1. Introduction

Position measurement of a macroscopic mechanical oscillator in the quantum regime is essential for obtaining insights into macroscopic quantum mechanics (MQM). Moreover such a quantum oscillator should be prepared over various scales to investigate the classical-quantum boundary. In particular, mass may play a critical role in generating such a boundary, mainly motivated by the gravitational decoherence [1, 2] among decoherence models [3]. In order to make an oscillator quantum for testing MQM [4–8], all classical noises are required to be less than the standard quantum limit (SQL), the limit of a continuous position measurement [9].

So far, in terms of the mass scale, the SQL at the mechanical resonance and quantum ground-state cooling have been achieved below the microgram scale [10–12]. At the heavier mass scales, reaching the SQL in the free-mass regime, which was studied originally for improving the sensitivity of gravitational-wave detectors, is of interest for generating macroscopic entanglement [8]. At the gram [13] and kilogram scale [14], there have been studies approaching the SQL. At the microgram and milligram scales, however, experiments using small oscillators [15] and a pendulum [16] are still far from the SQL either at the resonance or in the free-mass regime. For a pendulum, suspension thermal noise is one of the main obstacles. Thus optical support of an object [17, 18] instead of mechanical support would be promising because light does not introduce thermal noise. For a mirror, the ways of optical levitation have been proposed at the nanogram [19] and sub-milligram scale [20], yet its realization has not been reported at any scales including around the milligram scale.

In this paper, we propose a new configuration of optical levitation of a macroscopic mirror with a mass of sub-milligram. This enables reaching the SQL in the free-mass regime with respect to the vertical displacement. We consider a sandwich configuration with two Fabry-Pérot cavities that are aligned linearly and vertically. While using the optical cavities to vertically support the mirror has been previously proposed [19, 20], the difference is in the way used to trap the mirror horizontally. We use optical restoring forces derived from the geometry of our configuration instead of optical tweezers [19] or three pairs of double optical spring [20], which results in the minimum number of laser beams. The levitated mirror using our scheme will be applicable to the MQM test proposed in Ref. [8] at the milligram scale, and also be helpful to the study of reduction of quantum noise in gravitational-wave detectors.

2. Sandwich configuration

The sandwich configuration that we propose is composed of two cavities with three mirrors, shown in Fig. 1. At the center is the levitated mirror that is convex downward for the stability of its rotational motion, with the high reflective (HR) coating at the lower surface. At the upper and lower sides are fixed mirrors, each of which forms a Fabry-Pérot cavity with the common center mirror. The center mirror is supported by the radiation force from the lower cavity and is stabilized by the upper cavity for its horizontal motion. The relative positions of the center of curvature (COC) of the three mirrors is critical for the optical and mechanical stability of the system and must be arranged as shown in Fig. 1(a).

 

Fig. 1 Schematic of the sandwich configuration. The arcs and the centers of the sectors are the mirrors and their COCs, respectively. The blue and red lines are the circulating beams in the lower and upper cavities, respectively. (a) Balanced state. The vertical motion is constrained by the double optical spring. (b) Restoring torque due to the gravity. (c) Horizontal restoring force due to the upper cavity.

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A double optical spring [21, 22] is introduced into the cavities for vertical optical trapping without giving rise to suspension thermal noise. This requires at least two laser beams; one is blue-detuned and the other is red-detuned from the cavity resonance. Although levitation by using a single cavity is unstable, as explained in the next section, the sandwich configuration with two cavities is stable and allows for one cavity to be red detuned and the other to be blue detuned. Thus the sandwich configuration can be realized with the minimum two beams required by a double optical spring. The frequency of one laser beam compared with the other is shifted by an amount larger than the cavity linewidth so that each cavity can be regarded as being independent.

3. Stability

The motion of the levitated mirror is described in terms of 6 degrees of freedom; the position (x, y, z) of the COC of the mirror and the Euler angle (α, β, γ). Here the z-axis is set to be the vertical direction. A rotational symmetry about the z-axis of the system allows us to set α = γ = 0 without loss of generality. Also, it allows us to confine the motion within the xz plane, equivalent to setting y = 0. In this case, the stability of the mirror can be discussed by considering the mechanical response to small displacements (δx, δz, δ β) from its balanced state. The linear mechanical response is written as ^t(δF_x, δF_z, δN_β) = −K^t (δx, δz, δ β), where δF_x, _z is the restoring force in the x and z direction, δN_β is the restoring torque, and K is the 3 × 3 complex matrix with its real and imaginary parts representing the spring and damping effect, respectively. The condition for stability is that both the real and imaginary part of the three eigenvalues of K are positive.

