What Happens When Quantum Waves Intertwine in Nonlinear Spaces?

In the quantum realm, “coherent” states are not just fancy words: they are special states that behave almost like classical waves. Two families of such states, called SU(2) and SU(1,1), were compared using the same nonlinear two‑mode Hamiltonian.

Picture two balls rolling on a curved track: their paths depend on the track’s shape. Likewise, Wigner functions—probability‑like maps in phase space—show how these states evolve. For SU(2) the map squeezes in one direction, a phenomenon known as “squeezing”: the uncertainty of one observable shrinks while the other grows, respecting the uncertainty principle. SU(1,1) shows a similar effect, but in a hyperbolic setting where the curves are more open.

The most striking outcome is the emergence of Schrödinger’s cat states: superpositions of two opposite configurations, like a cat both alive and dead. Both symmetry groups generate such cats, yet with different shapes in their Wigner maps. In practice, the study reveals that despite mathematical differences, SU(2) and SU(1,1) share squeezing and cat‑state creation mechanisms, opening new ways to steer light and vibrations at the quantum level.

The Wigner functions W(α) are defined as the Weyl transform of the density operator ρ, providing a quasi‑probabilistic representation in the complex phase space α = x + iy. For the Lie groups SU(2) and SU(1,1), coherent‑state generation relies on the generators J_±, J_z and K_±, K_0, with commutation relations [J_+,J_-]=2J_z and [K_+,K_-]=‑2K_0. The nonlinear Hamiltonian considered is H = χ(a†b†ab) + γ(a†a – b†b), where a and b are the annihilation operators of the two modes. The interaction χ induces Kerr‑type terms, responsible for quadratic squeezing, while γ introduces energy imbalance between the modes. The time evolution U(t)=exp(‑iHt/ħ) acts on the basis vectors |j,m⟩ (SU(2)) and |k,n⟩ (SU(1,1)), generating superpositions of states with different m or n values. The Wigner dynamics is obtained via W(α,t)=Tr[ρ(t)Δ(α)], where Δ(α) is the parity operator. By analyzing higher‑order moments ⟨J_x^2⟩, ⟨K_y^2⟩, one observes uncertainty reduction (squeezing) accompanied by increased negativity of W, a hallmark of non‑classicality. Cat‑state formation is identified by two distinct peaks in the W distribution, separated by an interference region with negative values, typical of coherent superpositions. These features remain robust against small variations of χ and γ, suggesting applications in quantum metrology and continuous‑variable communication protocols.