Floquet approach to bichromatically driven cavity-optomechanical systems

Abstract

We develop a Floquet approach to solve time-periodic quantum Langevin equations in steady state. We show that two-time correlation functions of system operators can be expanded in a Fourier series and that a generalized Wiener-Khinchin theorem relates the Fourier transform of their zeroth Fourier component to the measured spectrum. We apply our framework to bichromatically driven cavity optomechanical systems, a setting in which mechanical oscillators have recently been prepared in quantum-squeezed states. Our method provides an intuitive way to calculate the power spectral densities for time-periodic quantum Langevin equations in arbitrary rotating frames. arXiv:1605.04749v2 [cond-mat.mes-hall] 1 Aug 2016

Quantum noise spectra for periodically driven cavity optomechanics

“…We note that Ref. [21] offers a different way to calculate the measured spectrum, but we come back to this point later in Sec. III C.…”

“…(21), the lth Fourier mode c (l) (ω) = T l0 (ω)c in (ω), so c(ω) = l∈Z c (l) (ω + lω d ). Reference [21] constructs the spectrum from the Fourier modes, where the role of the Kronecker δ correlation in Eq. (17) is played by matching of equal and opposite Fourier indices.…”

“…In particular, we have shown how cross correlations (and hence, how rotating parts of the cavity output spectrum) can be recovered naturally in modulated systems by using heterodyne detection. The same idea has been applied on the level of rotating mechanical quadratures using the Fourier components of the periodic spectrum [21], which we know from Appendix C are equivalent to unequal-frequency cross correlations of shifted operators. Note also that quantum cross-correlations have been measured, but for a standard optomechanical system [31].…”

“…We note that theâ(ω) solutions here denote the intracavity field (the actual detected cavity output field is then straightforwardly obtained by using the input-output relation a out (ω) =â in (ω) − √ κâ(ω) [24]. Such matrix methods have been used in previous studies of modulated optomechanics [19,21,25]. However, unlike the standard case, T couples frequencies which differ by multiples of the modulation frequency.…”

“…(ii) In Ref. [21], a Floquet ansatz is used so steady-state solutions are assumed to be periodic. This results in a Langevin equation for each Fourier component of the system operators which can be arranged into a matrix equation in Fourier space.…”