1. Introduction

In the last decade, remarkable breakthroughs have been achieved in the optical trapping of nanoparticles and their manipulation in vacuum [1]. In addition, in recent years, several studies have been conducted to cool the center-of-mass motion of nanoparticles to the quantum ground state [2–5], leading to new avenues for exploring the quantum mechanics of macroscopic objects [6]. Levitating nanoparticles also have promising applications in research on optomechanics, very weak forces and torques [7,8], dark matter [9,10], nonequilibrium physics [11,12], and nanoparticle–cold atom coupling [13]. We recently proposed a mixture of laser-cooled atoms and nanoparticles to investigate low-energy scattering [14].

In previous studies, objective lenses were generally used for optical trapping [2,4,5,15]. However, owing to their cylindrical configuration with multiple lenses, these lenses result in bulky systems and vacuum degradation as well as exhibit a high heat capacity. Furthermore, the multiple lenses configuration could cause optical misalignment when the pressure and/or temperature of the environment varies. A single aspheric lens with a large numerical aperture (NA) can overcome these limitations. In addition, a light beam collimated by an aspheric lens with a high transmittance exhibits a high power, which allows efficient trapping of nanoparticles. In a previous study, successful optical trapping of nanoparticles using a 1550-nm laser focused by an aspheric lens, resulting in the cooling of their center-of-mass motion to the quantum ground state, was reported [3]. In addition, two nanoparticles were simultaneously trapped, and their motion was cooled using an aspheric lens with a large NA (= 0.75) [16]. Shen et al. [17] demonstrated optical trapping with a metalens and 1064-nm laser. They successfully optically levitated a nanoparticle at a pressure of 2.0 × 10⁻⁴Torr (2.7 × 10⁻²Pa) without feedback cooling. The short laser wavelength resulted in a small beam diameter at the focus and increased the trapping frequency. Further, according to a previous study, the internal motion of nanocrystals, such as Yb³⁺:LiYF₄, can be effectively laser-cooled using a 1030-nm laser beam [18].

In this study, we optically trapped a nanoparticle in high vacuum using a single aspheric lens and 1030-nm laser. In contrast to the previous metalens-based approach, we achieved and maintained the trap in an even higher vacuum without implementing center-of-mass motion cooling. In addition, we improved the stability of the trap under high-vacuum conditions via uniaxial feedback cooling.

2. Theoretical description

At small oscillation amplitudes, an optically trapped nanoparticle harmonically oscillates in the potential owing to an optical gradient force. Simultaneously, Brownian motion occurs in air owing to collisions with gas molecules. The equation of motion for the x direction without feedback cooling is

According to the fluctuation–dissipation theorem, the autocorrelation function of the Langevin force can be expressed as: $\langle{F_{\textrm{fluct}}}(t ){F_{\textrm{fluct}}}({t + \tau } )\rangle= 2M{\mathrm{\Gamma }_0}{k_B}{T_0}\delta (t ),\; $ where ${k_B}$ is Boltzmann’s constant and ${T_0}$ is the temperature; this equation is the same for the y and z directions. Using Eq. (1), we can determine the power spectral density (PSD) of a particle (q = x, y, z) [19]:

The frequency of a harmonically oscillating particle can be expressed by: $\mathrm{\Omega }_0^{(q )} = \sqrt {k_{\textrm{trap}}^{(q )}/M} $, where $k_{\textrm{trap}}^{({x,y} )}$ is the trap stiffness, which acts as a coefficient when the gradient force is expressed as $F ={-} {k_{\textrm{trap}}}\textrm{}q.$ The trap stiffness can be calculated using paraxial and dipole approximations as:

The diffraction-limited beam waist can be expressed using the NA as ${\omega _0} = \lambda /({\pi \textrm{NA}} )$. Using Eqs. (3) and (4), the beam waist can be estimated from the ratio between the transverse and axial particle frequencies as follows [20]:

In this study, we controlled the particle motion by modulating the laser power. The trap stiffness owing to parametric feedback cooling [15,21] is

3. Experimental setup

The experimental setup is shown in Fig. 1. In this experiment, we used a Yb:YAG laser with an optical wavelength (λ) of 1030 nm (power P = 200 mW); the laser beam was focused by a single aspheric lens (AL1), with a large NA (SIGMAKOKI Co., Ltd. custom product; NA = 0.9; effective focal length = 6.67 mm; back focal length = 1.00 mm), mounted inside a vacuum chamber; the material of this lens was S-LAH88 (abbe number: 31.6, OHARA Inc.). For a high transmission and more efficient light collection, an effective diameter of 12 mm was used, and Strehl’s ratio was 0.995. To use the lens to its full potential, we measured and adjusted the diameter of the incident beam to 12.1 mm. A silica nanoparticle with a diameter of 156 nm (Microparticles, SiO2-R-0.15) was optically trapped. The nanoparticles were first diluted in ethanol and then loaded for trapping using an ultrasonic nebulizer (Omron, NE-U22) at ambient pressure. Once the particle was trapped near the focus, the chamber was evacuated.

 

Fig. 1. Schematic of the experimental setup (AOM: acousto-optic modulator, AL1: aspheric lens 1 (NA = 0.9), AL2: aspheric lens 2 (NA = 0.546), $\lambda /2:$ half-wavelength plate, PBS: polarizing beam splitter, DM: D-shaped mirror, PD: photo detector, PLL: phase-locked loop, G: Gain.) The inset figure is a picture of AL1 with a large NA of 0.9.

