Bogoliubov transformation
The Bogoliubov transformation is a linear canonical transformation in quantum mechanics and quantum field theory that mixes creation and annihilation operators to preserve the canonical commutation (for bosons) or anticommutation (for fermions) relations, enabling the diagonalization of quadratic Hamiltonians in interacting many-body systems. For bosonic modes, it typically takes the form \hat{b}_k = u_k \hat{a}_k + v_k \hat{a}^\dagger_{-k} with the condition |u_k|^2 - |v_k|^2 = 1, while for fermionic modes in contexts like superconductivity, it is \gamma_{k\uparrow} = u_k c_{k\uparrow} + v_k c^\dagger_{-k\downarrow} with |u_k|^2 + |v_k|^2 = 1.[1] This transformation introduces quasiparticle operators that describe collective excitations, simplifying the treatment of phenomena where interactions lead to phenomena like pairing or condensation.[1] Originally developed by Nikolai N. Bogoliubov in his 1947 microscopic theory of superfluidity for interacting Bose gases, the transformation provided a method to approximate the ground state and low-energy excitations of liquid helium-4 by accounting for quantum depletion of the condensate.[2] It was subsequently generalized and applied by Bogoliubov and collaborators, and independently by John George Valatin, in 1958 to the microscopic theory of superconductivity (also known as the Bogoliubov–Valatin transformation), where it diagonalizes the Bardeen-Cooper-Schrieffer (BCS) Hamiltonian to reveal the energy gap and paired electron states.[1][3] Beyond these foundational roles, the Bogoliubov transformation has broad applications across quantum physics. In Bose-Einstein condensates, it describes Bogoliubov quasiparticles as the elementary excitations above the condensate ground state. In quantum optics, it generates squeezed states and enables the analysis of parametric down-conversion processes.[4] It also plays a crucial role in relativistic quantum field theory, such as in the Unruh effect and Hawking radiation, where it relates inertial and accelerated observers' vacua through mode mixing in curved spacetimes.[4]General Mathematical Formulation
Canonical Transformations
The Bogoliubov transformation is defined as a linear isomorphism between sets of creation and annihilation operators that preserves the canonical (anti)commutation relations of bosonic or fermionic systems. For bosons, if \alpha and \alpha^\dagger satisfy [\alpha, \alpha^\dagger] = 1, the transformed operators \beta and \beta^\dagger take the general form \beta = u \alpha + v \alpha^\dagger and \beta^\dagger = u^* \alpha^\dagger + v^* \alpha, where u and v are complex coefficients. To ensure [\beta, \beta^\dagger] = 1, the condition |u|^2 - |v|^2 = 1 must hold. Similarly, for fermions, if c and c^\dagger satisfy \{c, c^\dagger\} = 1, the transformation \beta = u c + v c^\dagger preserves the anticommutation relation \{\beta, \beta^\dagger\} = 1 under the condition |u|^2 + |v|^2 = 1. These conditions are essential for maintaining the algebraic structure underlying quantum statistics, ensuring the transformation is canonical and induces a unitary evolution in the corresponding Fock space. For bosonic systems, the relation |u|^2 - |v|^2 = 1 reflects the symplectic nature of the transformation in phase space, where the operators correspond to position and momentum variables, preserving the Poisson bracket structure in the classical limit and the Heisenberg uncertainty principle quantum mechanically. In fermionic cases, the condition |u|^2 + |v|^2 = 1 aligns with the orthogonal group structure, guaranteeing unitarity and the Pauli exclusion principle. Both ensure the transformation can be implemented by a unitary operator on the Hilbert space, avoiding inconsistencies in expectation values or correlation functions. The Bogoliubov transformation was introduced by Nikolai Bogoliubov in his seminal work on the microscopic theory of superfluidity, where it was used to describe collective excitations in weakly interacting Bose gases.[5] This framework has since become foundational in many-body quantum physics, enabling the treatment of paired states in both bosonic and fermionic systems.Unified Matrix Description
