Analysis of Side-band Inequivalence

Abstract

Frequency shifts of red- and blue-scattered (Stokes/anti-Stokes) side-bands in quantum optomechanics are shown to be counter-intuitively inequal, resulting in an unexpected symmetry breaking. This difference is referred to as Side-band Inequivalenve (SI), which normally leans towards red, and being a nonlinear effect it depends on optical power or intracavity photon number. Also there exists a maximum attainable SI at an optimal operation point. The mathematical method employed here is a combination of operator algebra equipped with harmonic balance, which allows a clear understanding of the associated nonlinear process. This reveals the existence of three distinct operation regimes in terms of pump power, two of which have immeasurably small SI. Compelling evidence from various experiments sharing similar interaction Hamiltonians, including quantum optomechanics, ion/Paul traps, electrooptic modulation, Brillouin scattering, and Raman scattering unambiguously confirm existence of a previously unnoticed SI.

Nonlinear Limits to Optomechanical Thermometry

“…Interestingly, besides numerous experimental evidence presented in the preceding article [4] and its supplementary information, side-band inequivalence in frequency is clearly seen in the heterodyne thermometry measurement [5], where red and blue detuning frequencies are easily measureable to yield ∆ r = 3.633GHz ± 10.4kHz and ∆ b = −3.623GHz ± 10.4kHz. These numbers are equivalent to a side-band inequivalence of δ = 10MHz ± 20.8kHz or δ = 0.27%.…”

“…Following this particular notation of red and blue detunings respectively in ( 1) and ( 2), the red detuning ∆ r is mostly positive while blue detuning ∆ b is mostly negative. Additionally, δ is the side-band inequivalence [4,10] and represents a nonlinear symmetry breaking. We furthermore may define normalized dimensionless side-band inequivalence as…”

“…A full nonlinear analysis of side-band inequivalence [4,10] has been carried out in a recent study, showing that δ can be well approximated as…”

“…Extensive calculations using higher-order operators [10] lead to the compact result [4] nr − nb ≈ nδ .…”

“…The most prominent example of nonlinear effects is side-band inequivalence [4], which explains that every pair of side-bands are not exactly centered at ω ± jΩ, and there is actually a slight frequency difference. Furthermore, without considering the quantum occupancy effects and different scattering rates, the amplitudes of red and blue side-bands are not equal and also a bit different.…”