In time-dependent perturbation theory the main goal is to determine the time-evolution of a perturbed quantum system, with particular emphasis on calculating transition probabilities and modeling the irreversible decay of probability from a small quantum system coupled to a very large quantum system. The interaction picture will be used to discuss time-dependent perturbation.
Interaction Picture
Interaction Picture
Perturbation HamiltonianHS=H0S+HintS(t)
State vector∣ΨI(t)⟩=eℏiH0St∣ΨS(t)⟩
OperatorOI(t)=eℏiH0StOS(t)e−ℏiH0St
For the operator H0 itself, the interaction picture and Schrödinger picture coincide
H0I=eℏiH0StH0Se−ℏiH0St=H0S
which can be easily obtained from the fact that H0 and eℏiH0St commute. From now on, we will omit the superscript representing pictures of H0 .
For the density operator ρ(t),it is no different from any other operators.
Dynamics Transforming the Schrödinger equation into the interaction picture gives
iℏ∂t∂∣ΨI(t)⟩=HintI(t)∣ΨI(t)⟩
iℏ∂t∂OI(t)=[OI(t),H0]
A quantum state is evolved by the interaction Hamiltonian HintI in the interaction picture, but a operator is evolved by the unperturbed Hamiltonian H0 like in Heisenberg picture.
Expectation The expectation value of an operator in the interaction picture is a sandwich structure as well
A pure state ∣ψ⟩ is a representative of a equivalence class
{eiχ∣ψ⟩∣α∈R}
One can map the state space of a two-level system C2 to the unit 3D sphere S2 by
∣ψ⟩=cos2θ∣1⟩+eiϕsin2θ∣0⟩
n(θ,ϕ)=(sinθcosϕ,sinθsinϕ,cosθ)
where θ∈[0,π] and ϕ∈[0,2π).
Bloch vector
We set the basis of the state space to {∣1⟩,∣0⟩}, which means ∣1⟩=(10) and ∣0⟩=(01).
/* Note: This definition is different from the order of kets that are usually chosen. The advantage of this kind of choice is that the Hamiltonian will have the form H=21ℏω0σz instead of −σz, since we set ∣0⟩≡∣g⟩ and ∣1⟩≡∣e⟩. */
State
Coordinates on S2
(θ,ϕ)
Ket
Vector
∣+⟩z≡∣1⟩
(0,0,1)
(0,0)
∣1⟩
(10)
∣−⟩z=∣0⟩
(0,0,−1)
(π,0)
∣0⟩
(01)
∣+⟩x
(1,0,0)
(2π,0)
2∣1⟩+∣0⟩
21(11)
∣−⟩x
(−1,0,0)
(2π,π)
2∣1⟩−∣0⟩
21(1−1)
∣+⟩y
(0,1,0)
(2π,2π)
2∣1⟩+i∣0⟩
21(1i)
∣−⟩y
(0,−1,0)
(2π,23π)
2∣1⟩−i∣0⟩
21(1−i)
Theorem Antipodal points on the Bloch sphere corresponds to a pair of mutually orthogonal state vectors.
Rotation
A anticlockwise rotation around n^ by θ on the Bloch sphere is described by rotation operator
A unitary transformation (or frame change) can be expressed in terms of a time-dependent Hamiltonian H(t) and unitary operator U(t). Under this change, the Hamiltonian transforms as
H(t)→H~(t)=UHU†+iℏU˙U†
∣ψ(t)⟩→∣ψ~(t)⟩=U∣ψ(t)⟩
Maintaining the form of Schrodinger equation
iℏdtd∣ψ~(t)⟩=H~(t)∣ψ~(t)⟩
Warning: the expectation value of operators is no longer sandwich structure, the Hamiltonian, for example
There are two ways to comprehend the additional term iℏU˙U† in the expression of transformed Hamiltonian H~. The first way is to compare the situation with that in classical mechanics.
In classical mechanics, if an object undergoes circular motion with an angular frequency ω , then its centrifugal potential energy is E=21mω2R2. When switching to a reference frame rotating at the same angular frequency of ω , an observer in this reference frame will find that the object is completely stationary. Then this observer will consider its energy as 0. The centrifugal potential energy disappears when performing rotating frame transformation.
In quantum mechanics, the energy is described by the Hamiltonian. iℏU˙U† is that disappeared centrifugal potential energy. For example, in a two-level system H=ℏωg∣g⟩⟨g∣+ℏωe∣e⟩⟨e∣ , for a unitary transformation like
U=eiω1t∣g⟩⟨g∣+eiω2t∣e⟩⟨e∣
one can obtain with some calculation
iℏU˙U†=−ℏω1∣g⟩⟨g∣−ℏω2∣e⟩⟨e∣
therefore, the stationary part is transformed into
H~=UHU†+iℏU˙U†=ℏ(ωg−ω1)∣g⟩⟨g∣+ℏ(ωe−ω2)∣e⟩⟨e∣
We could draw a conclusion that the effect of U is to shift down the energy level of eigenstates by ω1 and ω2 .
The second way is to look back on Schrodinger equation. Still in our two-level system, supposing the initial state is
∣ψ(0)⟩=cg∣g⟩+ce∣e⟩
The state at arbitrary time t is then determined by iℏdtd∣ψ(t)⟩=H(t)∣ψ(t)⟩, thus
∣ψ(t)⟩=cge−iωgt∣g⟩+cee−iωet∣e⟩
where the energy of eigenstates, ℏωe and ℏωg, appear on the exponential of phase factor.
The unitary transformation ∣ψ(t)⟩→∣ψ~(t)⟩=U∣ψ(t)⟩ actually shift the kets to
And to examine whether the interaction part of the Hamiltonian is now time-independent under your transformation, just substitute the transformed ∣a~⟩,∣b~⟩,∣c~⟩ into the original expression and see if the oscillating factor of electric field is cancelled.