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arXiv:2412.00908 · Journal of Physics A Mathematical and Theoretical

Optomechanical systems with nonlinear interactions: photon blockade, collapse-revival effect and Fano-like resonance

A. P. Saǐko, Русецкий Григорий Александрович, Siarhei A. Markevich, Ryhor Fedaruk

Published Jan 28, 2026• 0 citations
arXivHTMLPDF

Abstract

Abstract Closed-form expressions for the average amplitude of the optical field in optomechanical systems are obtained, in which, in addition to the linear interaction, quadratic and cubic interactions of the vibrational mode of the mechanical resonator with the mode of the optical resonator are considered. In the framework of the non-secular perturbation theory, using the Bogoliubov averaging method, it is shown that the effects of photon blockade, collapse and revival of optical oscillations in such systems can be realized. The main contribution to the formation of revivals is provided by the Kerr self-action of the optical mode and the cross-Kerr interaction of the fourth degree in optical and mechanical amplitudes. The cross-Kerr interactions of the sixth and eighth orders reduce the amplitude of revivals. The influence of these cross-Kerr nonlinearities decreases with an increase in the decay rate of the optical mode. It is shown that the asymmetry of the spectral line of the optical field intensity in the cavity is most pronounced with an increase in the degree and magnitude of nonlinearity and is explained by Fano interference.

