Squeezed Cobosons: Composite Boson Squeezing

• Squeezed cobosons are quantum states of composite bosons formed by bound fermion pairs undergoing Bogoliubov transformations, yielding non-canonical commutation relations due to Pauli blocking.
• The squeezed state construction leads to modified quadrature uncertainties where bosonic coherence interplays with fermionic constraints, measurable via quadrature-noise spectroscopy.
• Finite-size effects and state-dependent bounds limit the achievable squeezing, with numerical analysis showing deviations from ideal bosonic behavior as occupation nears system capacity.

Squeezed cobosons are quantum states of composite bosons—specifically, bound pairs of two fermions such as electron–hole pairs—subjected to Bogoliubov transformations analogous to those defining squeezed states in canonical bosonic systems. Unlike elementary bosons, cobosons are fundamentally constrained by the Pauli exclusion principle, as their constituents are fermions. This compositeness leads to a non-canonical commutation algebra and modified uncertainty relations, making the analysis and physical implications of squeezed coboson states distinct from their elementary counterparts. Frenkel-like cobosons, corresponding to a flat Schmidt spectrum, exemplify physically relevant scenarios such as tightly bound pairs delocalized over a finite set of sites, as found in Frenkel excitons. Squeezed states of such cobosons offer a direct probe of the interplay between bosonic coherence and fermionic substructure through the measurement of quadrature fluctuations (Figueiredo et al., 29 Dec 2025).

1. Definition and Algebra of Composite Bosons

A composite boson (“coboson”) consists of two fermions whose combined state is described by a Schmidt decomposition: $$|\Psi\rangle = \sum_{k=1}^S \sqrt{\lambda_k}\;|a_k\rangle\,|b_k\rangle,\quad \sum_k \lambda_k = 1,$$ with the coboson creation operator

$$B^\dagger = \sum_{k=1}^S \sqrt{\lambda_k}\;a_k^\dagger\,b_k^\dagger.$$

For Frenkel-like cobosons, all Schmidt weights are flat ($\lambda_k = 1/N_s$), modeling pairs delocalized over $N_s$ sites. The creation operator simplifies to

$$B^\dagger = \frac{1}{\sqrt{N_s}}\sum_{k=1}^{N_s} e^{i\theta_k}\;a_k^\dagger\,b_k^\dagger.$$

Coboson operators obey a non-canonical commutation relation due to Pauli blocking: $[B,\,B^\dagger] = 1 - D,$ where $D = \sum_k(a_k^\dagger a_k + b_k^\dagger b_k)$ is positive semi-definite. Fock states are constructed as

$$|N\rangle = \frac{(B^\dagger)^N}{\sqrt{N!\,\chi_N}}\,|0\rangle,\quad \chi_N = \frac{N_s!}{N_s^N\,(N_s-N)!},$$

with the recursion

$$B^\dagger|N\rangle = F_{N+1}|N+1\rangle,\quad F_N = \sqrt{N\,\left(1-\frac{N-1}{N_s}\right)}.$$

This algebra underpins all subsequent squeezed-state analysis (Figueiredo et al., 29 Dec 2025).

2. Bogoliubov Construction of Squeezed Coboson States

The squeezed coboson operator is introduced analogously to quantum optics via a Bogoliubov transformation: $$\mathcal{B}_\xi = \cosh r\,B + e^{i\phi}\,\sinh r\,B^\dagger,\quad \xi = re^{i\phi}.$$ The non-canonical algebra persists: $$B^\dagger = \sum_{k=1}^S \sqrt{\lambda_k}\;a_k^\dagger\,b_k^\dagger.$$0 Squeezed coboson states $$B^\dagger = \sum_{k=1}^S \sqrt{\lambda_k}\;a_k^\dagger\,b_k^\dagger.$$1 are defined as right eigenstates of $$B^\dagger = \sum_{k=1}^S \sqrt{\lambda_k}\;a_k^\dagger\,b_k^\dagger.$$2: $$B^\dagger = \sum_{k=1}^S \sqrt{\lambda_k}\;a_k^\dagger\,b_k^\dagger.$$3 In the coboson Fock basis $$B^\dagger = \sum_{k=1}^S \sqrt{\lambda_k}\;a_k^\dagger\,b_k^\dagger.$$4, these states are expanded as

$$B^\dagger = \sum_{k=1}^S \sqrt{\lambda_k}\;a_k^\dagger\,b_k^\dagger.$$5

where the coefficients obey the tridiagonal recurrence

$$B^\dagger = \sum_{k=1}^S \sqrt{\lambda_k}\;a_k^\dagger\,b_k^\dagger.$$6

The construction preserves the essential features of squeezing while introducing coboson-specific constraints resulting from fermionic compositeness (Figueiredo et al., 29 Dec 2025).

3. Quadrature Operators and Modified Uncertainty Relations

Analogues of position and momentum quadratures are defined by

$$B^\dagger = \sum_{k=1}^S \sqrt{\lambda_k}\;a_k^\dagger\,b_k^\dagger.$$7

Due to $$B^\dagger = \sum_{k=1}^S \sqrt{\lambda_k}\;a_k^\dagger\,b_k^\dagger.$$8, their commutator is $$B^\dagger = \sum_{k=1}^S \sqrt{\lambda_k}\;a_k^\dagger\,b_k^\dagger.$$9. The Heisenberg–Robertson uncertainty relation for these quadratures becomes

$\lambda_k = 1/N_s$0

Within a squeezed coboson state $\lambda_k = 1/N_s$1 (for $\lambda_k = 1/N_s$2), the variances take the form

$\lambda_k = 1/N_s$3

where $\lambda_k = 1/N_s$4. The uncertainty product is

$\lambda_k = 1/N_s$5

These expressions reduce to canonical bosonic values in the limit $\lambda_k = 1/N_s$6, demonstrating that finite $\lambda_k = 1/N_s$7 encodes Pauli blocking and compositeness deviations (Figueiredo et al., 29 Dec 2025).

