Optomechanics with a position-modulated Kerr-type nonlinear coupling

Cavity optomechanics has proven to be a field of research rich with possibilities for studying motional cooling, squeezing, quantum entanglement, and metrology in solid-state systems. While to date most studies have focused on the modulation of the cavity frequency by the moving element, the emergence of new materials will soon allow us to explore the influences of nonlinear optical effects. We therefore study in this work the effects due to a nonlinear position-modulated self-Kerr interaction and find that this leads to an effective coupling that scales with the square of the photon number, meaning that significant effects appear even for very small nonlinearities. This strong effective coupling can lead to lower powers required for motional cooling and the appearance of multistability in certain regimes.

Figure 1

(a) Phonon number $n$ as a function of $E$ and $g_{\text{L}}$ for $g_{\text{NL}}=0$ and (b) as a function of $E$ and $g_{\text{NL}}$ for $g_L=0$. The color scale is capped at $n=0.1$ for clarity and any larger values are given by a deep red color. Note the different axis scaling for $E$. (c) Photon number $|{\alpha }_s{|}^2$ as a function of $E$ and $g_{\text{L}}$ for $g_{\text{NL}}=0$ and (d) as a function of $E$ and $g_{\text{NL}}$ for $g_L=0$. The color scale is capped below at $|{\alpha }_s{|}^2=1$, such that any smaller values are given by a deep blue color, and the point at which $|{\alpha }_s{|}^2=1$ is indicated by the gray dashed lines representing the boundary of the applicability of our model. The white areas in all panels correspond to parameter regimes where no stable steady-state solutions exist.

Figure 2

(a) Spectrum of a linear system with $g_{\text{L}}=0.1,g_{\text{NL}}=0,E=4{\mathrm{\Omega }}_m$ and (b) spectrum of the nonlinear system with $g_{\text{L}}=0.1,g_{\text{NL}}=0.01\kappa ,E=4{\mathrm{\Omega }}_m$ both as a function of $\mathrm{\Delta }$. (c) Spectrum of a linear system with $g_{\text{L}}=0.1\kappa ,g_{\text{NL}}=0,\mathrm{\Delta }={\mathrm{\Omega }}_m$ and (d) spectrum of the nonlinear system with $g_{\text{L}}=0.1,g_{\text{NL}}=0.01\kappa ,\mathrm{\Delta }={\mathrm{\Omega }}_m$ both as a function of $E$. The white dots correspond to the imaginary part of the eigenvalues of the drift matrix $A$.

Figure 3

(a) Photon number $|{\alpha }_s{|}^2$ and (b) phonon number $n$ as a function of $E$ for both the linear and nonlinear case. The thin red (dotted) line corresponds to the unstable solutions, while the thicker lines are the various stable solutions.

Figure 4

(a) Photon number $|{\alpha }_s{|}^2$, (b) phonon number $n$, and (c) position of the membrane $q_s$ as a function of $E$. The thin red line corresponds to the unstable solutions, while the thicker lines are the three stable solutions. (d) $g_{\text{NL}}{\left|{\alpha }_s\right|}^2$ and $g_{\text{L}}$ as a function of $E$.

Figure 5

(a) Phonon number $n$ as a function of $g_{\text{L}}$ and $g_{\text{NL}}$, where $E$ has been chosen so as to minimize the temperature for each pair. (b) Photon number $|{\alpha }_s{|}^2$ as a function of $g_{\text{L}}$ and $g_{\text{NL}}$ for the same steady-state solutions. The dashed white lines show where the photon number is equal to unity.