, show that (to first-order) the probability that a measurement on the spin at time t = t0 will indicate ms = 0 is:

You may find the following spin-1 matrix representations of z  and x :



and the following indefinite integral helpful:

(b) Without detailed calculation, explain why, in this example, second order perturbation theory is required to see a non-zero transition probability to the state |s = 1, ms  = -1).


2. Nonclassical light and compound systems. Consider a two-level atom with excited state |e)and ground state |g)in an external light field in a one-photon mode. The initial state of the atom is



The Hamiltonian describing the light-atom interaction reads

where a and a are the ladder operators. Assume now that the interaction takes place at time t, such that Ωt < 1 and you can consider the expansion of the time evolution operator up to second order. Show that the state of the light-atom compound system at a time t reads


3. Open Quantum Systems. Consider a two-level atom with excited state |e)and ground state |g)such that its Hamiltonian is H = 尸ω|e)(e|. The action of the environment interacting with the atom is described by the jump operators L1  = Γ|e)(g| and L2  = γ|g)(e|.

(a) Assuming that at t  =  0 the state of the atom is ρ(0)  =  |g)(g|, show that the probability of finding the atom in the excited state at time t , ρee (t)  =  (e|ρ(t)|e), is given by


(b) Find a relation between Γ and γ such that in the long-time limit ρee (o) equals the probability of finding the atom in its excited state when it is in thermal equilibrium at temperature T. Recall that in thermal equilibrium, a system with Hamiltonian H is described by the density matrix operator Express your answer as

and specify the value of C as a function of ω and kB T where kB  is the Boltzman constant.