Intro

• Light consists of photons which are bosons
• Bosons do not obey the Pauli Exclusion principle
• Since light itself does not have an electric charge, photons cannot directly interact.
• A laser beam can be thought of as a collection of photons all in the same quantum state

Cont.

• Photon-photon scattering is possible through an indirect mechanism as shown

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• This is a Feynman diagram where the red lines represent photons, the blue lines with e+ are electrons and the e- are positrons (anti-electrons)

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• The diagram represents 2 photons travelling anti-parallel to one another, decaying into 2 pairs of particle and anti-particles and recombining to form 2 new photons

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• Interestingly, this has never been observed experimentally

Cont.

• In QFT, bosons are the force carrying particles that mediate fermion interactions
• Specifically, we go from describing particles via wavefunctions (Hilbert space) to describing particles via wave-functionals (Fock space)
• The interactions between charged particles are then mediated by the exchange of (virtual) photons
• We consider the interactions to be perturbations in our QFT. Can solve for low order corrections with path integrals and Feynman diagrams

2^nd Quantization

• Let’s derive the free field Hamiltonian by quantizing an EnM field in a 1-D cavity.
• Start with source-less Maxwell equations

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• Now consider a 1-D cavity along z-axis with perfect conducting walls at Z=0 and Z=L

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Cont.

• Choose a polarization along x for E

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• Then the homogeneous EnM wave equation:

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becomes:

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• Solving by sep. of var.’s yields:

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• Giving solution:

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• Now using Ampere’s:

we obtain:

Cont.

• From Griffiths our classical field energy (and Hamiltonian) is:

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• Substituting our solutions yields:
• Similar to Harmonic oscillator
• The next step in quantizing our classical field theory is to promote our dynamical variables to operators

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• We can define Dirac ladder operators:

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• This implies:

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• And:

Quantum Nonlinear Optics

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• Want NLO’s at lowest energies possible
• QNLO the regime in which individual photons interact so strongly with each other that it affects propagation
• Essentially, we want to manipulate individual photons and then scale the process

Difficulties

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• The transition amplitude is (leading order)

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and taking a time derivative of the modulus squared gives the transition rate

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• This gives:

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Cont.

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• What we find is:
• The transition probability and thus the transition rates from state a to state b (and b to a) in the presence of our harmonic perturbation are equal
• This is how Einstein derived spontaneous emission

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• Saturation occurs when the number of particles in the excited state is the same as the number of particles in the ground state
• When this happens, absorption and emission vanish

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Optical Cavity

Single Atom Cavities

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Jaynes-Cummings model

• This is an energy diagram for our electromagnetic field coupled to our atom in a cavity

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• The rabi frequency due to a single photon interaction is responsible for splitting of the states

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Rabi Frequency

• The rabi problem is:

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• This allows for transitions between kets 1 and 2
• If at t=0 only state 1 is populated we obtain the transition probabilities:

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Atom Hamiltonian

• For a 2-state system we have unperturbed eigenkets:
• Thus,

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• Now, if we define:
• We see:
• So, we now have the free field and atomic Hamiltonian.
• Lastly, we need the Interaction term between photons and the atom

Interaction Hamiltonian

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• Equivalently the interaction term is

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• The first term represents a photon being absorbed and the atom being excited and the second represents a photon being emitted and the atom is de-excited

Cont.

• Free field, atomic and interaction Hamiltonian

• This Hamiltonian contains both quickly and slowly oscillating terms

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• To solve we ignore the quickly oscillating terms by the rotating wave approximation

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Turchette

• Here they are confining a cesium atom coupled to the cavity field

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• The cavity has length 56 µm and Gaussian waist of 35 µm

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• Their cavity finesse coefficient is

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• By sending in a linearly polarized probe beam they achieve phase shift

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QNLO Applications

• For a small photon number, strong interactions can achieve control of light fields photon-by-photon or implement photonic quantum gates
• QND single photon detectors
• Single-photon switches/transistors
• Deterministic photon-photon interactions

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