Higher-Order Interactions in Quantum Optomechanics: Analysis of Quadratic Terms

Abstract

This article presents a full operator analytical method for studying the quadratic nonlinear interactions in quantum optomechanics. The method is based on the application of higher-order operators, using a six-dimensional basis of second order operators which constitute an exactly closed commutators. We consider both types of standard position-field and the recently predicted non-standard momentum-field quadratic interactions, which is significant when the ratio of mechanical frequency to optical frequency is not negligible. This unexplored regime of large mechanical frequency can be investigated in few platforms including the superconducting electromechanics and simulating quantum cavity electrodynamic circuits. It has been shown that the existence of non-standard quadratic interaction could be observable under appropriate conditions.

Solution of Cross-Kerr Interaction Combined with Parametric Amplification

“…When combined with a parametric amplification term, then the total interaction Hamiltonian could be a lot more difficult to solve. So far, no exact solution to this problem has been reported to the best knowledge of the authors.Here, we demonstrate that the cross-Kerr interaction with parametric amplification could be exactly solvable using the method of higher-order operators [11][12][13][14][15][16], which has evolved out of the rich domain of quadratic optomechanics [17][18][19][20][21][22][23]. This method employs a different basis than the simple bath ladder operators, and quite recently has been independently also reported elsewhere [18].…”

Analysis of Side-band Inequivalence

“…This result is also well in complete agreement with the expression obtained earlier for the SI [15] in the limit of g 0 << Ω given as g 2 0 /Ω, considering that a factor of 1 2 must be added as a result of different definition of δ . The first immediate conclusion which can be obtained from (18) is that the SI δ is always positive, meaning that the detuning frequency of red-sideband should be always a bit larger in magnitude than the blue-sideband. This also agrees with the previous findings of higher-order operator algebra [15].…”

“…Based on (18), in the weakly-coupling limit of g 0 √ n < Γ, the normalized SI defined as δ = δ /Ω can be simply approximated as a function of mechanical frequency as δ (Ω)…”

“…Interestingly, in the limit of small optomechanical interaction rate g 0 and low optical power, one may use (18) to obtain the approximate proportionality dependence…”

“…While both time-reversal symmetry and energy conservation are fundamentally preserved in this scattering process, a nonlinear analysis of quantum optomechanics using the recently developed method of higher-order operators [15][16][17][18][19] necessitates a slight difference among detunings of blue and red-scattered photons, the amount of which was initially found to increase roughly proportional to the intracavity photon number n. Here, n is defined as the steady-state mean-value of the number operator n = â † â.…”