Higher-Order Interactions in Quantum Optomechanics: Analytical Solution of Nonlinearity

Abstract

A method is described to solve the nonlinear Langevin equations arising from quadratic interactions in quantum mechanics. While the zeroth order linearization approximation to the operators is normally used, here, first and second order truncation perturbation schemes are proposed. These schemes employ higher-order system operators, and then approximate number operators with their corresponding mean boson numbers only where needed. Spectral densities of higher-order operators are derived, and an expression for the second-order correlation function at zero time-delay has been found, which reveals that the cavity photon occupation of an ideal laser at threshold reaches √ 6 − 2, in good agreement with extensive numerical calculations. As further applications, analysis of the quantum anharmonic oscillator, calculation of Q-functions, analysis of quantum limited amplifiers, and nondemoliton measurements are provided.

Method of Higher-order Operators for Quantum Optomechanics

“…The interaction H OM is not quadratic, but is still cubic nonlinear. It is normally solved by a straightforward linearization [18][19][20], but can be also solved at the second-order accuracy using the higher-order operators described in the preceding article [66].…”

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

“…The exchange and conservation of momenta under standard quadratic interaction is uncertain and actually not quite obvious, since momentum operators do not show up in the interaction. (3) here is going to be based on the six-dimensional space spanned by the basis operators {A} T = {ĉ,ĉ † ,n,d,d † ,m} [34]. This basis can be easily seen to be the smallest possible set, with closed commutators, capable of describing the system modeled by (1).…”

“…Growing out of the context of quantum optomechanics, the method of higher-order operators developed by the au-thor [34] can address problems with any general combination of nonlinearity, stochastic input, operator quantities, and spectral estimation. Higher-order operators have been already used in analysis of nonlinear standard optomechanics [35], where its application has uncovered effects known as sideband inequivalence, quantities such as coherent phonon population, as well as corrections to the optomechanical spring effect, zero-point field optomechanical interactions, and a minimal basis with the highest-order which allows exact integration of optomechanical Hamiltonian subject to multiplicative noise input.…”

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].…”

“…Furthermore, a new type of symmetry breaking named as side-band inequivalence is also found using this algebraic method, which refers to inequal detunings in red-and blue-scattered photons [14][15][16]. Furthermore, a preliminary study of photon bunching and anti-bunching statistics applied to the lasing threshold has been carried out using this method [12], and it has been shown that around the lasing threshold, the cavity population of photons reaches the value of √ 6 − 2. In the context of superconducting quantum circuits, the interaction of two pump-probe microwave fields with the transmon qubits is effectively a cross-Kerr nonlinear interaction [24], and for all practical reasons it has to be followed immediately by a quantum-limited parametric amplifier stage.…”