In theoretical physics, the Bogoliubov transformation, also known as the Bogoliubov–Valatin transformation,[1][2][3] 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 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.
Operator eigenvalues calculated with the diagonalized Hamiltonian on the transformed state function thus are the same as before.
To find the conditions on the constants u and v such that the transformation is canonical, the commutator is evaluated, namely, It is then evident that
Since the form of this condition is suggestive of the hyperbolic identity the constants u and v can be readily parametrized as This is interpreted as a linear symplectic transformation of the phase space.
correspond to the orthogonal symplectic transformations (i.e., rotations) and the squeezing factor
[8] When calculating quantum field theory in curved spacetimes the definition of the vacuum changes, and a Bogoliubov transformation between these different vacua is possible.
Bogoliubov transforms are also used extensively in quantum optics, particularly when working with gaussian unitaries (such as beamsplitters, phase shifters, and squeezing operations).
The most prominent application is again by Nikolai Bogoliubov himself, this time for the BCS theory of superconductivity.
[8][9][10][11] 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
In particular, in the mean-field Bogoliubov–de Gennes Hamiltonian formalism with a superconducting pairing term such as
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
Also in nuclear physics, this method is applicable, since it may describe the "pairing energy" of nucleons in a heavy element.
[12] 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: All excited states are obtained as linear combinations of the ground state excited by some creation operators: One may redefine the creation and the annihilation operators by a linear redefinition: where the coefficients
, defined by the Hermitian conjugate equation, have the same commutators for bosons and anticommutators for fermions.
BCS wave function is an example of squeezed coherent state of fermions.
[13] The Hilbert space under consideration is equipped with these operators, and henceforth describes a higher-dimensional quantum harmonic oscillator (usually an infinite-dimensional one).
, defined by the Hermitian conjugate equation, have the same commutators for bosons and anticommutators for fermions.
BCS wave function is an example of squeezed coherent state of fermions.
and For boson operators, the commutation relations require and These conditions can be written uniformly as where where
Bogoliubov transformations are a crucial mathematical tool for understanding and describing fermionic condensates.
In a system where fermions can form pairs, the standard approach of filling single-particle energy levels (the Fermi sea) is insufficient.
The presence of a condensate implies a coherent superposition of states with different particle numbers, making the usual creation and annihilation operators inadequate.
The Hamiltonian of such a system typically contains terms that create or annihilate pairs of fermions, such as:
This Hamiltonian is not diagonal in terms of the original fermion operators, making it difficult to directly interpret the physical properties of the system.
Bogoliubov transformations provide a solution by introducing a new set of quasiparticle operators,
are chosen such that the Hamiltonian, when expressed in terms of the quasiparticle operators, becomes diagonal:
The Bogoliubov transformation reveals several key features of fermion condensates: The whole topic, and a lot of definite applications, are treated in the following textbooks: