Spin-orbit-coupling-induced backaction cooling in cavity optomechanics with a Bose-Einstein condensate

We report a spin-orbit-coupling-induced backaction cooling in an optomechanical system, composed of a spin-orbit-coupled Bose-Einstein condensate trapped in an optical cavity with one movable end mirror, by suppressing heating effects of quantum noises. The collective density excitations of the spin-orbit-coupling-mediated hyperfine states—serving as atomic oscillators equally coupled to the cavity field—trigger strongly driven atomic backaction. We find that the backaction not only revamps low-temperature dynamics of its own but also provides an opportunity to cool the mechanical mirror to its quantum-mechanical ground state. Further, we demonstrate that the strength of spin-orbit coupling also superintends dynamic structure factor and squeezes nonlinear quantum noises, like thermomechanical and photon shot noise, which enhances optomechanical features of the hybrid cavity beyond previous investigations. Our findings are testable in a realistic setup and enhance the functionality of cavity optomechanics with spin-orbit-coupled hyperfine states in the field of quantum optics and quantum computation.

Figure 1

(a) Schematic diagram of a spin-orbit (SO)-coupled ${}^{87}\mathrm{Rb}$ Bose-Einstein condensate (BEC) trapped inside a high-$Q$ Fabry-Pérot cavity with one moving end mirror, where the $\widehat{y}$ axis is the cavity axis and the $\widehat{x}$ axis is the direction of incident Raman beams. (b) Energy-level configuration of SO-coupled BEC.

Figure 2

(a–c) Eigenenergies spectrum $E_N$ of a spin-orbit (SO)-coupled Bose-Einstein condensate (BEC) as a function of quasimomentum $k_x/k_L$ for different ${\mathrm{\Omega }}_z/{\omega }_m$ and $\delta /{\omega }_m$, when $g_m/{\omega }_m=0.1$ and $\alpha /{\omega }_m=20\pi$. (It should be noted that we consider $k_y=k_z=0$ because SO coupling is occurring only in the direction of the $\widehat{x}$ axis.) The black curve represents dispersion at ${\mathrm{\Omega }}_z/{\omega }_m=0$ while magenta shaded curves (from darkest to lightest) correspond to ${\mathrm{\Omega }}_z/{\omega }_m=2,4,6,8$, respectively. (a,b,c) Behavior of dispersion $E_N$ for $\delta /{\omega }_m=-1,0,1$, respectively. (d,e) Dispersion $E_N$ vs $k_x/k_L$ and $g_m/{\omega }_m$, with $\alpha /{\omega }_m=30\pi$ and $\alpha /{\omega }_m=50\pi$, respectively, at ${\mathrm{\Omega }}_z={\omega }_m$ and $\delta /{\omega }_m=0$. The other parameters used are $U/{\omega }_m=5.5, \varepsilon /{\omega }_m=0.1,\kappa /{\omega }_m=0.1,{\gamma }_a/{\omega }_m=0.01,{\gamma }_m/{\omega }_m=0.05$, and mechanical mirror frequency ${\omega }_m\approx {\omega }_c-{\omega }_p$.

Figure 3

(a–c) DNS $S_{\uparrow }(\omega ,\mathrm{\Delta })$ as a function of $\mathrm{\Delta }/{\omega }_m$ and $\omega /{\omega }_m$ for $\alpha /{\omega }_m=0\pi ,150\pi$, and $250\pi$, respectively, when ${\mathrm{\Omega }}_z={\omega }_m,\delta /{\omega }_m=1$, and $G_a/{\omega }_m=28.5$. (Note: The color configuration corresponds to the strength of DNS.) Similarly, (d–f) demonstrate DNS $S_{\downarrow }(\omega ,\mathrm{\Delta })$ for $\alpha /{\omega }_m=0\pi ,150\pi$, and $250\pi$, respectively. Here $G_m/{\omega }_m=1.5,\mathrm{\Omega }/{\omega }_m=70.8,{\omega }_m=3.8\times 2\pi \mathrm{kHz}$ and the thermal reservoir temperature is taken as $T=300\mathrm{K}$.

Figure 4

(a) $S_m(\omega ,\mathrm{\Delta })$ vs normalized frequency $\omega /{\omega }_m$ at $\mathrm{\Delta }/{\omega }_m=1.8,G_m/{\omega }_m=10$, and $G_a/{\omega }_m=20$. The black curve is for $\alpha /{\omega }_m=0\pi$ and magenta curves from dark shade to light shade represent $\alpha /{\omega }_m=6\pi ,10\pi ,16\pi ,20\pi$, and $30\pi$, respectively. (b) $T_{\text{eff}}$ (in units of mK) vs $\mathrm{\Delta }/{\omega }_m$ at $\omega /{\omega }_m=0.1, G_m/{\omega }_m=10$, and $G_a/{\omega }_m=5$. Similarly, the black corresponds to $\alpha /{\omega }_m=0\pi$ while shaded curves from darkest to lightest represent $\alpha /{\omega }_m=60\pi ,90\pi ,120\pi ,150\pi$, and $170\pi$, respectively. (c,d) $T_{\text{eff}}$, as a function of $\alpha /{\omega }_m$ and $U/{\omega }_m$, at ${\mathrm{\Omega }}_z/{\omega }_m=0$ and 80, respectively. Here, $G_m/{\omega }_m=18,G_a/{\omega }_m=16$, and $\alpha /{\omega }_m=30\pi$.

