Bogoliubov-Transformed Coboson Operator

• The coboson operator is defined as a composite bosonic entity whose Bogoliubov transformation generalizes squeezing by incorporating Pauli-blocking corrections.
• It exhibits a non-canonical commutator structure that modifies quadrature fluctuations and uncertainty relations, reflecting finite-occupancy effects.
• Numerical and analytical methods reveal how eigenstate constructions and matrix representations can probe compositeness in tightly bound fermion pairs.

A Bogoliubov transformed coboson operator arises when applying a Bogoliubov transformation to the bosonic operators describing composite bosons (“cobosons”)—entities formed by strongly bound pairs of spin-$1/2$ fermions. While canonical bosonic operators obey simple commutation relations, coboson operators in realistic models (e.g., the Frenkel-like scheme) exhibit non-canonical algebra due to Pauli-blocking corrections. The Bogoliubov transformed coboson operator generalizes the squeezing operation to these non-ideal bosonic modes, resulting in significant modifications of the quadrature fluctuations and quantum uncertainty relations. This formalism connects the algebraic structure and observable signatures of compositeness, with direct relevance to excitonic systems and other tightly-bound fermionic pairs (Figueiredo et al., 29 Dec 2025, &&&1&&&).

1. Coboson Operator Algebra in the Frenkel-like Model

Two-fermion cobosons are constructed using creation $\mathrm{B}^\dagger$ and annihilation $\mathrm{B}$ operators. In the Frenkel-like scenario, where all Schmidt weights are equal, the coboson operators satisfy a non-canonical commutator

$$[\mathrm{B},\,\mathrm{B}^\dagger] = 1 - \mathrm{D},$$

where $\mathrm{D}$ is a positive operator quantifying the extent of Pauli-blocking corrections. This deviation from standard bosonic commutation relations underpins all consequences of compositeness in subsequent constructions. The parameter $\mathrm{D}$ is state-dependent and vanishes (formally) in the limit of infinite available pair states.

2. Bogoliubov Transformation: Formal Definition

To extend squeezing to composite bosons, the Bogoliubov transformation is applied directly to the coboson operators: $$\mathscr{B}_\xi \equiv \cosh r \cdot \mathrm{B} + e^{i\phi}\, \sinh r \cdot \mathrm{B}^\dagger,$$ where $\xi = r e^{i\phi}$ is the complex squeezing parameter. Denoting $u = \cosh r$, $v = e^{i\phi}\,\sinh r$, the operator can be compactly written as

$$\mathscr{B}_\xi = u\,\mathrm{B} + v\,\mathrm{B}^\dagger.$$

This structure parallels the canonical bosonic mode transformation, but is applied in a context with state-dependent deviations from the canonical algebra.

3. Squeezed Coboson States: Eigenstate Construction

Squeezed coboson states $|\alpha,\xi\rangle$ are defined as eigenstates of the Bogoliubov transformed coboson operator: $$\mathscr{B}_\xi\,|\alpha,\xi\rangle = \alpha\,|\alpha,\xi\rangle,$$ for $\alpha$ complex. These states are constructed by expansion in the coboson Fock basis. For a finite number $N_s$ of available fermion-pair states, the $N$-coboson state $|N\rangle$ and Pauli-blocking normalized creation coefficients are given by

$F_N = \sqrt{N \left(1 - \frac{N-1}{N_s}\right)}.$

The squeezed state is expressed as

$$|\alpha,\xi\rangle = \sum_{N=0}^{N_s} x_N\,|N\rangle.$$

The eigenvalue equation yields a three-term recurrence for the coefficients: $u F_{N+1}\,x_{N+1} + v F_N\,x_{N-1} = \alpha x_N,$ with boundary conditions $x_{-1} = x_{N_s+1} = 0$. The recurrence can be recast as a tridiagonal matrix eigenproblem $M x = \alpha x$, whose eigenvectors $x$ specify the state’s expansion.

4. Quadrature Fluctuations and Modified Uncertainty Relations

Hermitian quadrature operators for cobosons are defined as

$$\chi = \frac{\mathrm{B} + \mathrm{B}^\dagger}{\sqrt{2}},\qquad \pi = \frac{\mathrm{B} - \mathrm{B}^\dagger}{\sqrt{2}\,i}.$$

Quadrature variances in the squeezed state $|\alpha,\xi\rangle$ are computed via

$$(\Delta\chi)^2 = \langle \chi^2 \rangle - \langle \chi \rangle^2,\qquad (\Delta\pi)^2 = \langle \pi^2 \rangle - \langle \pi \rangle^2.$$

Using the inverse Bogoliubov transformation (at $\phi=0$), explicit expressions are

$$(\Delta\chi)^2 = e^{-2r}\, \frac{1-d}{2}, \qquad (\Delta\pi)^2 = e^{+2r}\,\frac{1-d}{2},$$

where $$d = \langle \mathrm{D} \rangle_{|\alpha,\xi\rangle} < 1$$ encodes the average Pauli-blocking effect. The quadrature commutator modifies the Heisenberg-Robertson uncertainty bound: $$[\chi, \pi] = i\,(1 - \mathrm{D}) \implies \Delta\chi\,\Delta\pi \geq \frac{1-d}{2}.$$ In composite boson systems ($d>0$), the uncertainty product falls below the canonical bosonic ½ limit, reflecting finite-occupancy corrections rather than any violation of quantum principles (Figueiredo et al., 29 Dec 2025).

