196
6 Effects of Quantisation
6.3 Benefits and Applications
Typically, quantum structures are at the heart of many optoelectronic applications
and quantum technologies of the first and second generation, as they enable nonclassical device concepts, charge-carrier localisation and tailored energy transitions.
A few examples are summarised below, mainly from the fields of laser physics and
engineering, optoelectronic devices, as well as light–matter interactions.
Naturally, potential-well systems can be prepared for fermions and bosons equally,
whereas the major difference comes from the occupancy of states and the corresponding nonclassical particle distribution functions. While fermions obeying the Fermi–
Dirac (FD) distribution function can only occupy a quantum state once owing to the
Pauli exclusion principle (exhibiting an anti-bunching effect in terms of statistics) and
are distinguishable in the same energy state, e.g. by spin, bosons can occupy the same
mode or energy state macroscopically (exhibiting a bunching effect). Their statistics
differ fundamentally from classical particles—described by Maxwell–Boltzmann
(MB) statistics—and that of non-integer spin particles (fermions). An overview on
the three distribution functions (see Fig. 6.3) is provided for the sake of completeness
(with μ = E 0 the chemical potential, for fermions also known as Fermi energy E F ):
f classical (E) =
1
exp
E−μ
k B T
, (MB)
(6.8)
f fermion (E) =
1
exp
E−μ
k B T
+ 1
, (FD)
(6.9)
f boson (E) =
1
exp
E−μ
k B T
− 1
.(BE)
(6.10)
Thus, bosonic behaviour is characterised by the Bose–Einstein (BE) distribution
function, which can give rise to an extreme aggregate state of bosonic matter referred
to as Bose–Einstein condensate. In such a regime occurring after a spontaneous
symmetry break (establishment of a single phase throughout the condensate), all
condensed bosons are described by the same macroscopic wave-function and, thus,
become indistinguishable.
The Bose–Einstein condensation effect is directly related to other intriguing manyparticle phenomena, such as superfluidity (cf. [36, 37]) and superconductivity (condensation of Cooper pairs, composite bosons). In fact, bosons in the same mode
with the same physical properties, such as energy, momentum, spin/polarisation and
phase, are indistinguishable by nature. Below a critical temperature, the majority
of Bose particles accumulates in the ground-state of the system through stimulated
scattering processes,
8 when their wave packages strongly overlap with each other.
Thereby, a quantum fluid with long-range order (spatial coherence) is established.
8 The more particles are in the final state, the higher the final-state scattering rate becomes.
6 Effects of Quantisation
6.3 Benefits and Applications
Typically, quantum structures are at the heart of many optoelectronic applications
and quantum technologies of the first and second generation, as they enable nonclassical device concepts, charge-carrier localisation and tailored energy transitions.
A few examples are summarised below, mainly from the fields of laser physics and
engineering, optoelectronic devices, as well as light–matter interactions.
Naturally, potential-well systems can be prepared for fermions and bosons equally,
whereas the major difference comes from the occupancy of states and the corresponding nonclassical particle distribution functions. While fermions obeying the Fermi–
Dirac (FD) distribution function can only occupy a quantum state once owing to the
Pauli exclusion principle (exhibiting an anti-bunching effect in terms of statistics) and
are distinguishable in the same energy state, e.g. by spin, bosons can occupy the same
mode or energy state macroscopically (exhibiting a bunching effect). Their statistics
differ fundamentally from classical particles—described by Maxwell–Boltzmann
(MB) statistics—and that of non-integer spin particles (fermions). An overview on
the three distribution functions (see Fig. 6.3) is provided for the sake of completeness
(with μ = E 0 the chemical potential, for fermions also known as Fermi energy E F ):
f classical (E) =
1
exp
E−μ
k B T
, (MB)
(6.8)
f fermion (E) =
1
exp
E−μ
k B T
+ 1
, (FD)
(6.9)
f boson (E) =
1
exp
E−μ
k B T
− 1
.(BE)
(6.10)
Thus, bosonic behaviour is characterised by the Bose–Einstein (BE) distribution
function, which can give rise to an extreme aggregate state of bosonic matter referred
to as Bose–Einstein condensate. In such a regime occurring after a spontaneous
symmetry break (establishment of a single phase throughout the condensate), all
condensed bosons are described by the same macroscopic wave-function and, thus,
become indistinguishable.
The Bose–Einstein condensation effect is directly related to other intriguing manyparticle phenomena, such as superfluidity (cf. [36, 37]) and superconductivity (condensation of Cooper pairs, composite bosons). In fact, bosons in the same mode
with the same physical properties, such as energy, momentum, spin/polarisation and
phase, are indistinguishable by nature. Below a critical temperature, the majority
of Bose particles accumulates in the ground-state of the system through stimulated
scattering processes,
8 when their wave packages strongly overlap with each other.
Thereby, a quantum fluid with long-range order (spatial coherence) is established.
8 The more particles are in the final state, the higher the final-state scattering rate becomes.