The Bogoliubov transformation for multimode systems can be expressed in a compact matrix notation by considering the column vector of operators \boldsymbol{\alpha} = \begin{pmatrix} \mathbf{a} \\ \mathbf{a}^\dagger \end{pmatrix}, where \mathbf{a} = (a_1, \dots, a_N)^T and \mathbf{a}^\dagger = (a_1^\dagger, \dots, a_N^\dagger)^T are the annihilation and creation operators for N modes, respectively. The transformed operators are given by \boldsymbol{\beta} = W \boldsymbol{\alpha}, where \boldsymbol{\beta} = \begin{pmatrix} \mathbf{b} \\ \mathbf{b}^\dagger \end{pmatrix} and W is a $2N \times 2N matrix of the block form W = \begin{pmatrix} U & V \\ V^* & U^* \end{pmatrix}, with U and V being N \times N complex matrices.[6][7] For the bosonic case, preservation of the canonical commutation relations [\beta_i, \beta_j^\dagger] = \delta_{ij} and [\beta_i, \beta_j] = 0 requires W to satisfy the paraunitary condition W \Sigma W^\dagger = \Sigma, where \Sigma = \operatorname{diag}(I_N, -I_N) is the indefinite metric tensor.[6] This condition implies the block relations U U^\dagger - V V^\dagger = I_N, U V^\dagger - V U^\dagger = 0, and their conjugates, ensuring the transformation belongs to the symplectic group \mathrm{Sp}(2N, \mathbb{C}).[7] Taking the determinant of the paraunitary condition yields \det(W) \det(W^\dagger) = 1, and given the structure of W in the real symplectic subgroup, the normalization follows as \det(W) = 1.[6] In the fermionic case, the transformation preserves the canonical anticommutation relations \{\beta_i, \beta_j^\dagger\} = \delta_{ij} and \{\beta_i, \beta_j\} = 0 under the unitary condition W^\dagger W = I_{2N}, with the matrix form adjusted to W = \begin{pmatrix} U & V \\ -V^* & U^* \end{pmatrix} to ensure consistency (note the minus sign, conventional in paired systems like BCS theory).[8] This leads to the block relations U U^\dagger + V V^\dagger = I_N and U V^\dagger - V U^\dagger = 0 (with conjugates), corresponding to the unitary group \mathrm{U}(2N) in the complex doubled space, preserving the structure.[9] The normalization involves \det(W) = \det(U + i V) \det(U - i V) or equivalent, distinguishing proper transformations.[10] The inversion of the transformation is given by \boldsymbol{\alpha} = W^{-1} \boldsymbol{\beta}. For the bosonic case, W^{-1} = \Sigma W^\dagger \Sigma, while for fermions, it follows from the unitary condition as W^{-1} = W^\dagger.[6][8] This matrix formulation facilitates numerical implementations and algebraic manipulations, such as in the diagonalization of quadratic Hamiltonians.[7]Bosonic Bogoliubov Transformations
Single-Mode Bosonic Case
In the single-mode bosonic case, the Bogoliubov transformation provides a canonical mapping between two sets of bosonic operators, mixing creation and annihilation operators for a single mode. Specifically, the transformed annihilation operator is given byb = u a + v a^\dagger,
where a and a^\dagger are the original bosonic operators satisfying the commutation relation [a, a^\dagger] = 1, and u, v \in \mathbb{C} are complex coefficients obeying the condition |u|^2 - |v|^2 = 1. This ensures the transformed operators b and b^\dagger also satisfy the bosonic commutation relation [b, b^\dagger] = 1. To verify this, compute
[b, b^\dagger] = u u^* [a, a^\dagger] + v v^* [a^\dagger, a] + u v^* [a, a] + v u^* [a^\dagger, a^\dagger] = |u|^2 - |v|^2 = 1,
since cross terms vanish under the original commutation rules.[11] The coefficients u and v can be parametrized using hyperbolic functions to satisfy the normalization condition explicitly:
u = \cosh \theta, \quad v = \sinh \theta \, e^{i \phi},
where \theta \geq 0 is a real squeezing parameter and \phi is a phase angle. This form arises naturally in the context of quadratic bosonic Hamiltonians and highlights the transformation's connection to squeezing operations in quantum optics. The inverse transformation is then a = u b - v b^\dagger, preserving the structure.[11] The vacuum state in the new basis, denoted |0'\rangle, is defined by the condition b |0'\rangle = 0. Expressed in the original Fock basis, this state takes the form of a squeezed vacuum:
|0'\rangle = \frac{1}{\sqrt{\cosh \theta}} \exp\left( \frac{1}{2} e^{i \phi} \tanh \theta \, (a^\dagger)^2 \right) |0\rangle,
where |0\rangle is the original vacuum. More explicitly in the Fock basis (for general phase \phi), it expands as
|0'\rangle = (\cosh \theta)^{-1/2} \sum_{n=0}^{\infty} \frac{ (\tanh \theta \, e^{i \phi})^n }{ \sqrt{2^{2n} (n!)^2 } } \sqrt{(2n)!} \, |2n\rangle.
