Quantum dot Quantum optics Nanocavity
Derivation free-field Hamiltonian9

A classical representation of a single-mode field satisfying Maxwell's equations is given by

  Equation 3.40 (3.40)


  Equation 3.41 (3.41)

V is the effective volume of a cavity, k is the wave number and q(t) is a time-dependant factor having the dimension of length.

The magnetic field inside the cavity is given by

  Equation 3.42 (3.42)


  Equation 3.43 (3.43)

that is the canonical momentum for a ‘article’ of unit mass. The Hamiltonian H, or the classical field energy, is given by

  Equation 3.44 (3.44)


  Equation 3.45 (3.45)

In the same way, it can be shown that the free-field Hamiltonian in operator form is given by

  Equation 3.46 (3.46)

where the operators of p and q satisfy the canonical commutation relation

  Equation 3.47 (3.47)

It is now convenient to introduce

  Equation 3.48 (3.48)

which are respectively the non-Hermitian annihilation and creation operators. From these equations it can be shown that

  Equation 3.49 (3.49)

and therefore

  Equation 3.50 (3.50)

As a result, the free-field Hamiltonian operator takes the form

  Equation 3.51 (3.51)

because by definition

  Equation 3.52 (3.52)

and because the creation and annihilation operator satisfy the commutation relation

  Equation 3.53 (3.53)

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