The vertical displacement δz and the rotation δ β do not change the positions of the beam spots on the center mirror and do not couple to the other degrees of freedom. The center mirror is subjected to the force $\delta F_z=-(K_{\mathrm{L}}^{\text{opt}}+K_{\mathrm{U}}^{\text{opt}})\delta z$ due to the double optical spring of the two cavities, where $K_I^{\text{opt}}=k_I^{\text{opt}}+im\omega {\gamma }_I^{\text{opt}}$, m is the mass of the mirror, ω/(2π) is the Fourier frequency, $k_I^{\text{opt}}$ is the optical spring constant and ${\gamma }_I^{\text{opt}}$ is the optical damping rate with I = L, U indicating the lower and upper cavities, respectively. The stability condition is that $k_{\mathrm{L}}^{\text{opt}}+k_{\mathrm{U}}^{\text{opt}}>0$ and ${\gamma }_{\mathrm{L}}^{\text{opt}}+{\gamma }_{\mathrm{U}}^{\text{opt}}>0$, which can be satisfied by adjusting the laser detuning from the cavity resonance. The mirror is, in turn, subjected to the torque δN_β = −mgR · δ β due to gravity, as shown in Fig. 1(b). Here g is the gravitational acceleration, and R is the radius of curvature (ROC) of the center mirror, the sign of which is defined to be positive when it is convex downward. To use the gravity as the restoring torque, the mirror must be convex downward, i.e., R > 0. The rotational motion would be damped by the residual gas instead of the lossless gravity.

On the other hand, the horizontal fluctuation of δx changes the positions of the beam spots, which could lead to an anti-spring effect resulting in the instability of the cavities. Due to the beam passing through the COCs, the direction of the radiation force with the magnitude of F_I tilts by the angle θ_I such that tan θ_I = δx/a_I, where a_I is the distance between the COCs of the mirrors of the cavity and is shown in Fig. 1(a). This yields the horizontal force $\delta F_{x,I}=F_I\sin {\theta }_I\simeq K_I^{\text{hor}}$ δx which includes the damping effect [23], where

To summarize the above, the matrix K is diagonalized:

The second order effects are discussed to clarify the trapping range. The change of the cavity length δz must be much smaller than the length detuning due to the optical spring, i.e., |δz| ≪ λ|δ_I|/ℱ_I, where λ is the wavelength of the laser and δ_I is the laser detuning normalized by the (amplitude) cavity decay rate. With regards to the horizontal range, three effects below could arise. First, the horizontal displacement δx also changes the cavity length by ${[a_I^2+{(\delta x)}^2]}^{1/2}-a_I\simeq (a_I/2){(\delta x/a_I)}^2$. Thus δx is required to be |δx| ≪ (2a_I λ|δ_I|/ℱ_I)^1/2. Second the vertical components of the radiation forces change as δF_z, _I/F_I = cos θ_I ≃ 1 − (δx/a_I)²/2, and are negligible when |δx| ≪ a_I. Third, the misalignment between the incident mode and the cavity mode causes the decrease in the intracavity power. Because the displacement of the beam waist of the cavity mode is approximately δx, the mode matching ratio decreases by (δx/w₀)²/2 + (δθ/θ₀)²/2, which must be much lower than unity. Here w₀ is the beam waist radius of the cavity mode, δθ (= δx/a) is the tilt of the beam and θ₀ = λ/(πw₀). Among the three conditions, the first one would be the most stringent.

4. Reaching the SQL

We focus on the vertical displacement of an optically levitated mirror to discuss the possibility of reaching the SQL. The mass m of the mirror determines its SQL in its free-mass regime as G_SQL(f) = 2ℏ/(mω²), where ℏ is the reduced Planck constant and f = ω/(2π) is the Fourier frequency. Note that in this paper we use single-sideband power spectra of the displacement including G_SQL(f). The SQL reaching frequency f_SQL, where the shot noise and the radiation pressure noise intersect, is determined by the power of the laser used for sensing the displacement [21]. However, m and f_SQL are not free parameters for the optical levitation because the mirror is supported by radiation pressure, namely $mg\approx 2P_{\mathrm{L}}^{\text{circ}}/c$, where $P_{\mathrm{L}}^{\text{circ}}$ is the intracavity power of the lower cavity. In the case that f_SQL is lower than the cavity pole, f_SQL is approximately given by

In order to reach the SQL, all classical noises must be below the SQL at f = f_SQL. In our case, the Brownian vibration of the mirror is the most critical among fundamental thermal noises. Therefore close attention must be paid to the levitated mirror including its material and shape. In particular, the aspect ratio of the mirror, which we define as diameter over thickness, is important and should be close to unity because a thinner mirror leads to a lower first resonant frequency and thus higher noise level of the Brownian vibration. In the case that the aspect ratio is close to unity and the displacement is sensed at the center of the mirror, its noise power spectrum below the mechanical resonant frequency of the mirror is given by [24–26]

Table 1. Parameters for reaching the SQL. The suffix indicates s for the substrate, Ta for the TiO₂:Ta₂O₅ coating layer, Si for the SiO₂ coating layer, L for the lower cavity and U for the upper cavity.

