Squeezed states for Frenkel-like two-fermion composite bosons

TL;DR

This work extends squeezed-state concepts to composite bosons (cobosons) formed by two spin-$\tfrac{1}{2}$ fermions, focusing on Frenkel-like cobosons with a flat Schmidt distribution. It defines squeezed cobosons as eigenstates of a Bogoliubov-transformed coboson operator $\mathcal{B}_{\xi}=\cosh r\,B+e^{i\varphi}\sinh r\,B^{\dagger}$ and derives explicit quadrature variances, revealing a state-dependent Heisenberg–Robertson bound due to Pauli blocking, with $[\hat{\chi},\hat{\pi}]=i(1-D)$ and $d=\langle D\rangle$. The paper shows $\Delta\hat{\chi}^2=\frac{e^{-2r}}{2}(1-d)$ and $\Delta\hat{\pi}^2=\frac{e^{+2r}}{2}(1-d)$, highlighting finite-size effects that cause saturation of fluctuations at large squeezing. Numerical results for finite $N_s$ illustrate how compositeness constrains achievable squeezing and provides an operational signature of internal fermionic structure in systems such as exciton polaritons.

Abstract

We investigate squeezed states of composite bosons (cobosons) formed by pairs of spin-$1/2$ fermions, with emphasis on Frenkel-like cobosons. While squeezing for standard bosonic modes is well established, its extension to cobosons requires accounting for Pauli blocking and the resulting non-canonical commutation algebra. Building on earlier constructions of coboson coherent states, we define squeezed cobosons as eigenstates of a Bogoliubov transformed coboson operator and derive explicit expressions for the associated quadrature variances. We show that the underlying fermionic structure leads to state-dependent modifications of the Heisenberg--Robertson uncertainty bound, which may fall below the canonical bosonic limit without implying any violation of uncertainty principles. Numerical results based on finite-dimensional matrix representations illustrate how these effects constrain the attainable squeezing. Our framework is relevant to composite boson systems such as tightly bound electron-hole pairs and provides a physically transparent setting to probe compositeness through observable quadrature fluctuations.

Figures (3)

• Figure 1: $(\Delta\hat{\chi})^2$ calculated for the eigenstate of the squeeze operator indexed by $N_s=n$, and different values of $r$. Those values are superposed to the value of the uncertainties of this quadrature for usual bosonic modes, $\frac{\mathrm{e}^{-2r}}{2}$agarwal2013QO. For $(\Delta\hat{\chi})^2$ the fact that the exponential decreases very fast gives a seemingly good agreement.
• Figure 2: $(\Delta\hat{\pi})^2$ calculated for the eigenstate of the squeeze operator indexed by $N_s=n$, and different values of $r$. Those values are superposed to the value of the uncertainties of this quadrature for usual bosonic modes, $\frac{\mathrm{e}^{2r}}{2}$agarwal2013QO. For $(\Delta\hat{\pi})^2$ the fact that the exponential increases very fast gives a picture of the deviation from the usual bosonic behavior.
• Figure 3: $\Delta\hat{\chi}\Delta\hat{\pi}$ calculated for the eigenstate of the squeeze operator indexed by $N_s=n$, the number of pairs, and different values of $r$. The insets show the behavior near the limits, $1/2$ and zero, that are dictated by the finite $N_s.$