Lecture 12: Periodic Perturbation and Fermi’s Golden Rule

Recall first:

Quantum-transition probabilities from 1st-order time-dependent perturbation theory

Simple example first: constant perturbationSimple example first: constant perturbation

where is called the transition frequency between the two levels

Time-dependence of transition probability for a fixed pair of initial and final states

At a fixed time t, the transition probability At a fixed time t, the transition probability VS the energy difference between initial and final statesVS the energy difference between initial and final states

Roughly speaking, the transition prob. is significant for ΔE/hbar ~ 1/Δt. Hence ΔE Δt ~ hbar

∆

Periodic perturbation

Plug this time-dependence into the general formula

Continued on the next slide

let us assume that the driving frequency is quite close to the absolute value of the transition frequency, then:

If second term dominates (almost zero denominator)

If first term dominates (almost zero denominator)

1st term 2nd term

continued on the next slide

Continuing the previous slide: For convenience, we set

Take the case of as an example:

This oscillation behavior does exist in the exact solution!

But is there anything missing because of the 1st-orderperturbation theory? Answer: multi-photon transitions are not captured.e.g.: two-photon transitions need second-order perturbation theory etc)

Rotating Wave Approximation (RWA)Rotating Wave Approximation (RWA)

In our derivations for the transition probability, we dropped either of the two terms. This is equivalent to drop one of the two terms (see above) of the time-periodic potential. This approximation is called rotating wave approximation (RWA), a well-known and important procedure in quantum optics.

Comment I:

Comment II: Excitation VS stimulated de-excitation

m

nExcitation probability:

n

m

An oscillating field can also bring quantum states down to lower levels! (think abouttime-reversal symmetry)

Example: Hydrogen atom in the presence of a radiation fieldExample: Hydrogen atom in the presence of a radiation field

Find the 1st-order perturbation result for quantum probabilities on n=2 states at an arbitrary time, assuming that at t = 0 the atom is in its ground state.

Time periodic perturbation:

One example of matrix element calculations:One example of matrix element calculations:

=

The total transition probability to all final states in a continuum

continued on the next slide

What if the final state is in a continuum?

plug in

Continuing the previous slide:

assumptions wemade here

First-order perturbation result

Assumed that driving frequency isclose to the transition frequency

Assumed that the transition matrixelements and density of states are slowly varying functions of energy

:Note:

Hmmm… are you sure? By addingmany oscillating terms you get alinearly increasing function?

Fermi’s Golden RuleFermi’s Golden Rule

The rate of transitions to a continuum of final states is proportional to the square of the transition matrix element between the initial and final states.

The rate of transitions to a continuum is also proportional to the density of states, evaluated at an energy that differs from the initial state energy by , where ω is the driving frequency of a periodic perturbation.

(same conclusions can be found if you consider transitions to lower levels)

SummarySummary

First-order perturbation theory is already very useful to describe First-order perturbation theory is already very useful to describe quantum transitions induced by periodic fields. quantum transitions induced by periodic fields.

Fermi’s golden rule is also derived from our first-order perturbation Fermi’s golden rule is also derived from our first-order perturbation theory, under the assumptions that the transition matrix elements theory, under the assumptions that the transition matrix elements and density of states are slowly varying functions.and density of states are slowly varying functions.