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A charge-coupled device camera was set up to observe the particle trapped in the chamber. To measure the center-of-mass motion of a trapped nanoparticle, a set of balanced photodetectors were mounted, which collected the light scattered by the nanoparticle. Another aspheric lens (AL2, Thorlabs, AL1512-C, NA = 0.546) was placed in the chamber to collect and collimate the scattered light. The center-of-mass motion of the particle was measured by analyzing the interference between the light scattered from the trapped particle and the unscattered light. A D-shaped mirror was used to split the light vertically, and we detected the motion in the x direction using a balanced photodetector (Thorlabs, PDB440-C). The detected signal was DC blocked (Thorlabs, EF599) and amplified (NF Corporation, SA-200F3), and subsequently, sent in two directions. One signal was sent to the data acquisition system to calculate and record the PSD, while the other signal was sent to a digital lock-in amplifier (Zürich Instruments, HF2LI) to generate a feedback signal. To stabilize the trap under high vacuum, we controlled the particle motion in the x direction via parametric feedback cooling [15,21]. In this process, an external oscillator was locked to the motion of the particle using a PLL, and the laser intensity was modulated using an acousto-optic modulator (AOM) with a controlled phase shift relative to the particle motion.

4. Results and discussion

Optical trapping using a single aspheric lens

Figure 2 shows the PSD of the particle’s motion for 0.1 s at pressures of 3.0 × 10³ and 4.5 × 10⁻³Pa; the sampling rate of the calculated data was 8 × 10⁵ Sa/s. We found that the trapping frequencies were $\mathrm{\Omega }_0^{(x )} = 2\pi \times 129\,\textrm{kHz}$, $\mathrm{\Omega }_0^{(y )} = 2\pi \times 147\,\textrm{kHz}$, and $\mathrm{\Omega }_0^{(z )} = 2\pi \times 50$ kHz, and the other sharp peaks were due to electrical noise. The oscillation mode of 2$\mathrm{\Omega }_0^{(z )}$ was caused by the anharmonicity arising from the higher-order terms in the trap potential. Figure 3 shows the trap-frequency dependence of the laser power with theoretical fitting. The trapping frequency depends on the square root of the laser power. In other words, the trap stiffness was affected by the laser power; therefore, the shape of the trap potential could be controlled by modulating the laser power—this is underlying mechanism of controlling the temperature of the center-of-mass motion via parametric feedback cooling. In addition, using Eq. (5), the beam waist was estimated to be $\omega _0^{(x )} = \textrm{}$612(9) nm and $\omega _0^{(y )} = \textrm{}$705(17) nm in the x and y directions, respectively. The results show that the beam is not focused up to the diffraction limit. The phase plane of the incident beam is disturbed owing to deflection by the AOM. The difference between the beam waists in the x and y directions can be attributed to the polarization direction.

 

Fig. 2. PSD of the mechanical motions of the levitated particle. The blue, orange, and gray lines represent a pressure of 3.0 × 10³Pa, a pressure of 4.5 × 10⁻³Pa, and the noise floor of our measurement system, respectively.

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Fig. 3. Trap frequency dependence on trapping laser power. The black dotted lines are the results of fitting with a square root function.

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4.2 Trapping a nanoparticle with feedback cooling

We used parametric feedback cooling to stabilize the particle in high vacuum. Generally, cooling is initiated at a pressure of 100 Pa, and the feedback phase, which is the difference between the phase of the particle’s motion and that of the feedback signal, is optimized. Figure 4(a) shows the PSD without cooling at 150 Pa and with cooling at 5.8 × 10⁻⁴Pa. In the inset of Fig. 4(a), a comparison is shown by focusing on the area around the oscillation mode in the x direction. The PSD area with cooling was smaller than that without cooling, and the area was proportional to the effective temperature of the center-of-mass motion, confirming that cooling was achieved. The effective temperature for cooling was calculated to be 2.3 K using the calibration techniques described in a previous report [19]; additionally, the frequency shifted by $\delta f = 5\; \textrm{kHz}$. We believe that dehydration of the particle occurs during the vacuuming process and the frequency is shifted mainly due to the change in the density and the refractive index of the particle.

 

Fig. 4. Parametric feedback cooling. (a) (red spectrum) PSD obtained without feedback cooling at 297 K at a pressure of 150 Pa; (blue spectrum) PSD obtained with cooling at 2.3 K at a pressure of 5.8 × 10⁻⁴Pa; (gray spectrum) represents the noise floor of our measurement system. The inset shows the PSD extracted around the x peak. (b) Effective temperature when the feedback phase is varied with a step of 10°. The blue and orange dots represent the temperatures at pressures of 130 and 50 Pa, respectively.

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We measured the dependence of the feedback phase on temperature at approximately 100 and 50 Pa. At a constant modulation depth, we varied the feedback phase and observed that the highest efficiency was obtained at a feedback phase of ${\theta _{\textrm{fb}}} = 120^\circ $ (Fig. 4(b)). In addition, the temperature change at 50 Pa was greater than that at 100 Pa. Equation (4) indicates that the effect of feedback control is more significant at low pressures because of the reduced damping rate ${\mathrm{\Gamma }_0}$.

When the pressure reached approximately 20 Pa with cooling, splitting of the oscillation mode peak was observed because of the coupling between the oscillators in the PLL and particle oscillations [22]; this phenomenon was effectively eliminated by reducing the feedback gain.

5. Conclusions

In conclusion, we demonstrated optical trapping and control of a nanoparticle using a single aspheric lens and 1030-nm laser beam. The particle was trapped with parametric feedback cooling at a pressure of 5.8 × 10⁻⁴Pa. In contrast to a conventional objective lens, optical levitation with a single high-NA lens in high vacuum provides more freedom for designing experimental systems owing to its small package and low heat capacity. This approach is expected to open new avenues for studying the fundamentals of physics, e.g., the interactions between ultracold atoms and nanoparticles [14].