Figure 5

(a,b) DNS of optical field leaking out of cavity $S_{\text{out}}(P,\omega )$ (in units of W/Hz), vs $P/P_{\mathrm{cr}}$ and $\omega /{\omega }_m$, for $\alpha /{\omega }_m=0\pi$ and $80\pi$, respectively. (c–f) Dynamic structure factor (in units of 1/Hz) corresponding to the out-going optical mode DNS vs $\omega /{\omega }_m$ for different values of $P/P_{\mathrm{cr}}$. SO coupling $\alpha /{\omega }_m=0\pi$ ($80\pi$) for (c,d), while the strength of SO coupling is $\alpha /{\omega }_m=80\pi$ for (e,f). Here $G_m/{\omega }_m=1.5,G_a/{\omega }_m=0.9,U/{\omega }_m=5.5$, and $\delta /{\omega }_m=0$.

Figure 6

The dynamic structure factor $S_D(k,\omega )$ (in units of 1/Hz) as a function of $\omega /{\omega }_m$ for different strengths of $\alpha /{\omega }_m$ at $U=5.5{\omega }_m$ (a) and atom-atom interactions $U/{\omega }_m$ at $\alpha =10\pi {\omega }_m$ (b). The input field power ratio is fixed to $P/P_{\mathrm{cr}}=4$ and the remaining coupling strengths are the same as in Fig. 5. In (a), the green shaded curves from darkest to lightest correspond to the SO coupling $\alpha /{\omega }_m=0\pi ,60\pi ,100\pi$, and $140\pi$, respectively. Similarly, in (b), green curves (from dark shade to light shade) carry the influence of atom-atom interactions with strengths $U/{\omega }_m=5.5,7.5,9.5$, and 11.5, respectively.

Figure 7

The dynamics of $S_{\uparrow }(\omega ,\mathrm{\Delta })$, as a function of detuning $\mathrm{\Delta }/{\omega }_m$ and frequency $\omega /{\omega }_m$, under the effects of many-body interactions $U/{\omega }_m$. (a–c) $S_{\uparrow }(\omega ,\mathrm{\Delta })$ with atom-atom interaction $U=5.5{\omega }_m,7.5{\omega }_m$, and $9.5{\omega }_m$, respectively. Here, the strength of SO coupling is kept constant at $\alpha =10\pi {\omega }_m$. One can observe that the strength of atom-atom interactions influences $S_{\uparrow }(\omega ,\mathrm{\Delta })$ in a similar way as SO coupling does. By increasing $U/{\omega }_m$, the area underneath $S_{\uparrow }(\omega ,\mathrm{\Delta })$ is decreased, which leads to the cooling of the atomic mode [23]. The remaining parameters are the same as in Fig. 2.

Figure 8

Density-noise spectrum (DNS) $S_m(\omega ,\mathrm{\Delta })$ (in units of W/Hz) for a mechanical mirror of the system vs normalized effective detuning $\mathrm{\Delta }/{\omega }_m$ and frequency $\omega /{\omega }_m$ under the influence of spin-orbit (SO) coupling $\alpha /{\omega }_m$ of atomic spin states and normalized atom-atom interactions $U/{\omega }_m$ among atomic states. (a) $S_m(\omega ,\mathrm{\Delta })$ in the absence of SO coupling $\alpha =0\pi {\omega }_m$ with atom-atom interactions $U=5.5{\omega }_m$. The color configuration corresponds to the strength of mechanical mirror DNS $\left[S_m(\omega ,\mathrm{\Delta })\right]$. (b,c) Behavior of $S_m(\omega ,\mathrm{\Delta })$ under the influence of $\alpha =60\pi {\omega }_m$ and $120\pi {\omega }_m$, respectively. The dynamics of $S_m(\omega ,\mathrm{\Delta })$ under the effects of many-body interactions $U/{\omega }_m$ are illustrated in (d–f). (d–f) $S_m(\omega ,\mathrm{\Delta })$ as a function of $\mathrm{\Delta }/{\omega }_m$ and $\omega /{\omega }_m$ with atom-atom interaction $U=5.5{\omega }_m,7.5{\omega }_m$, and $9.5{\omega }_m$, respectively, while the strength of SO coupling is kept constant at $\alpha =10\pi {\omega }_m$. The atom-field coupling is considered as $G_a=4.1{\omega }_m$ while the mirror-field coupling is taken as $G_m=1.5{\omega }_m$. The remaining parameters, used in numerical calculations, are the same as in Fig. 2.

Figure 9

(a) Effective temperature $T_{\text{eff}}$ of a mechanical mirror under the influence of $G_a/{\omega }_m$. The SO coupling strength is now considered as $\alpha =100\pi {\omega }_m$ and many-body interaction is kept as $U=5.5{\omega }_m$. The black curve represents $G_a=20{\omega }_m$ while magenta curves from dark shade to light shade are for atom-field coupling $G_a=23{\omega }_m,26{\omega }_m,29{\omega }_m,31{\omega }_m$, and $34{\omega }_m$, respectively. Similarly, (c) deals with the behavior effective temperature $T_{\text{eff}}$ of a mechanical mirror under the influence of $U/{\omega }_m$ at $\alpha =100\pi {\omega }_m$ and $G_a=20{\omega }_m$. Similarly, the black curve represents $U=5{\omega }_m$ while magenta curves from dark shade to light shade represent atom-atom interactions $U=6{\omega }_m,7{\omega }_m,8{\omega }_m,9{\omega }_m$, and $10{\omega }_m$, respectively. The other parameters used in numerical calculation are the same as in Fig. 2.