This state represents a basic example of a squeezed vacuum, where quadrature fluctuations are reduced in one direction at the expense of the other, enabling applications in precision measurements.[12][11]
Multimode Bosonic Case
In the multimode bosonic case, the Bogoliubov transformation generalizes the single-mode formulation to an arbitrary number of coupled bosonic modes, capturing collective excitations such as those in extended quantum systems. This extension is essential for describing phenomena involving inter-mode interactions, where the transformation mixes annihilation and creation operators across multiple modes while preserving bosonic commutation relations [ \vec{b}_i, \vec{b}_j^\dagger ] = [\delta_{ij}](/page/Delta).[13] The multimode transformation is expressed in vector form as \vec{b} = U \vec{a} + V \vec{a}^\dagger, where \vec{a} and \vec{b} are vectors of annihilation operators for the original and transformed modes, respectively, and U, V are complex matrices of appropriate dimensions. To maintain the canonical commutation relations, these matrices must satisfy the bosonic symplectic conditions: U U^\dagger - V V^\dagger = I and U V^T = V U^T. These orthogonality relations ensure the transformation is canonical and unitary in the extended phase space, allowing for the diagonalization of multimode quadratic forms.[13] A representative example is the two-mode squeezing transformation in quantum optics, where the matrices U and V mix operators from two spatial or temporal modes to generate entangled squeezed states, such as the two-mode squeezed vacuum |\psi\rangle = \sum_n \frac{(\tanh r)^n}{\cosh r} |n\rangle_a |n\rangle_b, with squeezing parameter r. This transformation, parameterized by U = \cosh r \, I and V = \sinh r \, \sigma (where \sigma couples the modes), produces non-classical correlations useful for quantum information tasks.[13][14] In inhomogeneous bosonic systems, such as trapped Bose gases, the multimode amplitudes are determined by the Bogoliubov-de Gennes (BdG) equations, which form a coupled set of eigenvalue problems for the quasiparticle wavefunctions u_j(\mathbf{r}) and v_j(\mathbf{r}): \begin{pmatrix} \mathcal{L} & \mathcal{M} \\ -\mathcal{M}^* & -\mathcal{L}^* \end{pmatrix} \begin{pmatrix} u_j \\ v_j \end{pmatrix} = E_j \begin{pmatrix} u_j \\ v_j \end{pmatrix}, where \mathcal{L} and \mathcal{M} incorporate the mean-field potential and interactions, yielding the matrices U and V from the eigenvectors. These equations extend the uniform case to spatially varying densities, enabling the study of localized excitations. Numerically, the matrices U and V are obtained by solving the generalized eigenvalue problem associated with the system's dynamical matrix, typically using iterative methods like the Lanczos algorithm for large mode numbers to compute the positive-energy eigenvectors while enforcing the symplectic constraints. This approach scales efficiently for systems up to hundreds of modes, providing the transformation parameters for simulation of quasiparticle dynamics.Fermionic Bogoliubov Transformations
Single-Mode Fermionic Case