4. Numerical Analysis and Finite-Size Effects

Numerical evaluation is performed using $\lambda_k = 1/N_s$8 matrix representations of $\lambda_k = 1/N_s$9 and $N_s$0 in the Fock basis. The nonzero off-diagonal elements are determined by the $N_s$1 factors: $N_s$2 Diagonalization of the tridiagonal $N_s$3 matrix yields eigenvectors $N_s$4 needed for expectation values. For all $N_s$5, $N_s$6 as a function of $N_s$7 closely follows the ideal $N_s$8 form, indicating squeezing is robust in this channel. In contrast, $N_s$9 for large $$B^\dagger = \frac{1}{\sqrt{N_s}}\sum_{k=1}^{N_s} e^{i\theta_k}\;a_k^\dagger\,b_k^\dagger.$$0 deviates substantially from canonical anti-squeezing, saturating at a finite value determined by $$B^\dagger = \frac{1}{\sqrt{N_s}}\sum_{k=1}^{N_s} e^{i\theta_k}\;a_k^\dagger\,b_k^\dagger.$$1 due to Pauli blocking. The state-dependent bound $$B^\dagger = \frac{1}{\sqrt{N_s}}\sum_{k=1}^{N_s} e^{i\theta_k}\;a_k^\dagger\,b_k^\dagger.$$2 for $$B^\dagger = \frac{1}{\sqrt{N_s}}\sum_{k=1}^{N_s} e^{i\theta_k}\;a_k^\dagger\,b_k^\dagger.$$3 interpolates between $$B^\dagger = \frac{1}{\sqrt{N_s}}\sum_{k=1}^{N_s} e^{i\theta_k}\;a_k^\dagger\,b_k^\dagger.$$4 (elementary bosons, $$B^\dagger = \frac{1}{\sqrt{N_s}}\sum_{k=1}^{N_s} e^{i\theta_k}\;a_k^\dagger\,b_k^\dagger.$$5) and $$B^\dagger = \frac{1}{\sqrt{N_s}}\sum_{k=1}^{N_s} e^{i\theta_k}\;a_k^\dagger\,b_k^\dagger.$$6 ($$B^\dagger = \frac{1}{\sqrt{N_s}}\sum_{k=1}^{N_s} e^{i\theta_k}\;a_k^\dagger\,b_k^\dagger.$$7), highlighting how compositeness constraints emerge as occupation approaches system capacity (Figueiredo et al., 29 Dec 2025).

5. Physical Significance and Experimental Implications

Squeezed Frenkel-like cobosons arise naturally in systems of tightly bound electron–hole pairs (excitons) or exciton–polaritons. Quadrature-squeezed states can be implemented using established optical and microcavity techniques. Measurement of quadrature variances, and their deviation from canonical bosonic formulas, provides direct information about the expectation value $$B^\dagger = \frac{1}{\sqrt{N_s}}\sum_{k=1}^{N_s} e^{i\theta_k}\;a_k^\dagger\,b_k^\dagger.$$8, and thus accesses the degree of coboson compositeness and underlying fermionic structure. Quadrature-noise spectroscopy, therefore, emerges as a powerful method for probing many-body Pauli correlations and the breakdown of perfect bosonicity in composite systems, both in condensed-matter and ultracold atomic settings (Figueiredo et al., 29 Dec 2025).

6. Summary of Core Results and Mathematical Structure

The central properties of squeezed cobosons, along with their mathematical distinctions from elementary bosons, are summarized as follows:

Algebraic Relation	Squeezed Cobosons	Elementary Bosons (Limit)
$$B^\dagger = \frac{1}{\sqrt{N_s}}\sum_{k=1}^{N_s} e^{i\theta_k}\;a_k^\dagger\,b_k^\dagger.$$9	$[B,\,B^\dagger] = 1 - D,$0	$[B,\,B^\dagger] = 1 - D,$1
$[B,\,B^\dagger] = 1 - D,$2, $[B,\,B^\dagger] = 1 - D,$3	$[B,\,B^\dagger] = 1 - D,$4, $[B,\,B^\dagger] = 1 - D,$5	$[B,\,B^\dagger] = 1 - D,$6, $[B,\,B^\dagger] = 1 - D,$7
Uncertainty product	$[B,\,B^\dagger] = 1 - D,$8	$[B,\,B^\dagger] = 1 - D,$9

All quadrature variances are globally rescaled by the factor $D = \sum_k(a_k^\dagger a_k + b_k^\dagger b_k)$0, which reflects Pauli blocking as $D = \sum_k(a_k^\dagger a_k + b_k^\dagger b_k)$1 increases with coboson occupation. Finite $D = \sum_k(a_k^\dagger a_k + b_k^\dagger b_k)$2 imposes an intrinsic upper limit on attainable squeezing, and the uncertainty bound is state-dependent. Measurement of quadrature statistics thus provides a quantitative diagnostic of compositeness effects in coboson systems (Figueiredo et al., 29 Dec 2025).