Bogoliubov transformation

In theoretical physics, the Bogoliubov transformation, named after Nikolay Bogolyubov, is a unitary transformation from a unitary representation of some canonical commutation relation algebra or canonical anticommutation relation algebra into another unitary representation, induced by an isomorphism of the commutation relation algebra. The Bogoliubov transformation is often used to diagonalize Hamiltonians, which yields the steady-state solutions of the corresponding Schrödinger equation. The solutions of BCS theory in a homogeneous system, for example, are found using a Bogoliubov transformation. The Bogoliubov transformation is also important for understanding the Unruh effect, Hawking radiation, pairing effects in nuclear physics, and many other topics.

Single bosonic mode example

Consider the canonical commutation relation for bosonic creation and annihilation operators in the harmonic basis

$\left[{\hat {a}},{\hat {a}}^{\dagger }\right]=1~.$

Define a new pair of operators

${\hat {b}}=u{\hat {a}}+v{\hat {a}}^{\dagger }$

${\hat {b}}^{\dagger }=u^{*}{\hat {a}}^{\dagger }+v^{*}{\hat {a}}~,$

where the latter is the hermitian conjugate of the first.

The Bogoliubov transformation is the canonical transformation mapping the operators ${\hat {a}}$ and ${\hat {a}}^{\dagger }$ to ${\hat {b}}$ and ${\hat {b}}^{\dagger }$. To find the conditions on the constants u and v such that the transformation is canonical, the commutator is evaluated, viz.

$\left[{\hat {b}},{\hat {b}}^{\dagger }\right]=\left[u{\hat {a}}+v{\hat {a}}^{\dagger },u^{*}{\hat {a}}^{\dagger }+v^{*}{\hat {a}}\right]=\cdots =\left(|u|^{2}-|v|^{2}\right)\left[{\hat {a}},{\hat {a}}^{\dagger }\right].$

It is then evident that $\,|u|^{2}-|v|^{2}=1$ is the condition for which the transformation is canonical.

Since the form of this condition is suggestive of the hyperbolic identity

$\cosh ^{2}x-\sinh ^{2}x=1$,

the constants u and v can be readily parametrized as

$u=e^{i\theta _{1}}\cosh r$

$v=e^{i\theta _{2}}\sinh r~.$

Applications

The most prominent application is by Nikolai Bogoliubov himself in the context of superfluidity. Other applications comprise Hamiltonians and excitations in the theory of antiferromagnetism. When calculating quantum field theory in curved space-times 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.

Fermionic mode

For the anticommutation relation

$\left\{{\hat {a}},{\hat {a}}^{\dagger }\right\}=1$,

the same transformation with u and v becomes

$\left\{{\hat {b}},{\hat {b}}^{\dagger }\right\}=(|u|^{2}+|v|^{2})\left\{{\hat {a}},{\hat {a}}^{\dagger }\right\}$

To make the transformation canonical, u and v can be parameterized as

$u=e^{i\theta _{1}}\cos r\,\!$

$v=e^{i\theta _{2}}\sin r\,\!.$

Applications

The most prominent application is again by Nikolai Bogoliubov himself, this time for the BCS theory of superconductivity . The point where the necessity to perform a Bogoliubov transform becomes obvious is that in mean-field approximation the Hamiltonian of the system can be written in both cases as a sum of bilinear terms in the original creation and destruction operators, involving finite  $\,\langle a_{i}^{+}a_{j}^{+}\rangle$-terms, i.e. one must go beyond the usual Hartree–Fock method (-> Hartree–Fock-Bogoliubov method). Also in nuclear physics this method is applicable since it may describe the "pairing energy" of nucleons in a heavy element.

Multimode example

The Hilbert space under consideration is equipped with these operators, and henceforth describes a higher-dimensional quantum harmonic oscillator (usually an infinite-dimensional one).

The ground state of the corresponding Hamiltonian is annihilated by all the annihilation operators:

$\forall i\qquad a_{i}|0\rangle =0$

All excited states are obtained as linear combinations of the ground state excited by some creation operators:

$\prod _{k=1}^{n}a_{i_{k}}^{\dagger }|0\rangle$

One may redefine the creation and the annihilation operators by a linear redefinition:

$a'_{i}=\sum _{j}(u_{ij}a_{j}+v_{ij}a_{j}^{\dagger })$

where the coefficients $\,u_{ij},v_{ij}$ must satisfy certain rules to guarantee that the annihilation operators and the creation operators $a_{i}^{\prime \dagger }$, defined by the Hermitian conjugate equation, have the same commutators for bosons and anticommutators for fermions.

The equation above defines the Bogoliubov transformation of the operators.

The ground state annihilated by all $a'_{i}$ is different from the original ground state $|0\rangle$ and they can be viewed as the Bogoliubov transformations of one another using the operator-state correspondence. They can also be defined as squeezed coherent states. BCS wave function is an example of squeezed coherent state of fermions.

Literature

The whole topic, and a lot of definite applications, are treated in the following textbooks:

• J.-P. Blaizot and G. Ripka: Quantum Theory of Finite Systems, MIT Press (1985)
• A. Fetter and J. Walecka: Quantum Theory of Many-Particle Systems, Dover (2003)
• Ch. Kittel: Quantum theory of solids, Wiley (1987)