[Paper Review] Momentum-Field Interactions Beyond Standard Quadratic Optomechanics

Sina Khorasani|arXiv (Cornell University)|Jan 5, 2018

Mechanical and Optical Resonators3 citations

TL;DR

This paper introduces a higher-order operator method to analyze nonlinear momentum-field interactions in optomechanics beyond standard quadratic optomechanics, revealing a non-standard quadratic interaction involving field and mirror momenta. The key result is the discovery of sideband inequivalence due to this interaction, which becomes significant in sideband-resolved cavities and enables new symmetry-breaking phenomena without cavity heating under resonant pumping.

ABSTRACT

This chapter summarizes the recent progress in the theory and analytical tools of quadratic optomechanical interactions, as one of the prominent domains of contemporary nonlinear quantum optics. Emphasis has been put here first to show what types of nonlinear interactions do exist, and what physical interpretations follow each. The standard quadratic interactions between light and mechanical motion is expressed as the product of cavity light intensity and squared mirror position. However, there exists a non-standard quadratic optomechanical interaction as well, which assumes a mathematically different form and appears as the squared product of field and mirror momenta. This non-standard type of quadratic interaction originates from two corrections: the momentum exchange and conservation among mirror and field, as well as relativistic corrections due to different mechanisms. Both these types of non-standard interactions become relevant when the ratio of mechanical to optical frequency is no longer negligible. Next, we turn to the solution technique of such interactions, and introduce a formal higher-order operator method to tackle the nonlinear evolution of quantum systems. This enables one to accurately study any type of quantum nonlinear interaction using the analysis tools of linear algebra. In order to employ the analytical power of higher-order operator method, one first needs to identify a closed Lie algebra, which should satisfy closedness property under commutation either exactly or approximately, and is referred to as the basis. Having the basis of higher-order operators known, one may proceed to construct the corresponding Langevin equations, which can be now conveniently analyzed using the existing mathematical toolbox of linear algebra to yield the spectral densities, moments, and expectation values.

Motivation & Objective

• To identify and analyze non-standard quadratic optomechanical interactions beyond the conventional intensity-position coupling.
• To develop a higher-order operator formalism capable of treating nonlinear quantum interactions using linear algebra tools.
• To investigate the physical implications of momentum-based interactions, including their role in symmetry breaking and sideband inequivalence.
• To clarify the conditions under which quadratic interactions dominate or mask standard optomechanical coupling.
• To determine the practical limits of observing sideband inequivalence in realistic optomechanical systems.

Proposed method

• Introduces a formal higher-order operator method based on constructing a closed Lie algebra of higher-order operators to describe nonlinear quantum interactions.
• Identifies two types of non-standard quadratic interactions: one from momentum exchange and conservation, and another from relativistic corrections, both becoming relevant when mechanical-to-optical frequency ratio is non-negligible.
• Derives generalized Langevin equations from the higher-order operator basis, enabling spectral density, moment, and expectation value calculations using linear algebra techniques.
• Applies the method to analyze sideband inequivalence, showing that red- and blue-sideband frequency shifts differ due to nonlinear coupling.
• Uses the operator basis to quantify interaction strength via photon-phonon cross-population, which scales quadratically with pump power.
• Demonstrates that on-resonance pumping can mask standard optomechanical coupling due to dominant quadratic interaction, even with non-zero $ g_0 $.

Experimental results

Research questions

• RQ1 What types of non-standard quadratic optomechanical interactions exist beyond the standard intensity-position coupling?
• RQ2 How do momentum exchange and relativistic corrections contribute to the emergence of a new quadratic interaction form?
• RQ3 In what parameter regimes does the non-standard quadratic interaction dominate over the standard optomechanical coupling?
• RQ4 How does the higher-order operator method enable the analysis of nonlinear quantum dynamics using linear algebra tools?
• RQ5 What are the observable signatures of this new interaction, particularly in sideband spectra?

Key findings

• A non-standard quadratic optomechanical interaction exists, mathematically expressed as the squared product of field and mirror momenta, arising from momentum conservation and relativistic corrections.
• This interaction becomes significant when the ratio of mechanical to optical frequency is no longer negligible, particularly under strong on-resonant pumping.
• The interaction strength increases quadratically with pump power, allowing observation of nonlinear effects without significant cavity heating, as phonon population saturates near unity.
• Sideband inequivalence emerges as a key signature, with a frequency shift difference estimated as $ \delta\Omega \approx g_0^2\bar{n}/\Omega $, bounded to $ 10^{-6} $ to $ 10^{-4} $ in practical systems.
• On-resonant pumping can effectively mask the standard optomechanical interaction due to dominance of the quadratic coupling, even when $ g_0 \neq 0 $.
• The higher-order operator method enables accurate analysis of nonlinear quantum systems by constructing a basis closed under commutation, allowing use of linear algebra for spectral and moment calculations.