3.1 Where Strong Interactions with Light Matters
65
3.1.1 Basics of Light–Matter Systems
Obviously, light–matter interaction goes beyond semiconductor physics and can be
quantum-mechanically very fundamental, as atom–cavity physics shows. To give
an idea of the formalism (after [107]), the example of a two-level atom interacting
with a single field mode shall be highlighted. Therefor, one typically considers the
interaction in dipole approximation, giving the prominent Hamiltonian H 1 = −er ·
E(r), with spatial vector r and electric field vector E. For stationary fields, this can
be rewritten as
H 1 = g(σ + + σ − )( f + f
†
),
(3.1)
with σ ± the Pauli spin matrices corresponding to atom excitation (+) and deexcitation (−), f and f
† the photon annihilation and creation operator, respectively,
and g the coupling strength. This is the fundamental light–matter interaction term,
which implies two energy conserving and two non-conserving processes (see e.g.
[107]). In dipole and rotating wave approximation (neglecting rapidly oscillating
terms), the interaction term is reduced to the energy conserving processes and the
Hamiltonian for the atom–field system becomes
H = H 0 + H 1 =
ω ab
2
σ z + ω f
† f + g( f σ + + σ − f
†
),
(3.2)
with the transition frequency ω ab , the optical frequency ω, the Pauli spin matrix σ z .
This is the Jaynes–Cummings Hamiltonian with atom, field and interaction term,
with the zero energy level set half-way between the two atomic levels a and b, so
that unperturbed the two-level system’s energy is ±
ω ab
2
. The interesting interaction
processes are f σ + , i.e. one photon gets absorbed and atom correspondingly excited,
and σ − f
† , the opposite process where one photon is emitted and the atom de-excited.
It is the starting point for the description of quantum Rabi oscillations.
In other words, strong interactions between a radiation field (optical mode) and
an emitter (electronic mode) can lead to reversible energy exchange (periodic oscillations), which manifests itself in the hybridisation of the coupled-oscillator system
and gives rise to new eigen-states in the strong-coupling regime, i.e. mixed states
originating from a linear superposition. Indeed, two bosons (oscillators with ω ab and
ω) form composite bosons (oscillators with ω ± ). For exact resonance conditions,
the new states represent a half–half hybrid. The detuning-dependent fractions of
the constituents (i.e. mode and excitation) are commonly expressed by the Hopfield
coefficients [85] or amplitudes of the dressed states in the diagonalised Hamiltonian.
To give examples from semiconductor physics, hybridisation of a single exciton with
the empty vacuum field can be achieved in special microcavities (see Sect. 4.3), or a
quantum-well exciton ensemble with a planar-microcavity mode, or a macroscopic
65
3.1.1 Basics of Light–Matter Systems
Obviously, light–matter interaction goes beyond semiconductor physics and can be
quantum-mechanically very fundamental, as atom–cavity physics shows. To give
an idea of the formalism (after [107]), the example of a two-level atom interacting
with a single field mode shall be highlighted. Therefor, one typically considers the
interaction in dipole approximation, giving the prominent Hamiltonian H 1 = −er ·
E(r), with spatial vector r and electric field vector E. For stationary fields, this can
be rewritten as
H 1 = g(σ + + σ − )( f + f
†
),
(3.1)
with σ ± the Pauli spin matrices corresponding to atom excitation (+) and deexcitation (−), f and f
† the photon annihilation and creation operator, respectively,
and g the coupling strength. This is the fundamental light–matter interaction term,
which implies two energy conserving and two non-conserving processes (see e.g.
[107]). In dipole and rotating wave approximation (neglecting rapidly oscillating
terms), the interaction term is reduced to the energy conserving processes and the
Hamiltonian for the atom–field system becomes
H = H 0 + H 1 =
ω ab
2
σ z + ω f
† f + g( f σ + + σ − f
†
),
(3.2)
with the transition frequency ω ab , the optical frequency ω, the Pauli spin matrix σ z .
This is the Jaynes–Cummings Hamiltonian with atom, field and interaction term,
with the zero energy level set half-way between the two atomic levels a and b, so
that unperturbed the two-level system’s energy is ±
ω ab
2
. The interesting interaction
processes are f σ + , i.e. one photon gets absorbed and atom correspondingly excited,
and σ − f
† , the opposite process where one photon is emitted and the atom de-excited.
It is the starting point for the description of quantum Rabi oscillations.
In other words, strong interactions between a radiation field (optical mode) and
an emitter (electronic mode) can lead to reversible energy exchange (periodic oscillations), which manifests itself in the hybridisation of the coupled-oscillator system
and gives rise to new eigen-states in the strong-coupling regime, i.e. mixed states
originating from a linear superposition. Indeed, two bosons (oscillators with ω ab and
ω) form composite bosons (oscillators with ω ± ). For exact resonance conditions,
the new states represent a half–half hybrid. The detuning-dependent fractions of
the constituents (i.e. mode and excitation) are commonly expressed by the Hopfield
coefficients [85] or amplitudes of the dressed states in the diagonalised Hamiltonian.
To give examples from semiconductor physics, hybridisation of a single exciton with
the empty vacuum field can be achieved in special microcavities (see Sect. 4.3), or a
quantum-well exciton ensemble with a planar-microcavity mode, or a macroscopic