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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.

Contents

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 a canonical transformation of these operators. To find the conditions on the constants u and v such that the transformation remains canonical, the commutator is expanded, 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 can be seen that | u | 2 − | v | 2 = 1 is the condition for which the transformation is canonical. Since the form of this condition is reminiscent of the hyperbolic identity, the constants u and v can be parameterized as

u = e^{i \theta_1} \cosh r \,\!
v = e^{i \theta_2} \sinh r \,\! .

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 \,\! .

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^\dagger_j)

where the coefficients uij,v ij must satisfy certain rules to guarantee that the annihilation operators and the creation operators a^{\prime\dagger}_i, defined by the Hermitian conjugate equation, have the same commutators.

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.

In physics, the Bogoliubov transformation is important for understanding of the Unruh effect, Hawking radiation and BCS theory, among many other things.

See also

References

  • 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)

External links

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