Chapter 1: Two-level atom

1.1 Maxwell equations

1.2 Material polarization

1.3 Quantum (semi-classical) approach to polarization

of a two-level atom and density matrix

1.4 Two-level atom in e/m field

1.5 Two-level atom: dynamics – Rabi Oscillations

1.6 Linear and nonlinear susceptibilities of a two level

atom (saturation effects)

1.7 Maxwell-Bloch equations in optical cavities

(lasers and bistability)

Book: Robert Boyd, ‘Nonlinear Optics’,

Chapter 6: Nonlinear Optics in the 2 level approximation

Chapter 7.3: Optical bistability

1.1 Maxwell equations

Current and charges located

outside the material. They are

assumed zero in this course

The external set drives the material, while the internal one

is a material itself, which responds to the driving

Materials we consider are nonmagnetic.

Magnetic response of atoms is usually

much smaller than their electric response.

below M=0

so called constitutive equations

M

Expanding curl of curl we find

Expanding curl of curl we find

Reduction of Maxwell equations to the amplitude equation

(with the amplitude slowly varying in time)

+

+

Let’s apply electric field to an atom for a very

short period of time, so that its electron

cloud shifts and then switch the field off

E exists

only at t=0

The cloud starts oscillating.

time

Dipole moment (polarization)

of a single atom

While electron oscillates x\ne 0 and hence P and its 2nd

time derivative \ne 0, which creates non zero electric field

oscillating at the frequency of the electron oscillations.

Classically

N is the density of atoms

1.2 Material polarization

this makes

the asymmetry

of the electron

cloudEven to even, or odd to odd transitions, can not be associated

with the asymmetric shift of the cloud (only with symmetric

pulsations, which do not induce dipole moment )

Constraints on wavefunctions required for dipole oscillations

1.3 Quantum (semi-classical) approach to polarization

of a two-level atom and density matrix

Our aim is to learn how to express polarization in terms of E.

Nonlinear optics: P(E) and frequency response

a

b

Is the Hamiltonian of a bare (i.e. no external field) atom

When an atom is in the process of emitting or absorbing

It is in a superposition state

are the time independent constants

As we’ve seen above for a dipole to be created by e/m field,

We need to consider a superposition of wavefunctions with

opposite parity, i.e.

Dipole moment operator

Atom in superposition states creates a polarization, which

Is an expectation value of its dipole moment

or vice versa, then

Density Matrix

Matrix element of the dipole moment

Coherence or polarization

populations

Population inversion of a single atom

Total population inversion, N density of atoms

P_0 is slowly varying in time relative to

because

Note, that there are 2 terms inside the expression for

polarization. One oscillates with positive frequency,

the other with negative

an atom interacting with nearly resonant field

1.4 Two-level atom in e/m field

The force on a charge

The corresponding potential energy

Interaction potential

a

b

Now, f’s have to be functions of time

Make a substitution into the Schrodinger equation,

Project the result onto U_a and U_b. While taking the

Integrals consider that

Detuning between the field frequency and

Frequency of the atomic transition

Note, that \vec E is real

1.5 Dynamics of 2 level atom, Rabi oscillations

Population inversion and

polarization in absence of the

optical field have the finite life

time, because atom interacts

with its surrounding:

vacuum fluctuations

(spontaneous emission) and

collisions with other atoms.

Therefore peaks are broadened

No interaction

with surrounding Realistic case

-1

1

If the electromagnetic (EM) wave is resonant with the atoms transition

frequency, it brings the atom from the ground state to its excited state.

When the atom is in its excited state, then another passing photon can

de-excite the atom, so that a photon is emitted.

Thus, the probability of finding the atom in the excited state initially

increases and at some point becomes one and then drops again. Since

the atom continues to interact with the EM wave, it emits and absorbs

photons periodically. This process of periodic emission and re-

absorption is called Rabi oscillations. The Rabi frequency is the rate at

which the population difference between the ground and excited states

oscillates and it is proportional to the strength of the EM field.

Rabi frequency equals to frequency splitting of the optical field seeing in

the interaction of a monochromatic wave with a two level atom.

The above theory and discussion is valid providing

Generation of new frequencies as a result of light atom

Interaction, is an example of nonlinear interaction. In linear

Optics the input spectrum=output spectrum. It is an example of

Four-wave mixing

Differentiate P_0 and M and demonstrate that

We introduce relaxation terms phenomenologically.

Decay of the polarization and inversion happens due

collisions with other atoms, spontaneous photons etc

M_0 is the equilibrium inversion achieved in the absence

of the optical field. If it is assumed that the ground state ‘a’

does not experience a spontaneous decay, while ‘b’ does

decay into ‘a’, then the total population of all levels in all atoms

is 1*N. At the large time we are expecting for a single atom

In the absence of the field to have M_0= -1.

In general and for many atoms M_0 is different from – N .

1.6 Bloch equations. Linear and nonlinear

Susceptibilities of a two level atom. Saturation effects.

Dropping vectors for brevity

In the approximation, when time derivatives are zero,

You have to demonstrate, that

where

This a definition of atomic susceptibility, which is a

coefficient of proportionality between the field

and polarization

1.7 Maxwell-Bloch equations in cavities

1.7a Lasers

Linear cavity

Ring cavity

Standing

waves

Cavity frequency

Travelling waves

Light out

Detuning between the

atomic and cavity

resonances

For these condition, the time derivatives of Q and M,

can be assumed zero

Laser threshold

1.7b Bistability in optical cavities (coherent pump)

An optical cavity is bistable, if for the same pump F, there exist

two different values of the intracavity field ‘b’

Detuning between the

cavity resonance and pump

Light out

Light in

Nonlinear

crystal

Pattern formation in bistable optical cavities