Problem set 8
Problem 1: Dynamics in the rotating frame
a. Let |ψ̃(t)i = U † (t)|ψ(t)i where |ψ(t)i is the solution of the Schrödinger equation
d|ψi
i dt = H|ψi (~ = 1). Show that the “rotating wavefunction” |ψ̃(t)i evolves according to a
†
U . If we choose U (t) = exp(−iHt)
Schrödinger equation with Hamiltonian H̃ = U † HU +i dU
dt
what would H̃ be?
b. How would one do a measurement in Pauli X-basis in the rotating frame when this
rotating frame is given by some U † (t) = exp(iωZt)?
Problem 2: Rabi Oscillations
~ t).
We consider an atom with two electronic levels in a time-changing electric field E(r,
The two-level Hamiltonian of the atom itself is H0 = − ω20 Z = − ω20 (|gihg| − |eihe|) where
|ei is the excited and |gi is the groundstate. In the so-called dipole approximation we
approximate the effect of the electric field as
V = −JX cos(ωt).
(1)
where J determines the strength of the coupling and X = |eihg|+|gihe|. The full Hamiltonian
is H = H0 + V .
a. Assume an ansatz for the Schrödinger equation of the form |ψ(t)i = αg (t)e−iH0 t |gi +
αe (t)e−iH0 t |ei and obtain differential equations for αg (t), αe (t). Make the rotating wave
approximation, that is, neglect in the differential equations for αg (t) and αe (t) the fast
changing term proportional to exp ±i(ω + ω0 )t. Let ∆ = ω0 − ω (called the detuning) which
is assumed to be much smaller than ω + ω0 .
b. Assume that αe (t) = eiλt and show that one can solve for λ± = 21 (∆ ±
√
∆2 + J 2 ).
Thus a general solution for αe (t) is αe (t) = α+ eiλ+ t + α− eiλ− t with λ± = 12 (∆ ± ΩR ) where
√
ΩR = ∆2 + J 2 is called the Rabi frequency.
c. What is the expression for the probability Pe (t) to find the atom in the state |ei at
a time t when we start it in the initial state |gi? What is the frequency of the oscillatory
behavior?
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Problem 3: Jaynes - Cummings Hamiltonian
The Hamiltonian approximately describing atom-field interaction which does take into
account the quantum state of the electromagnetic field is the Jaynes-Cummings Hamiltonian.
We start with the Hamiltonian
H = Hatom + Hf ield + Hint .
where Hatom =
ω0
(|eihe| − |gihg|),
2
(2)
Hf ield = ω(a† a + 12 ) and Hint = Ω2 (a + a† )(σ+ + σ− ) where
σ± = 12 (X ± iY ).
a. When |ω − ω0 | ω + ω0 , argue that one can make a rotating wave approximation and
approximate this Hamiltonian by the so-called Jaynes-Cummings Hamiltonian
HJC = Hatom + Hf ield +
Ω
(aσ+ + a† σ− ).
2
(3)
b. Show that the energy eigenstates of the Jaynes-Cummings Hamiltonian at resonance
ω0 = ω are
1
√ (|e, ni ± |g, n + 1i) n = 0, 1, . . .
2
(4)
1 √
E±,n = (n + 1)ω ± Ω n + 1.
2
(5)
with eigenenergies equal to
(Hint: analyze the effect of the Hamiltonian in the 2 x 2 subspaces spanned by |e, ni and
|g, n + 1i).
c. Imagine that at time t = 0 the system starts in the state |ei ⊗ |αi where |αi is a
coherent state with very high mean photon number n̄ = |α|2 (corresponding to the classical
limit). Argue that in this limit we can approximate the time evolution by Rabi oscillations
as in Problem 2. That is, the field approximately remains in the state |αi and the atom
√
rotates with Rabi-frequency Ωcl = Ω n̄.
2