The single-mode fermionic Bogoliubov transformation mixes the annihilation and creation operators of a single fermionic mode to diagonalize quadratic Hamiltonians with particle-hole symmetry. The quasiparticle annihilation operator is expressed as \gamma = u \, c + v \, c^\dagger, where c and c^\dagger are the original fermionic operators satisfying \{c, c^\dagger\} = 1 and \{c, c\} = 0, and u, v are complex coefficients obeying the normalization condition |u|^2 + |v|^2 = 1. This condition arises from requiring the transformed operator to preserve the canonical anticommutation relation \{\gamma, \gamma^\dagger\} = 1, ensuring \gamma behaves as a valid fermionic quasiparticle operator.[15][16] The transformation exhibits a close relation to Majorana fermions through real linear combinations of the operators. With u and v taken as real, the combination \gamma + \gamma^\dagger = (u + v)(c + c^\dagger) is proportional to a real Majorana operator, defined as the Hermitian part \eta_1 = (c + c^\dagger)/\sqrt{2}, while the imaginary part involves \eta_2 = -i(c - c^\dagger)/\sqrt{2}. These Majorana operators satisfy \{\eta_i, \eta_j\} = \delta_{ij} and \eta_i^\dagger = \eta_i, underscoring the inherent particle-hole mixing in the single-mode case.[17] The quasiparticle operators obey the fermionic anticommutation relations \{\gamma, \gamma^\dagger\} = 1 and \{\gamma, \gamma\} = 0, which follow directly from the normalization and the original operators' statistics. The corresponding vacuum state, annihilated by \gamma (i.e., \gamma |\Omega\rangle = 0), can be interpreted in the field-theoretic sense as a filled Dirac sea, where all negative-energy states below the Fermi level are occupied to ensure stability. In the context of pairing mechanisms, this vacuum manifests as a coherent paired state, a superposition that correlates the unoccupied and occupied configurations of the mode.[15][16] In many theoretical models, the phases of u and v are chosen to be real (e.g., u = \cos\theta, v = \sin\theta for some mixing angle \theta) to exploit symmetries in the Hamiltonian and simplify the diagonalization, aligning the transformation with time-reversal or particle-hole invariance.[15] This single-mode formulation serves as the foundational building block and can be represented in a unified 2×2 matrix form for extension to multimode scenarios.[16]Multimode Fermionic Case
In the multimode fermionic case, the Bogoliubov transformation generalizes to a linear mixing of creation and annihilation operators across multiple fermionic modes, preserving the canonical anticommutation relations \{ \gamma_i, \gamma_j^\dagger \} = \delta_{ij} and \{ \gamma_i, \gamma_j \} = 0. The transformation takes the vector form \vec{\gamma} = U \vec{c} + V \vec{c}^\dagger, where \vec{c} and \vec{c}^\dagger are vectors of original annihilation and creation operators for N modes, and U, V are N \times N complex matrices satisfying the orthonormality conditions U U^\dagger + V V^\dagger = I and U V^T + V U^T = 0 to ensure the transformed operators \vec{\gamma} obey fermionic statistics.[9] This multimode framework is central to the Bogoliubov-de Gennes (BdG) formalism, which describes quasiparticle excitations in inhomogeneous superconductors by coupling electron-like and hole-like components across momentum-space modes. In the BdG approach, the mean-field Hamiltonian for a superconductor is expressed in a doubled Nambu-Gorkov space, leading to an eigenvalue problem that diagonalizes the quadratic form via the multimode transformation. The resulting BdG equations are \begin{pmatrix} H_0 & \Delta \\ \Delta^\dagger & -H_0^* \end{pmatrix} \begin{pmatrix} u_k \\ v_k \end{pmatrix} = E_k \begin{pmatrix} u_k \\ v_k \end{pmatrix}, where H_0 is the single-particle Hamiltonian (e.g., kinetic energy plus potential), \Delta is the pairing potential matrix coupling modes k and k', and u_k, v_k are the electron and hole wavefunction components for quasiparticle mode k.