DOI
10.1088/1751-8121/ae3ee7

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Abstract Closed-form expressions for the average amplitude of the optical field in optomechanical systems are obtained, in which, in addition to the linear interaction, quadratic and cubic interactions of the vibrational mode of the mechanical resonator with the mode of the optical resonator are considered. In the framework of the non-secular perturbation theory, using the Bogoliubov averaging method, it is shown that the effects of photon blockade, collapse and revival of optical oscillations in such systems can be realized. The main contribution to the formation of revivals is provided by the Kerr self-action of the optical mode and the cross-Kerr interaction of the fourth degree in optical and mechanical amplitudes. The cross-Kerr interactions of the sixth and eighth orders reduce the amplitude of revivals. The influence of these cross-Kerr nonlinearities decreases with an increase in the decay rate of the optical mode. It is shown that the asymmetry of the spectral line of the optical field intensity in the cavity is most pronounced with an increase in the degree and magnitude of nonlinearity and is explained by Fano interference.
Few-photon optomechanical effects not only are important physical evidence for understanding the radiation-pressure interaction between photons and mechanical oscillation, but also have wide potential applications in modern quantum technology. Here we study the few-photon optomechanical effects including photon blockade and generation of the Schr\"odinger cat states under the assistance of a cross-Kerr interaction, which is an inherent interaction accompanied by the optomechanical coupling in a generalized optomechanical system. By exactly diagonalizing the generalized optomechanical Hamiltonian and calculating its unitary evolution operator, we find the physical mechanism of the enhancement of photon blockade and single-photon mechanical displacement. The quantum properties in this generalized optomechanical system are studied by investigating the second-order correlation function of the cavity field and calculating the Wigner function and the probability distribution of the rotated quadrature operator for the mechanical mode. We also study the influence of the dissipations on the few-photon optomechanical effects.
The nonlinear dynamics of a mechanical resonator in an optomechanical system with linear, quadratic and cubic photon-vibration interactions (with respect to mechanical displacements) in a modulated driving field under conditions of adiabatic elimination of the optical field is studied. Based on the constructed bifurcation diagrams of the mechanical coordinate and the largest Lyapunov exponent as a function of the modulation amplitude, as well as power spectra, phase portraits and Poincare sections, regions of regular and chaotic dynamics of the optomechanical system are identified. It is also shown that for a certain modulation amplitude in the presence of all three types of interactions, chaotic dynamics of the mechanical resonator (oscillator) is realized, which is replaced by quasi-periodic oscillations in the absence of cubic interaction, and the system returns to chaotic behavior if only linear interaction remains. This non-monotonic dependence of chaotic dynamics on the order of nonlinearity originates from the interplay between parametric driving and effective potential reshaping and manifests that nonlinearity does not always enhance chaos. For an optomechanical system in a membrane-in-the-middle configuration, where only quadratic photon-vibration interaction is present, it is demonstrated that at small modulation amplitudes the mechanical oscillator exhibits quasi-periodic motion in each of the wells of a symmetric two-minimum potential, whereas large modulation amplitudes lead to chaotic motion, involving interwell transitions.
We study the use of the self-Kerr and cross-Kerr nonlinearities to realize strong photon blockade in a weakly driven, four-mode optomechanical system. According to the Born—Oppenheimer approximation, we obtain the cavity self-Kerr coupling and the inter-cavity cross-Kerr coupling, adiabatically separated from the mechanical oscillator. Through minimizing the second-order correlation function, we find out the optimal parameter conditions for the unconventional photon blockade. Under the optimal conditions, the strong photon blockade can appear in the strong or weak nonlinearities.
The nonlinearity is essential for creation of non-classical states of the cavity or mechanical resonator such as squeezed or cat states. A microwave cavity can be made nonlinear by, for instance, adding Josephson junctions. The mechanical resonator is inherently nonlinear. The radiation pressure interaction between cavity and mechanical resonator is also inherently nonlinear but typically under strong drive of the cavity interaction can be linearized. However, if the optomechanical system is in the strong coupling regime nonlinear quantum effects become observable. These three cases provide the motivation for our studies. We start with backaction analysis of the classical regime of a dc SQUID (superconducting quantum interference device) with an embedded mechanical resonator. Then, we perform quantum analysis to understand how the asymmetry of two Josephson junctions influences the coupling strength of optomechanical interactions such as radiation pressure, cross-Kerr and single-photon beam splitter. Using variational method and self-consistent harmonic approximation we estimate the influence of the radiation pressure coupling and Kerr nonlinearity of the cavity on the effective frequency and dissipation of the cavity. The second part of this thesis focuses on the nonlinearity of the mechanical resonator coupled to the optical/microwave cavity. We study in details the nonlinear optomechanical response and draw the response map to summarize all results of the overcoupled and undercoupled cavity for the red and blue sidebands additionally to the weak and strong drive powers. In the end, w e confirm the developed theory by using it to numerically fit the experimental results.
We discuss the Fano resonance and amplification in a quadratically coupled optomechanical system with a nonlinear Kerr medium. We find that effective cavity detuning and the strength of the Kerr nonlinearity lead to the appearance of the Fano resonance in the output probe field at room temperature, respectively. We show that the Kerr nonlinearity causes the asymmetric line shape to appear in the intensities of the output Stokes field and the second-order upper sideband generation. We also show that the Kerr nonlinearity can amplify the intensities of the output Stokes field and the second-order upper sideband generation, and the amplification factor is related to the strength of the Kerr nonlinearity. The findings of this study have potential applications in optical switching and frequency conversion.
Abstract We explore the photon blockade in optomechanical systems with a position-modulated Kerr-type nonlinear coupling, i.e. H i n t ∼ a ^ † 2 a ^ 2 ( b ^ 1 † + b ^ 1 ) . We find that the Kerr-type nonlinear coupling can enhance the photon blockade greatly. We evaluate the equal-time second-order correlation function of the cavity photons and find that the optimal photon blockade does not happen at the single photon resonance. By working within the few-photon subspace, we get an approximate analytical expression for the correlation function and the condition for the optimal photon blockade. We also find that the photon blockade effect is not always enhanced as the Kerr-type nonlinear coupling strength g 2 increases. At some values of g 2 , the photon blockade is even weakened. For the system we considered here, the second-order correlation function can be smaller than 1 even in the unresolved sideband regime. By numerically simulating the master equation of the system, we also find that the thermal noise of the mechanical environment can enhance the photon blockade. We give out an explanation for this counter-intuitive phenomenon qualitatively.