Beyond squeezing à la Virasoro algebra

Abstract

The generalization of squeezing is realized in terms of the Virasoro algebra. The higher-order squeezing can be introduced through the higher-order time-dependent potential, in which the standard squeezing operator is generalized to higher-order Virasoro operators. We give a formula that describes the number of particles generated by the higher-order squeezing when a parameter specifying the degree of squeezing is small. Formula (18) shows that the higher the order of squeezing, the larger the number of generated particles.

A Formalism Useful to Study Beyond Squeezing in Non-linear Quantum Optics

“…We consider a simple case of NL-material with non-linear susceptibility, χ (2) and χ (4) . For the laser beam with a resonance energy ω L , two photons γ 1 and γ 2 are relevant, having energies ω L /4 and ω L /2, respectively.…”

“…( 25) depends on the waves, since it depends on the r and γ ′ whose dependency on ω may not be ignored. We consider a crystal, having NL dielectric constants χ (2) and χ (4) . Generalization to more general cases with χ (i) (i = 3, 5, 6, • • • ) is also possible.…”

“…Corresponding to the mode expansion of E in Eq. ( 8), the mode expansion of the vector potential A and its canonical conjugate Π A , are as follows 4 :…”

“…In the following, we study a simple NL case of having only χ (2) and χ (4) , and hence the real interaction Hamiltonian takes the following form:…”

“…In this paper, we develop a formalism to study the phenomena a more generally "beyond the squeezing mechanism", for example to see beyond the squeezing induced by the anharmonic coupling λ 4 (t) in QM or by the non-linear dielectric constant χ (4) in QO. The more general case of beyond the squeezing, having the higher-order interactions, was studied separately in [4], by referring to the Virasoro algebra, in which a general formula is derived, which gives the number of particles produced, under a condition of the number of particles produced is large.…”