Nonclassical mechanical states in an optomechanical micromaser analog

P. D. Nation

Phys. Rev. A 88, 053828 – Published 19 November 2013

Abstract

Here we show that quantum states of a mechanical oscillator can be generated in an optomechanical analog of the micromaser in the absence of any atomlike subsystem, thus exhibiting single-atom masing effects in a system composed solely of oscillator components. In the regime where the single-photon coupling strength is on the order of the cavity decay rate, a cavity mode with at most a single-excitation present gives rise to sub-Poissonian oscillator limit-cycles that generate quantum features in the steady state just above the renormalized cavity resonance frequency and mechanical sidebands. The merger of multiple stable limit-cycles markedly reduces these nonclassical signatures. Varying the cavity-resonator coupling strength, corresponding to the micromaser pump parameter, reveals transitions for the oscillator phonon number that are the hallmark of a micromaser. The connection to the micromaser allows for a physical understanding of how nonclassical states arise in this system and how best to maximize these signatures for experimental observation.

Received 23 August 2013

DOI:https://doi.org/10.1103/PhysRevA.88.053828

Authors & Affiliations

P. D. Nation^*

Department of Physics, Korea University, Seoul 136-713, Korea

^*pnation@korea.ac.kr

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Issue

Vol. 88, Iss. 5 — November 2013

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Images

Figure 1
Conventional driven optomechanical setup where the position of a mechanical resonator ( $\hat{b}$ ) with frequency $ω_{m}$ and energy damping rate $Γ_{m}$ is coupled parametrically via radiation pressure to an optical cavity ( $\hat{a}$ ) at frequency $ω_{c}$ . Here, the cavity is assumed to be driven by a laser with amplitude $E$ and frequency $ω_{p}$ . The damping due to coupling with the laser mode is given by $κ$ .Reuse & Permissions
Figure 2
(a) Steady-state cavity energy as a function of detuning and normalized coupling strength $g_{0} / κ$ . The dashed line shows the renormalized cavity frequency. (b) Steady-state energy of the corresponding mechanical oscillator mode, including the first three oscillator sidebands. The detuning value at which the sidebands occur follows the frequency pulling of the cavity (dashed). (c) Nonclassical ratio for the mechanical oscillator. Contours (solid) show the regions where the nonclassical ratio is larger than $0.1 %$ . This region includes portions of the red-detuned side of the cavity response, below the renormalized cavity frequency, and the bistable region is enclosed in the dashed lines. (d) Log-scale plot of the mechanical resonators’ Fano factors.Reuse & Permissions
Figure 3
(a) Wigner distribution for the state at $Δ = 0$ and $g_{0} / κ = 1.35$ with the largest nonclassical ratio $η ≃ 6 %$ . (b) The number state probability distribution for the state in panel (a) (solid red line) together with a coherent state of the same amplitude (blue bars). Dashed lines show the corresponding semiclassical limit-cycle amplitudes found using Eqs. (7) and (8). The inset shows the small peak in the distribution for the larger limit-cycle that gives $F = 5.2$ . (c) Wigner function at $Δ = - 0.43$ and $g_{0} / κ = 2.4$ , located in the bistable regime with $η ≃ 2 %$ . (d) Phonon distribution function (solid red line) for the state in panel (c) showing four stable limit-cycles with $F = 10.4$ and an equal amplitude coherent state (blue bars).Reuse & Permissions
Figure 4
Oscillator order parameter (black squares), mean phonon number (solid-green line), and Fano factor (dashed red line) as a function of $g_{0} / κ$ at $Δ = 0$ . The shaded region indicates where $η \geq 1 %$ .Reuse & Permissions
Figure 5
Distributions for the lower (red bars) and upper (green bars) limit-cycles along with corresponding coherent distributions, (blue solid) and (black solid), respectively, of the same amplitude. Coherent states are normalized to the probabilities of each limit-cycle. The inset shows the Fano factors corresponding to the lower (black line) and upper (gray line) limit-cycles individually, as well as the Fano factor (dashed-black line) and nonclassical ratio $η$ (dashed gray line) for the overall state. Above $g_{0} / κ ≃ 1.3$ , the two limit-cycles begin to merge, and the individual limit-cycle Fano factors are approximate values.Reuse & Permissions
Figure 6
Nonclassical ratio for the states in Fig. 3a (black line) and Fig. 3c (dashed black line), as well as two states on the first mechanical sideband, $Δ = 1$ and $g_{0} / κ = 1.4$ (gray line) and $Δ = 0.5$ and $g_{0} / κ = 2.6$ (dashed gray line), as a function of the bath temperature ${\bar{n}}_{th}$ . The inset shows the Fano factors for the $Δ = 0$ limit-cycle above threshold seen in the larger figure for bath temperatures ${\bar{n}}_{th} = 0$ (black), 1 (dashed black line), 3 (gray), and 5 (dashed gray line).Reuse & Permissions

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