Bogoliubov transformation

In theoretical physics, the Bogoliubov transformation, also known as the Bogoliubov–Valatin transformation, was independently developed in 1958 by Nikolay Bogolyubov and John George Valatin for finding solutions of BCS theory in a homogeneous system. The Bogoliubov transformation is an isomorphism of either the canonical commutation relation algebra or canonical anticommutation relation algebra. This induces an autoequivalence on the respective representations. The Bogoliubov transformation is often used to diagonalize Hamiltonians, which yields the stationary solutions of the corresponding Schrödinger equation. The Bogoliubov transformation is also important for understanding the Unruh effect, Hawking radiation, Davies-Fulling radiation (moving mirror model), pairing effects in nuclear physics, and many other topics.

The Bogoliubov transformation is often used to diagonalize Hamiltonians, with a corresponding transformation of the state function. Operator eigenvalues calculated with the diagonalized Hamiltonian on the transformed state function thus are the same as before.

Single bosonic mode example
Consider the canonical commutation relation for bosonic creation and annihilation operators in the harmonic oscillator basis
 * $$\left [ \hat{a}, \hat{a}^\dagger \right ] = 1.$$

Define a new pair of operators
 * $$\hat{b} = u \hat{a} + v \hat{a}^\dagger,$$
 * $$\hat{b}^\dagger = u^* \hat{a}^\dagger + v^* \hat{a},$$

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 $$\hat{a}$$ 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, namely,
 * $$\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
 * $$u = e^{i \theta_1} \cosh r,$$
 * $$v = e^{i \theta_2} \sinh r.$$

This is interpreted as a linear symplectic transformation of the phase space. By comparing to the Bloch–Messiah decomposition, the two angles $$\theta_1$$ and $$\theta_2$$ correspond to the orthogonal symplectic transformations (i.e., rotations) and the squeezing factor $$r$$ corresponds to the diagonal transformation.

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 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).

Fermionic mode
For the anticommutation relations
 * $$\left\{ \hat{a}, \hat{a}\right\} = 0, \left\{ \hat{a}, \hat{a}^\dagger \right\} = 1,$$

the Bogoliubov transformation is constrained by $$uv=0, |u|^2+|v|^2=1$$. Therefore, the only non-trivial possibility is $$u=0, |v|=1,$$ corresponding to particle–antiparticle interchange (or particle–hole interchange in many-body systems) with the possible inclusion of a phase shift. Thus, for a single particle, the transformation can only be implemented (1) for a Dirac fermion, where particle and antiparticle are distinct (as opposed to a Majorana fermion or chiral fermion), or (2) for multi-fermionic systems, in which there is more than one type of fermion.

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. In particular, in the mean-field Bogoliubov–de Gennes Hamiltonian formalism with a superconducting pairing term such as $$\Delta a_i^+a_j^+ + \text{h.c.}$$, the Bogoliubov transformed operators $$b, b^\dagger$$ annihilate and create quasiparticles (each with well-defined energy, momentum and spin but in a quantum superposition of electron and hole state), and have coefficients $$u$$ and $$v$$ given by eigenvectors of the Bogoliubov–de Gennes matrix. 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:


 * $$a'_i = \sum_j (u_{ij} a_j + v_{ij} a^\dagger_j),$$

where the coefficients $$u_{ij},v_{ij}$$ 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 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.

Unified matrix description
Because Bogoliubov transformations are linear recombination of operators, it is more convenient and insightful to write them in terms of matrix transformations. If a pair of annihilators $$(a, b)$$ transform as

\begin{pmatrix} \alpha\\ \beta \end{pmatrix} = U \begin{pmatrix} a\\ b \end{pmatrix} $$

where $$U$$is a $$2\times2$$ matrix. Then naturally

\begin{pmatrix} \alpha^\dagger\\ \beta^\dagger \end{pmatrix} = U^* \begin{pmatrix} a^\dagger\\ b^\dagger \end{pmatrix} $$

For Fermion operators, the requirement of commutation relations reflects in two requirements for the form of matrix $$U$$



U= \begin{pmatrix} u & v\\ -v^* & u^* \end{pmatrix} $$

and



$$
 * u|^2 + |v|^2 = 1

For Boson operators, the commutation relations require

U= \begin{pmatrix} u & v\\ v^* & u^* \end{pmatrix} $$

and



$$
 * u|^2 - |v|^2 = 1

These conditions can be written uniformly as

U \Gamma_\pm U^\dagger = \Gamma_\pm $$ where

\Gamma_\pm = \begin{pmatrix} 1 & 0\\ 0 & \pm1 \end{pmatrix} $$ where $$\Gamma_\pm$$ applies to Fermions and Bosons, respectively.

Diagonalizing a quadratic Hamiltonian using matrix description
Bogoliubov transformation lets us diagonalize a quadratic Hamiltonian

\hat{H} = \begin{pmatrix} a^\dagger & b^\dagger \end{pmatrix} H \begin{pmatrix} a \\ b \end{pmatrix} $$ by just diagonalizing the matrix $$\Gamma_\pm H$$. In the notations above, it is important to distinguish the operator $$\hat{H}$$ and the numeric matrix $$H$$. This fact can be seen by rewriting $$\hat{H}$$ as

\hat{H} = \begin{pmatrix} \alpha^\dagger & \beta^\dagger \end{pmatrix} \Gamma_\pm U (\Gamma_\pm H) U^{-1} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} $$

and $$\Gamma_\pm U (\Gamma_\pm H) U^{-1}=D$$ if and only if $$U$$ diagonalizes $$\Gamma_\pm H$$, i.e. $$U (\Gamma_\pm H) U^{-1} = \Gamma_\pm D$$.

Useful properties of Bogoliubov transformations are listed below.