All-creation and all-annihilation time-dependent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetric bosonic Hamiltonians: An infinite squeezing degree at a finite time

Abstract

Here we introduce the all-creation and all-annihilation time-dependent (TD) $\mathcal{PT}$-symmetric bosonic Hamiltonians, which in the interaction picture are described only by creation or annihilation operators. These Hamiltonians are defined from the most general TD $\mathcal{PT}$-symmetric quadratic bosonic Hamiltonian, describing a cavity mode under linear and parametric amplifications. After presenting a general ansatz for the derivation of the TD Dyson map and metric operators, we solve the Schr\"odinger equations for both the $\mathcal{PT}$-symmetric Hamiltonian and its Hermitian counterpart. We then compute analytically the squeezing degree coming from the all-annihilation Hamiltonian and compare the result with that coming from an ordinary Hermitian Hamiltonian, showing a crucial result for interferometric procedures: instead of the asymptotic divergence of the squeezing degree that takes place for a Hermitian parametric pumping, the all-annihilation amplification leads to a divergence of that quantity at a finite controllable time.