Suppressing Recoil Heating in Levitated Optomechanics Using Squeezed Light

We theoretically show that laser recoil heating in free-space levitated optomechanics can be arbitrarily suppressed by shining squeezed light onto an optically trapped nanoparticle. The presence of squeezing modifies the quantum electrodynamical light-matter interaction in a way that enables us to control the amount of information that the scattered light carries about a given mechanical degree of freedom. Moreover, we analyze the trade-off between measurement imprecision and back-action noise and show that optical detection beyond the standard quantum limit can be achieved. We predict that, with state-of-the-art squeezed light sources, laser recoil heating can be reduced by at least $60\mathrm{\%}$ by squeezing a single Gaussian mode with an appropriate incidence direction, and by $98\mathrm{\%}$ by squeezing a properly mode-matched mode. Our results, which are valid both for motional and librational degrees of freedom, will lead to improved feedback cooling schemes as well as boost the coherence time of optically levitated nanoparticles in the quantum regime.

Popular Summary

Nanoparticles levitating in vacuum are a cornerstone in both fundamental and applied sciences. They offer a bridge between the quantum and classical realms and also hold promise as ultrasensitive detectors for forces and inertia. Recent experiments have showcased the ability of lasers to levitate and “cool” these nanoparticles down to their fundamental quantum state. However, a challenge emerges: the lasers that trap and cool the nanoparticles also introduce “decoherence”—unwanted noise that quickly erases the delicate quantum states. Turning these lasers off isn't a solution, as one risks losing the particle. The alternative of using different forces, like electrostatics, to secure the particle comes with its own set of challenges.

We propose an innovative, light-based solution: instead of replacing the trapping beam, supplement it with a special kind of quantum light. We theorize that by introducing a light beam in a “vacuum-squeezed” state (which has even less noise than a regular vacuum) alongside the original trapping laser, we can counteract and even eliminate decoherence. The secret lies in aligning this new beam flawlessly with the light scattered by the nanoparticle due to the trapping laser—the source of decoherence. By employing a standard lens, one could potentially reduce this decoherence by up to 60%, making it feasible to validate our predictions in current experiments. Furthermore, our research predicts this added quantum light could also enhance the precision of measuring particle motion.

This exploration lights the path for crafting intricate quantum states and forging the next generation of quantum sensors.

Figure 1

(a) We consider a subwavelength particle (green) trapped by a focused laser beam (red arrow) and interacting with an electromagnetic mode in a squeezed vacuum state (blue arrow). The functions $A_s({\mathrm{\Omega }}_{\mathbf{k}},{\mathit{\epsilon }}_{\mathbf{k}})$ and $A_z({\mathrm{\Omega }}_{\mathbf{k}},{\mathit{\epsilon }}_{\mathbf{k}})$ respectively describe the angular and polarization distributions of the squeezed beam and of the photons scattered inelastically by the motion along $z$ in the absence of squeezing. (b) Wigner function of the squeezed light mode at frequency $\omega$; see Appendix pp3 for details.

Figure 2

(a) Recoil heating rate for the mechanical motion along the $z$ axis as a function of the degree of squeezing for ${\varphi }_s-2\arg ({\xi }_z)=0$ (lower three lines) and ${\varphi }_s-2\arg ({\xi }_z)=\pi$ (upper three lines). Here ${\mathrm{\Gamma }}_z^{(0)}$ is the recoil heating rate in the absence of squeezing. (b) Recoil heating rate versus squeezing phase for $r_s=1.73$ (15 dB). In both (a) and (b) we show the perfect overlap case ($|{\xi }_z|=1$, dashed) and the case of squeezed light propagating along the negative $z$ axis and focused by a lens of numerical apertures $0.8$ (green) and $0.5$ (purple); see the text for details.

Figure 3

(a) Modification of the recoil heating of the motion along the $z$ axis is maximized for a squeezed beam counterpropagating with respect to the light. (b) Differential scattering cross section $d{\sigma }_z/d{\mathrm{\Omega }}_{\mathbf{k}}$ in the absence of squeezing (information radiation pattern). The value of the function is encoded in the radial distance to the origin and the color scale encodes the relative amplitude within each panel. (c) Differential scattering cross section for squeezed light with a Gaussian spatial profile of different numerical apertures. Upper and lower rows correspond to 13 dB and ${\varphi }_s-2\arg ({\xi }_z)=0$ and to 3 dB and ${\varphi }_s-2\arg ({\xi }_z)=\pi$, respectively.

Figure 4

(a) Minimum detectable signal for $\omega \ll {\mathrm{\Omega }}_{\mu }$ in units of its SQL as a function of $4{\mathrm{\Gamma }}_{\mu }^{(0)}/{\mathrm{\Omega }}_{\mu }$, in the absence of squeezing (dashed) and for 15 dB squeezing and perfect overlap, $|{\xi }_{\mu }|=1$, along different quadratures (colored). Inset: Wigner function of the input light for these curves (see Appendix pp3 for details). The black circle indicates the Wigner function size in the absence of squeezing. (b) Minimum signal $\underset{u}{min}(S_{\mu ,\min }/S_{\mu ,\mathrm{SQL}}){|}_{{\varphi }_s-2\arg ({\xi }_{\mu })=3\pi /2}$ as a function of the degree of noise reduction $e^{2r_s}$ and mode overlap $|{\xi }_{\mu }|$. The upper ticks mark the value of the overlap $|{\xi }_z|$ corresponding to a squeezed beam with Gaussian angular profile and $\text{NA}=0.3$, $0.5$, $0.7$, and $0.9$. Constant-value isolines are marked by dashed curves.

Figure 5

(a) Our formalism can be directly extended to the libration of an anisotropic particle around its equilibrium orientation along the laser polarization. (b) Differential scattering cross section for the libration along $y$ in the absence of squeezing. (c) Recoil heating for the libration along $y$ as a function of the degree of squeezing for ${\varphi }_s-2\arg ({\xi }_y)=0$ (upper three lines) and ${\varphi }_s-2\arg ({\xi }_y)=\pi$ (lower three lines). The black dashed curves correspond to the perfect overlap case $|{\xi }_y|=1$ and are identical to the black dashed lines in Fig. 2, whereas colored curves to a squeezed beam focused by a lens of numerical apertures $\text{NA}=0.5$ (purple) and $\text{NA}=0.8$ (green). (d) Differential scattering cross section for the libration along $y$, for a copropagating squeezed light beam [see panel (a)] focused by lenses of different numerical apertures. Upper and lower rows correspond to 13 dB and ${\varphi }_s-2\arg ({\xi }_{\mu })=0$ and to 3 dB and ${\varphi }_s-2\arg ({\xi }_{\mu })=\pi$, respectively.