Spontaneous emission in Casimir-Rabi oscillations through a weak optomechanical coupling

TL;DR

The paper tackles spontaneous emission in dynamical Casimir effect–driven Casimir-Rabi oscillations under weak optomechanical coupling, deriving a high-order $3^{\text{rd}}$-order effective Hamiltonian that enables resonant $|0,3\rangle \leftrightarrow |2,0\rangle$ transitions. It employs quantum trajectory methods to study open-system emission, revealing multiphoton and multiphonon bundle emissions with purities around $0.92$–$0.95$ at small coupling and low dissipation, and analyzes how temperature and dissipation affect these emissions via $n(\omega_m)$ and related rates. Finite-temperature effects increase total emissions but reduce emission purities, while higher coupling strengthens the effective interaction yet challenges the rotating-wave approximation. An experimental path based on a SQUID-cavity system is proposed, suggesting that with state-of-the-art lifetimes in 3D cavities and high-quality mechanical resonators, the predicted Casimir-Rabi dynamics and multiphoton/phonon emissions could be observed.

Abstract

The dynamical Casimir effect (DCE) describes the energy conversion from a mechanical motion to the electromagnetic fields. When the mechanical oscillator is in a mechanically excited state, the free evolution due to the DCE produces radiation in the vacuum, in analogy with the spontaneous emission from an excited atom. In this manuscript, we investigate such a spontaneous radiation process by employing the quantum trajectory approach. When the dissipation rate of the system is very low, there can be a reversible energy exchange between the mirror in the excited state and the vacuum field, and this reversible exchange is called vacuum Casimir-Rabi oscillations. Multiple quantum trajectory simulations of this process show that the number of trajectories responsible for the generation of radiation can reach a significant value when the mechanical dissipation rate is less than the photon dissipation rate. We also find that two-photon (two/three-phonon) bundle emission occurs in photon (phonon) emission. In comparison to pure two-photon and three-phonon free dissipation, the probability of two-photon bundle emission and two-phonon bundle emission are observed to be marginally elevated as a consequence of the presence of the DCE. This pattern may assist in developing a deeper comprehension of the physical characteristics of photon and phonon emission in the DCE.

Figures (14)

• Figure 1: (a) Schematic of a typical optomechanical system where one of the mirrors in an optical cavity can vibrate at frequency $\omega_{m}$. The resonance frequency of the cavity is $\omega_{c}$. (b) All the possible virtual paths for the transition $\ket{0,3}\rightarrow \ket{2,0}$.
• Figure 2: Eigenvalues $E_{5}/ \omega_{m}$ and $E_{6}/ \omega_{m}$ as a function of the ratio between the cavity frequency $\omega_{c}$ and the mechanical frequency $\omega_{m}$. The continuous curves are the eigenvalues of $H_{s}$ in Eq. (\ref{['eq1']}) and the dashed lines describe the eigenenergies of the $H_{\rm eff}$ in Eq. (\ref{['eq3']}). When energy level splitting is minimal, $\omega_{c}=1.5000105\omega_{m}$. Other parameter is $g=0.001\omega_{m}$.
• Figure 3: (a) Fidelity $F(t)$ defined in Eq. (\ref{['eq4']}) (b) Population $P(t)$ vs the time for various values of the coupling rate: $g=0.001\omega_{m}$ (purple solid curve), $g=0.005\omega_{m}$ (black solid curve), $g=0.01\omega_{m}$ (orange solid curve), and $g=0.05$ (blue solid curve). Other parameter is $\omega_{c}=1.5000105\omega_{m}$. The initial state of the system is $\ket{0,3}$.
• Figure 4: Examples of a single quantum trajectory, numerically obtained by studying the open quantum dynamics. It shows the time evolution of the mean excitation number of the radiation field $\langle a^{\dagger}a \rangle$ (red solid curve) and of the mechanical oscillator $\langle b^{\dagger}b \rangle$ (blue dotted curve). The black arrows in both panels indicate that the system has undergone a quantum jump. In panel (a), the system emits two photons outward with an emission interval less than the photon lifetime. In panel (b), the system emits three phonons outward with an emission interval less than the phonon lifetime. In both panels, the system is initialized in $\ket{0,3}$ and other parameters are $g=0.001\omega_{m}$, $\omega_{c}=1.5000105\omega_{m}$, and $\gamma=\gamma_{a}=\gamma_{b}=10^{-9}\omega_{m}$.
• Figure 5: (a) and (c) show a single quantum trajectory obtained from the effective Hamiltonian in Eq. (\ref{['eq3']}) at different coupling strengths, respectively. The numerical result in (b) and (d) are obtained from the system Hamiltonian in Eq. (\ref{['eq1']}). In both panels, the system is initialized in $\ket{0,3}$ and other parameters are $\omega_{c}=1.5000105\omega_{m}$ and $\gamma=\gamma_{a}=\gamma_{b}=10^{-9}\omega_{m}$.