[18] The eigenvalue spectrum features positive energies E_k > 0 for particle-like excitations and negative energies -E_k < 0 for hole-like excitations, with the physical quasiparticle spectrum given by the positive branch due to particle-hole symmetry in the fermionic system. This structure arises because the transformation mixes particle and hole sectors, yielding a doubled set of eigenvalues symmetric around zero, and the ground state is the vacuum of the \gamma operators annihilating all positive-energy modes.[18] A representative example is singlet pairing in s-wave superconductors, where the off-diagonal pairing matrix V couples opposite momenta such that V_{k, -k} \neq 0 while V_{k, k'} = 0 for k' \neq -k, reflecting time-reversal symmetry and momentum conservation in the uniform case. This form simplifies the BdG equations to decoupled pairs of k and -k modes, yielding quasiparticle energies E_k = \sqrt{\xi_k^2 + |\Delta_k|^2}, where \xi_k is the normal-state dispersion and \Delta_k the s-wave gap.[18] The single-mode fermionic case can be viewed as a local approximation to this multimode structure in position space.[9]Diagonalization of Quadratic Hamiltonians
Bosonic Hamiltonians
Quadratic bosonic Hamiltonians arise in many-body systems and quantum optics, describing non-interacting quasiparticles after applying a Bogoliubov transformation. In the multimode case, such a Hamiltonian takes the general form H = \vec{a}^\dagger H_0 \vec{a} + \frac{1}{2} \left( \vec{a}^\dagger M \vec{a}^\dagger + \vec{a} M^* \vec{a} \right), where \vec{a} is the column vector of annihilation operators for N modes satisfying [\vec{a}_i, \vec{a}_j^\dagger] = \delta_{ij}, H_0 is an N \times N Hermitian matrix, and M is an N \times N symmetric complex matrix.[7] This form captures both kinetic-like terms (diagonal in creation-annihilation) and pairing terms (anomalous contributions mixing creation with creation or annihilation with annihilation).[19] To diagonalize H using a bosonic Bogoliubov transformation, one introduces new quasiparticle operators \vec{b} via \vec{b} = U \vec{a} + V \vec{a}^\dagger and \vec{b}^\dagger = \vec{a}^\dagger U^\dagger + \vec{a} V^\dagger, where U and V are N \times N matrices satisfying the bosonic canonical commutation relations U U^\dagger - V V^\dagger = I and U V^T = V U^T (symplectic conditions). Substituting into H yields a diagonal form H = \sum_k E_k b_k^\dagger b_k + C, where E_k > 0 are the quasiparticle energies and C is a constant (vacuum energy shift), provided the original Hamiltonian is stable (i.e., bounded from below).[7] The matrices U and V are determined by solving the eigenvalue problem for the doubled $2N \times 2N matrix \mathcal{H} = \begin{pmatrix} H_0 & M \\ M^* & H_0 \end{pmatrix}, assuming H_0 is real symmetric for simplicity in many cases.[19] The diagonalization procedure involves finding a paraunitary matrix W (satisfying W^\dagger \Sigma W = \Sigma with \Sigma = \operatorname{diag}(I, -I)) such that W^\dagger \mathcal{H} W = \operatorname{diag}(E, -E), where E = \operatorname{diag}(E_1, \dots, E_N) contains the positive eigenvalues. The columns of W = \begin{pmatrix} U \\ V \end{pmatrix} corresponding to positive eigenvalues define the transformation coefficients, ensuring the quasiparticle operators \vec{b} obey the same commutation relations as \vec{a}.[7] The eigenvalues E_k must be real and positive for the spectrum to be bounded below, with the vacuum energy constant given by C = -\frac{1}{2} \sum_k E_k.[19] This process leverages Williamson's theorem for symplectic diagonalization of positive-definite quadratic forms.