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We show an example set of realistic parameters for reaching the SQL that satisfies the stability conditions mentioned above, in Table 1. A 0.2 mg mirror is used, with an aspect ratio of 3, a radius of 0.35 mm and a ROC of 30 mm. Its substrate and coating are respectively made of fused silica (SiO₂) and alternating titania-doped tantala/silica (TiO₂:Ta₂O₅/SiO₂) layers with optimal thickness [26], which have been proved to have low loss angle values and low optical absorption [27–29]. A laser of λ = 1064 nm is adopted, with input powers of 13 W and 4 W for the lower and upper cavities, which are blue and red detuned for the double optical spring, respectively. Cavity finesses are chosen to be 100, which is relatively small compared with previous proposals, to increase the ratio G_SQL/G_coa as much as possible. The distances between the COCs are respectively 5.0 mm and 1.3 mm for the lower and upper cavities, generating a positive spring effect for the horizontal stability. Here the effective ROC of the levitated mirror for the upper cavity is modified to R/n_s due to the HR surface being at the lower side, where n_s is the refractive index of the substrate. For these parameters, the trapping ranges of the system are calculated to be |δx| ≪ (2a_I λ|δ_I|/ℱ_I)^1/2 = 0.6 μm and |δz| ≪ λ|δ_I|/ℱ_I = 50 pm.

For the vertical displacement of the mirror, which is measured by using the reflected light of the lower cavity as the change of the lower cavity length, we obtain the noise power spectra in Fig. 2 using the parameters in Table 1. The coating Brownian noise is the largest classical source and is lower than the SQL level below 100 kHz. When other technical noises, which will be discussed below, are reduced to levels smaller than the SQL, the 0.2 mg mirror can reach the SQL at f_SQL = 23 kHz in its free-mass regime, where the vertical displacement is 2.2 × 10⁻¹⁹ m/Hz^1/2. The mirror’s mechanical resonances that contribute to the noise in the displacement sensing are estimated by finite element analysis to have a lowest frequency of 3.1 MHz. This is far from 23 kHz, and thus does not affect the thermal noise level there.

 

Fig. 2 Noise spectra of the 0.2 mg mirror, using parameters in Table 1. The mirror reaches the SQL at 23 kHz, corresponding to the vertical displacement of 2.2 × 10⁻¹⁹ m/Hz^1/2. The resonance at 340 Hz is due to the double optical spring.

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5. Discussion

With regards to other noise sources, the most significant technical noises would be classical noises of the laser, consisting of frequency noise and intensity noise. For a Fabry-Pérot cavity, the power spectrum of frequency noise is given by $G_{\text{freq}}^{1/2}=l_{\mathrm{L}}\times \delta f_{\mathrm{a}}/f_{\mathrm{a}}$, where l_L, f_a and δ f_a are the lower cavity length, the laser frequency and the fluctuation of the laser frequency, respectively. The requirement is δ f_a to be on the order of 0.1 mHz/Hz^1/2 at 23 kHz to make the frequency noise smaller than the thermal noise level, which is challenging but within our reach [30]. The classical intensity noise, in turn, must be stabilized at the level of the shot noise. Although active stabilization is difficult for high input power, passive stabilization at the input stage would be possible by using a cavity with its cavity pole lower than f_SQL to filter out the intensity noise at f = f_SQL.

A thermal noise will arise due to the viscous damping of the residual gas. Its damping rate is given by γ_gas = SP/(Cm) × [m_mol/(k_B/T)]^1/2 [31], where S, P, C and m_mol are the one-side area of the mirror, the air pressure of the residual gas, a constant close to unity depending on the shape of the mirror and the mean mass of the gas molecules, respectively. Assuming that C = 1, we have γ_gas = 7 × 10⁻⁸ Hz for a vacuum pressure of P = 10⁻⁵ Pa, giving a sufficiently small noise level to be negligible. Seismic vibration noise is known to have the typical spectrum of (10⁻⁷/f ²) m/Hz^1/2 above 0.1 Hz [32] and would be crucial at low frequency. However, f_SQL is so high that it can be fully suppressed at f = f_SQL using a single suspension of the whole system with its resonant frequency of 1 Hz.

The heat balance of the levitated mirror should be mentioned because the extreme increase in temperature worsens the thermal noise level and can possibly deform the levitated mirror. The mirror receives heat due to the absorption of the beams inside the cavities as well as environmental radiation (room temperature), and releases heat via its own radiation. The absorption is dominant for the coating (TiO₂:Ta₂O₅/SiO₂), which is < 0.34 ppm at 1064 nm [29]. The total absorption is then estimated to be < 0.14 mW, corresponding to the temperature rise of < 20 K above the room temperature. Another issue concerning the laser is that the high intensity of the circulating laser beams could damage the coating. In our case, the peak intensities are 14 kW/mm² (lower light) and 2.3 kW/mm² (upper light), well within the typical damage threshold of a few MW/mm².

6. Conclusion

We proposed a new configuration, sandwich configuration, utilizing optical levitation of a macroscopic mirror for reaching the SQL. Our configuration gives a stable levitation with the minimum two beams. By investigating the critical noise sources, we showed that reaching the SQL with our system can be achieved with current technology. Our system enables sufficient sensitivity to create macroscopic entanglement and test MQM.