Experimental Realization of a Thermal Squeezed State of Levitated Optomechanics

We experimentally squeeze the thermal motional state of an optically levitated nanosphere by fast switching between two trapping frequencies. The measured phase-space distribution of the center of mass of our particle shows the typical shape of a squeezed thermal state, from which we infer up to 2.7 dB of squeezing along one motional direction. In these experiments the average thermal occupancy is high and, even after squeezing, the motional state remains in the remit of classical statistical mechanics. Nevertheless, we argue that the manipulation scheme described here could be used to achieve squeezing in the quantum regime if preceded by cooling of the levitated mechanical oscillator. Additionally, a higher degree of squeezing could, in principle, be achieved by repeating the frequency-switching protocol multiple times.

Figure 1

Experimental implementation of squeezing levitated optomechanics. (a) Schematic of the squeezing setup. The paraboloidal trapping system interferes the $E_{\mathrm{div}}$ divergence field and the $E_{\mathrm{scat}}$ Rayleigh scattered field from the trapped particle. The grey region is for homodyne detection, as well as pulse application. (b) Negative square pulse for squeezing generation, as seen by the photodetector. Two different pulse durations are shown, ${\tau }_2$ and ${\tau }_4$. (c) Root-mean-square position as a function of time, $z_{\mathrm{rms}}(t)=\sqrt{⟨(z-⟨z⟩{)}^2⟩}$, obtained from 1500 pulse sequences applied to the same particle. Oscillations for $t<0$ are due to bandpass filtering. (d) Time dependence of the mean position $⟨z⟩$ (the center of the thermal distribution). This quantity also shows oscillations at ${\omega }_1$ after the squeezing pulse.

Figure 2

Experimentally measured phase-space distributions of the mechanical state before and after the squeezing pulse. The average displacement of the state has been subtracted [see Fig. 1]. (a)–(c) Density plots of the phase-space distributions for $z$ motion, at three different times, for a pulse duration $\tau =3/10T_2$. (a) State of the particle motion before the pulse is applied. The former is well approximated by a Gaussian distribution, as is typical for a thermal state. (b) Phase-space distribution shortly after the pulse has been applied (at the time $t_0$): note how it presents clear signatures of squeezing. (c) Phase-space distribution at the time $t_0+\frac{1}{4}{T}_1$. The squeezed state rotates in phase space while squeezing degrades with time, mainly due to background gas collisions that tend to restore the initial thermal distribution.

Figure 3

Quantitative analysis of the squeezing effect. (a) Squeezing factor $\lambda$ as a function of pulse duration $\tau$ (measured in microseconds). This is extracted by comparing the minor axes of the phase-space ellipses (see, e.g., Fig. 2) before and after the pulse. The theoretical fit to the data (the blue line) has been done according to Eq. (10) in the Supplemental Material [17]. (b) Lorentzian fit to the power spectral density (PSD) of the $z$ motion. This is used to extract the radius and the mass of the particle, as well as the collisional damping rate $\mathrm{\Gamma }$ according to Eq. (9) in the Supplemental Material [17].