[7] A representative example is the single-mode harmonic oscillator with a squeezing term: H = \omega a^\dagger a + \frac{\kappa}{2} (a^{\dagger 2} + a^2), where \omega > 0 and |\kappa| < \omega for stability. Here, H_0 = \omega and M = \kappa (real symmetric). The doubled matrix is \mathcal{H} = \begin{pmatrix} \omega & \kappa \\ \kappa & \omega \end{pmatrix}, with eigenvalues \pm E where E = \sqrt{\omega^2 - \kappa^2}.[7] The Bogoliubov transformation is b = u a + v a^\dagger with u = \cosh r, v = \sinh r, and \tanh 2r = \kappa / \omega, yielding H = E b^\dagger b - \frac{E}{2}.[19] This illustrates how the transformation mixes creation and annihilation operators to uncouple the modes into free quasiparticles.Fermionic Hamiltonians
Quadratic fermionic Hamiltonians arise in models of superconductivity and fermionic superfluidity, where interactions lead to pairing terms that couple particle creation and annihilation operators. A canonical example is the BCS-like Hamiltonian, expressed asH = \sum_k \varepsilon_k c_k^\dagger c_k + \sum_k \left( \Delta_k c_k^\dagger c_{-k}^\dagger + \Delta_k^* c_{-k} c_k \right),
where c_k^\dagger and c_k are fermionic creation and annihilation operators satisfying anticommutation relations \{c_k, c_l^\dagger\} = \delta_{kl}, \varepsilon_k represents the single-particle kinetic energy relative to the chemical potential, and \Delta_k is the pairing potential. This form captures the essential physics of Cooper pair formation while remaining quadratic in the operators, allowing exact diagonalization. To diagonalize this Hamiltonian, a fermionic Bogoliubov transformation is applied, introducing quasiparticle operators \gamma_k = u_k c_k + v_k c_{-k}^\dagger and \gamma_{-k} = u_k c_{-k} - v_k c_k^\dagger, with real coefficients satisfying u_k^2 + v_k^2 = 1 to preserve anticommutation relations. Substituting these into the Hamiltonian yields the Bogoliubov-de Gennes (BdG) equations as an eigenvalue problem for the coefficients:
\begin{pmatrix} \varepsilon_k & \Delta_k \\ \Delta_k^* & -\varepsilon_k \end{pmatrix} \begin{pmatrix} u_k \\ v_k \end{pmatrix} = E_k \begin{pmatrix} u_k \\ v_k \end{pmatrix}.
The solutions give eigenvalues E_k = \pm \sqrt{\varepsilon_k^2 + |\Delta_k|^2}, with corresponding eigenvectors determining u_k and v_k. These quasiparticles \gamma_k diagonalize the Hamiltonian, transforming it into
H = \sum_k E_k \gamma_k^\dagger \gamma_k + \text{constant},
where the constant term arises from the ground-state energy shift, typically -\sum_k E_k / 2. The BdG spectrum features both positive and negative eigenvalues due to the structure of the pairing terms, reflecting the electron-hole mixing in the quasiparticle excitations. However, particle-hole redundancy in the fermionic description implies that negative-energy states are not independent but correspond to the positive-energy excitations of the conjugate modes; specifically, if (u_k, v_k, E_k) is an eigenstate, so is (v_k^*, -u_k^*, -E_k) for the particle-hole partner.[20] To obtain a complete set of positive-energy quasiparticle operators, only the positive branch E_k \geq 0 is retained, ensuring the diagonal form expresses the Hamiltonian solely in terms of non-negative excitation energies above the ground state. This redundancy is a hallmark of the BdG formalism, preventing double-counting of degrees of freedom in the doubled Nambu space.[20] In multimode scenarios, such as inhomogeneous superconductors, the transformation generalizes to position-dependent coefficients, but the core algebraic structure remains analogous to the momentum-space case.