For squeezing orders n≥3, the generalized squeezing operator produces oscillatory dynamics that almost return the vacuum state to itself, with the effect shrinking as n increases.
The $k$-photon quantum Rabi model
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abstract
A generalization of the quantum Rabi model is obtained by replacing the linear (dipole) coupling between the two-level system and the radiation mode by a non-linear expression in the creation and annihilation operators, corresponding to multi-photon excitations. If each spin flip involves $k$ photons, it is called the "$k$-photon" quantum Rabi model. While the formally symmetric Hamilton operator is self-adjoint in the case $k=2$, it is demonstrated here that the Hamiltonian is not self-adjoint for $k\ge 3$. Therefore it does not generate a unitary time evolution and is unphysical. This result cannot be obtained by numerical calculations in finite-dimensional spaces which attempt to approximate an unbounded operator by a finite-rank operator.
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Properties and dynamics of generalized squeezed states
For squeezing orders n≥3, the generalized squeezing operator produces oscillatory dynamics that almost return the vacuum state to itself, with the effect shrinking as n increases.