Two-timescale stochastic Langevin propagation for classical and quantum optomechanics

M. J. Akram, E. B. Aranas, N. P. Bullier, J. E. Lang, and T. S. Monteiro

Phys. Rev. A 98, 063827 – Published 19 December 2018

Abstract

Interesting experimental signatures of quantum cavity optomechanics arise because the quantum back-action induces correlations between incident quantum shot noise and the cavity field. While the quantum linear theory of optomechanics (QLT) has provided vital understanding across many experimental platforms, in certain new setups it may be insufficient: analysis in the time domain may be needed, but QLT obtains only spectra in frequency space; and nonlinear behavior may be present. Direct solution of the stochastic equations of motion in time is an alternative, but unfortunately standard methods do not preserve the important optomechanical correlations. We introduce two-timescale stochastic Langevin (T2SL) propagation as an efficient and straightforward method to obtain time traces with the correct correlations. We show that T2SL, in contrast to standard stochastic simulations, can efficiently simulate correlation phenomena such as ponderomotive squeezing and reproduces accurately cavity sideband structures on the scale of the applied quantum noise and even complex features entirely submerged below the quantum shot noise imprecision floor. We investigate nonlinear regimes and find that, where comparison is possible, the method agrees with analytical results obtained with master equations at low temperatures and in perturbative regimes.

Received 3 August 2018

DOI:https://doi.org/10.1103/PhysRevA.98.063827

Physics Subject Headings (PhySH)

Optomechanics

Quantum cavities

Stochastic differential equations

Atomic, Molecular & Optical

Authors & Affiliations

M. J. Akram, E. B. Aranas, N. P. Bullier, J. E. Lang, and T. S. Monteiro^*

Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom

^*t.monteiro@ucl.ac.uk

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Vol. 98, Iss. 6 — December 2018

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Images

Figure 1
(a) T2SL reproduces accurately quantum ponderomotive squeezing effects. Comparison with analytical expressions of quantum linearized theory of optomechanics (QLT) is excellent. The presence of correlations $S_{corr} (ω)$ between optical field quadratures dependent on the homodyne phase angle result in PSD values below the quantum noise floor. Parameters and results are similar to experimentally observed behavior in quantum squeezing regimes seen in [26], using both a damping and readout optical beams. Left panels illustrate cuts through the color map. The shot noise floor $\equiv 1$ . The color map is in a logarithmic scale where white lines denote the shot noise floor and dark blue indicates noise below the shot noise floor. (b) T2SL, surprisingly, perfectly reproduces sideband structures on the scale of the quantum shot noise kicks. The figure illustrates an example with complex sideband structure completely below the quantum shot noise floor. In levitated experiments with hybrid electro-optical traps both the optomechanical coupling $g_{1} (t) = 2 {\bar{g}}_{1} sin ω_{d} t$ and the mechanical frequency $ω_{M} (t) = {\bar{ω}}_{M} + 2 ω_{2} cos (2 ω_{d} t)$ are subject to slow modulations $ω_{d} ≪ ω_{M}$ , resulting in split-sideband peaks, split by $2 ω_{d}$ at $\pm ω = ω_{M} \pm ω_{d}$ . The QLT for this case was developed in [19, 33]: with increasing $ω_{2} / ω_{d}$ , the $ω_{M} + ω_{d}$ peak is suppressed, then reappears. In squeezing regimes, this second peak is fully submerged below the quantum noise floor: the phase change around the suppression point is well described by T2SL; shot noise $\equiv 1$ so features in the right panels are all below this. Parameters are listed in Appendix pp4.
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Figure 2
Illustration of the use of filtering in the time domain. T2SL provides a correlated time trace, thus allowing temporal processing of the calculated heterodyne or homodyne signal, unlike quantum linearized theory of optomechanics (QLT) which directly returns spectra in the frequency domain. Ths figure illustrates the temporal filtering technique termed r-heterodyne, developed and experimentally tested in [17], which extracts homodyne features from a heterodyne-detected spectrum using a temporal filter function. A comparison between the filtered T2SL time trace and the predicted spectrum $S_{\hat{a} {\hat{a}}^{†}} (Ω + ω) + S_{{\hat{a}}^{†} \hat{a}} (Ω - ω) + S_{{\hat{a}}^{†} {\hat{a}}^{†}} (ω) e^{2 i θ} + S_{\hat{a} \hat{a}} (ω) e^{- 2 i θ}$ shows excellent agreement, showing the usefulness of T2SL correlated time traces for allowing the development of new temporal filtering and signal processing methods. Parameters are the same as in Fig. 1.
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Figure 3
T2SL for nonlinear quantum optomechanics. T2SL is used in a regime combining linear and position-squared nonlinearity using the parameters of Ref. [26], where the frequency of the mechanical oscillator is $ω_{M} / 2 π = 1.53$ MHz. (a) A set of homodyne PSDs (for $θ = 0.05 π$ in the left panels; as a function of $θ$ in the right panels) for a system with optomechanical interaction $H_{int} = g_{1} ({\hat{a}}^{†} + \hat{a}) \hat{x} + g_{2} ({\hat{a}}^{†} + \hat{a}) {\hat{x}}^{2}$ with parameter similar to Fig. 1 apart from the additional nonlinear $g_{2}$ term. The top panel, for $g_{2} = 0$ , shows a comparison with QLT. It shows the interaction of ponderomotive squeezing with nonlinearity with increasing $g_{2}$ . Note that the central nonlinear peak (left panels) for weaker nonlinearity $g_{2} ≲ 10^{- 2} g_{1}$ can be fully submerged below the quantum shot noise floor, indicated by red line. (b) Corresponding color maps where black maps the region below quantum shot noise for different values of nonlinear position-squared coupling. Remaining parameters are the same as in Fig. 1.
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Figure 4
T2SL for nonlinear quantum optomechanics. T2SL may be employed in regimes of extremely low thermal occupancies $n_{th} ≲ 1$ of the mechanical oscillator. Here we have pure position-squared nonlinearity (linear coupling $g_{1} = 0)$ and optomechanical interaction $H_{int} = g_{2} (\hat{a} + {\hat{a}}^{†}) {\hat{x}}^{2}$ and bath temperatures $T_{B} = 0$ , so it is possible to compare with analytical perturbative results [6] (black line). The figure shows the PSD of the cavity output and compares with Eq. (7) of [6] for two values of detuning. Remaining parameters are listed in Appendix pp4.
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