Editing Bogoliubov transformation (section)

== Single bosonic mode example ==

Consider the canonical [[Commutator|commutation relation]] for [[bosonic]] [[creation and annihilation operators]] in the [[harmonic oscillator]] basis
:<math>\left [ \hat{a}, \hat{a}^\dagger \right ] = 1.</math>
Define a new pair of operators
:<math>\hat{b} = u \hat{a} + v \hat{a}^\dagger,</math>
:<math>\hat{b}^\dagger = u^* \hat{a}^\dagger + v^* \hat{a},</math>
for complex numbers ''u'' and ''v'', where the latter is the [[Hermitian conjugate]] of the first.

The Bogoliubov transformation is the canonical transformation mapping the operators <math>\hat{a}</math> and <math>\hat{a}^\dagger</math> to <math>\hat{b}</math> and <math>\hat{b}^\dagger</math>. To find the conditions on the constants ''u'' and ''v'' such that the transformation is canonical, the commutator is evaluated, namely,
:<math>\left [ \hat{b}, \hat{b}^\dagger \right ]
     = \left [ u \hat{a} + v \hat{a}^\dagger , u^* \hat{a}^\dagger + v^* \hat{a} \right ]
     = \cdots = \left ( |u|^2 - |v|^2 \right ) \left [ \hat{a}, \hat{a}^\dagger \right ]. </math>
It is then evident that <math>|u|^2 - |v|^2 = 1</math> is the condition for which the transformation is canonical.

Since the form of this condition is suggestive of the [[Hyperbolic function|hyperbolic identity]] 
:<math>\cosh^2 x - \sinh^2 x = 1,</math>
the constants {{mvar|u}} and {{mvar|v}} can be readily parametrized as
:<math>u = e^{i \theta_1} \cosh r,</math>
:<math>v = e^{i \theta_2} \sinh r.</math>
This is interpreted as a [[Symplectic vector space|linear symplectic transformation]] of the [[phase space]]. By comparing to the [[Symplectic matrix#Diagonalisation and decomposition|Bloch–Messiah decomposition]], the two angles <math>\theta_1</math> and <math>\theta_2</math> correspond to the orthogonal symplectic transformations (i.e., rotations) and the [[Squeeze operator|squeezing factor]] <math>r</math> corresponds to the diagonal transformation.

===Applications===
The most prominent application is by [[Nikolai Bogoliubov]] himself in the context of [[superfluidity]].<ref>N. N. Bogoliubov: ''On the theory of superfluidity'', J. Phys. (USSR), 11, p. 23 (1947), (Izv. Akad. Nauk Ser. Fiz. 11, p. 77 (1947)).</ref><ref>{{cite web |last1=Bogolubov [sic] |first1=N. |title=On the theory of Superfluidity |url=http://ufn.ru/pdf/jphysussr/1947/11_1/3jphysussr19471101.pdf |website=Advances of Physical Sciences |publisher=Lebedev Physical Institute |access-date=27 April 2017}}</ref> Other applications comprise [[Hamiltonian (quantum mechanics)|Hamiltonians]] and excitations in the theory of [[antiferromagnetism]].<ref name="Kittel">See e.g. the textbook by [[Charles Kittel]]: ''Quantum theory of solids'', New York, Wiley 1987.</ref> When calculating quantum field theory in curved spacetimes the definition of the vacuum changes, and a Bogoliubov transformation between these different vacua is possible. This is used in the derivation of [[Hawking radiation]]. Bogoliubov transforms are also used extensively in quantum optics, particularly when working with gaussian unitaries (such as beamsplitters, phase shifters, and squeezing operations).