5. Numerical Matrix Construction and Squeezing Behavior

Numerical studies involve the finite $(N_s+1)\times(N_s+1)$ matrix representations of $\chi$ and $\pi$ in the basis $\{|N\rangle\}$, with off-diagonal elements given by $F_N$ structure. Diagonalization of the tridiagonal $\mathscr{B}_\xi$ matrix provides the eigenvector $x$ for chosen $\alpha$ (in practice, typically $\alpha=0$ for squeezed vacuum).

Empirically, the quadrature variances exhibit the following behaviors:

• $(\Delta\chi)^2$ as a function of squeezing parameter $r$ tracks closely $e^{-2r}/2$ even for moderate $N_s$, since squeezing reduces $\chi$-fluctuations more quickly than Pauli-blocking sets in.
• $(\Delta\pi)^2$ initially follows $e^{2r}/2$ up to moderate $r$, then saturates below the bosonic limit as $r$ increases, indicating the impact of Pauli-blocking that caps further growth of $\pi$-fluctuations.
• The uncertainty product $\Delta\chi\,\Delta\pi = (1-d)/2$ interpolates between the canonical $½$ (for large $N_s$ or small $d$) and a lower bound set by finite $N_s$.

6. Physical Implications for Composite Boson Systems

In physical contexts such as tightly bound electron–hole pairs (excitons) or other composite boson systems, the deviations from ideal bosonic squeezing reflect intrinsic compositeness. Observable quadrature noise spectra reveal these deviations especially at strong squeezing. The reduction of the uncertainty product serves as a probe for finite-occupancy and Pauli-blocking effects rather than any fundamental limitation of quantum uncertainty. This framework provides a physically transparent methodology for detecting compositeness through quadrature fluctuations (Figueiredo et al., 29 Dec 2025).

7. Generalization via Bogoliubov Transformations in Quadratic Bosonic Hamiltonians

The more abstract framework for Bogoliubov transformations is furnished by Nam, Napiórkowski, and Solovej (Nam et al., 2015), in which quadratic bosonic Hamiltonians $H$ on Fock spaces are diagonalized by these transformations. Writing original bosonic operators $a_j$, $a_j^\dagger$ and transformed ones $b_i$, $b_i^\dagger$,

$b_i = \sum_j U_{ij}\,a_j + V_{ij}\,a_j^\dagger,$

with CCR preservation subject to constraints on $U$ and $V$. For bilinear coboson operators,

$$B^\dagger = \sum_{p,q} \phi_{pq}\, a_p^\dagger a_q^\dagger,$$

the transformed coboson operator in the new basis is

$$B'{}^\dagger = \frac{1}{2}\left[ b^\dagger (U\phi U^T) b^\dagger + 2 b^\dagger (U\phi V^T) b + b (V\phi V^T) b \right].$$

Commutation relations and normalization are altered; one obtains $[B', B'{}^\dagger] = 1 + \Delta'$, where $\Delta'$ quantifies the residual deviation from canonical bosonic algebra. Normalization of the one-coboson state involves the vacuum expectation value

$$\langle0|\,B' B'{}^\dagger\,|0\rangle = \mathrm{Tr}\left[(U\phi U^T)(U\phi U^T)^\dagger\right] + \mathrm{Tr}\left[(V\phi V^T)(V\phi V^T)^\dagger\right].$$

A plausible implication is that the Bogoliubov transformation generally renders the coboson's algebra non-canonical, directly connecting the operator-theoretic transformation and the many-body compositeness mechanism (Nam et al., 2015).

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Table: Key Structural Elements in Bogoliubov-Transformed Coboson Theory

Operator / Parameter	Structural Role	Physical Consequence
$\mathrm{B}$, $\mathrm{B}^\dagger$	Original coboson creation/annihilation	Encodes pairwise fermionic composition
$\mathrm{D}$	Pauli-blocking deviation	Non-canonical commutator, finite occupancy effects
$\mathscr{B}_\xi$	Bogoliubov-transformed coboson	Enables squeezing, modifies algebra
$d = \langle \mathrm{D} \rangle$	State-dependent deviation	Adjusts uncertainty relations
$N_s$	Number of available pair states	Bounds occupation, controls $F_N$

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The theory of Bogoliubov transformed coboson operators is foundational for understanding non-ideal squeezing, compositeness, and quantum fluctuations in systems beyond ideal bosonic models. Mathematical formalism, matrix construction, and physical implications are tightly interwoven and form the basis for experimental probes of composite boson behavior (Figueiredo et al., 29 Dec 2025, Nam et al., 2015).