Detection of radiation torque exerted on an alkali-metal vapor cell

We have developed a torsion balance to detect the rotation of a cell containing spin-polarized gaseous atoms to study angular momentum transfer from gaseous atoms to solid. A cesium vapor cell was hung from a thin wire in a vacuum chamber, and irradiated from the bottom with circularly polarized light tuned to the $D_2$ transition to polarize cesium atoms in the cell. By varying the light helicity at the resonance frequency of the torsion balance, we induced forced rotational oscillation of the cell and detected radiation torque exerted on the cesium vapor cell through the cesium atoms inside. The torque was particularly large when both hyperfine levels of cesium atoms were optically pumped with application of a longitudinal magnetic field. Further detailed study will provide new insights into spin-transfer processes at the gas-solid interface.


INTRODUCTION
Circularly polarized light transports angular momentum and exerts torque on an irradiated object. In 1936, Beth demonstrated that a hung half waveplate rotated when circularly polarized light passed through it and flipped the light helicity [1]. From a quantum viewpoint, each photon in left (right) circularly polarized light has a spin component ofh (−h) in the direction of light propagation (e.g., [2,3]), and transfer of the photon spins to an object leads to positive (negative) radiation torque on the object.
With regard to spin transfer from photons to materials, photon spins can be effectively transferred and stored as atomic spins in gaseous atoms by optical pumping [4]. The spinpolarized atoms fly to and collide with the walls of a gas container (often called a cell). The spin polarization relaxes almost completely following a single collision with surfaces, such as glass and metal [5]. Some coating materials, such as paraffin, suppress spin relaxation to a great extent [6], but the polarized atomic gas eventually loses its spin angular momenta.
Although spin relaxation mechanisms at surfaces are not completely understood for all types of surfaces [7], this spin relaxation is considered to involve the transfer of spin angular momenta from the gaseous atoms to the cell, from the viewpoint of the conservation of angular momentum. Under specific conditions, the spin polarization of alkali-atom nuclei in alkali-metal hydrides was achieved by spin transfer from an optically pumped alkali-metal vapor [8]. There are various types of spin relaxation and transfer processes, but any type of spin transfer must rotate the cell macroscopically.
Here, we have developed a torsion balance to detect the macroscopic rotation of a cell containing spin-polarized gaseous atoms, to study angular momentum transfer from gaseous atoms to solid. Dilute cesium (Cs) atoms in a vapor contained in an evacuated cell hung from a thin wire were spin-polarized by absorbing circularly polarized light and undergoing spontaneous emission repeatedly during optical pumping. The spin polarization was maintained under a uniform longitudinal magnetic field until the atoms collided with the cell surface, where they lost their spin polarization. We varied the helicity of the circularly polarized light at the resonance frequency of the torsion balance and clearly showed that optical pumping of Cs atoms exerted torque on the cell. The induced rotational oscillation was enhanced when the atomic gas achieved high polarization under appropriate conditions of light polarization, light frequency, longitudinal magnetic field, and atomic density of the gas. Further quantitative and detailed studies will hopefully reveal spin-transfer processes at the gas-solid interface from a mechanical viewpoint. Figure 1 shows a schematic representation of the experimental setup. It is basically a torsion balance made up of a gas-containing cell hung with a thin wire in a vacuum. The cell contained a Cs vapor in equilibrium with a small amount of Cs metal. Cs metal has a relatively high melting temperature of 28 • C, and Cs atomic densities in the vapor are 6 × 10 10 cm −3 at 30 • C and 5 × 10 11 cm −3 at 50 • C [9]. The ground state of a Cs atom has a 1/2 electron spin and a 7/2 nuclear spin; there is no orbital angular momentum. The ground state splits into two hyperfine levels labeled as total angular momenta F = 4 and 3,

EXPERIMENTAL APPARATUS
respectively. Cs atoms thus store up to 4h angular momentum in the ground state.
The Cs vapor cell was cylindrical, 20 mm in outer diameter and 63 mm in outer length, with a short (5 mm), pinched-off stem. The cell was made of quartz glass. The glass wall was 0.5 mm thick to reduce the weight and inertia momentum of the cell, which were 5 g and 5 × 10 −7 kg m 2 , respectively.
The cell was hung with a short tungsten (W) wire 10 µm in diameter and 0.7 cm in length. The wire was connected to a wheel hung with a long W wire 10 µm in diameter and 83 cm in length. The long wire was connected to the top flange of the vacuum chamber.
The wheel had a relatively large mass and a large moment of inertia; 17 g and 7 × 10 −6 kg m 2 , respectively. The longer wire and larger momentum of inertia of the wheel resulted in a lower resonance frequency of rotational oscillation of the wheel than the cell attached to the short wire. Thus, the wheel acted to isolate vibration from noise input through the top flange of the chamber [10]. The wheel was made of black-anodized aluminum. We avoided using magnetic materials even for small screws contained in the wheel, to reduce the magnetic influence from the outside. The wheel could be made of dielectric material, which may be problematic when charged up, but might further reduce the magnetic influence of eddy currents.
The top flange holding the long wire was electrically rotatable, and was used to damp the rotational oscillation of the wheel; otherwise it would have kept oscillating out of the range of the detector (30 mrad in the current setup; see below) even for 1 week in a vacuum chamber. Once the rotational oscillation entered the detector range by manual control of the rotatable flange, it was further damped by a PC-based servo system. This servo system enabled rapid recovery from strong external disturbances, such as earthquakes. We turned off the servo and allowed the cell to oscillate freely during measurements.
The vacuum chamber made of stainless steel was evacuated with an ion pump down to 1 × 10 −5 Pa and placed on an optical table (table top dimensions This and other oscillation frequency components were manifested in the Fourier transform spectrum shown in Fig. 2  at 29 mHz corresponded to the rotational oscillation of the cell hung from the wheel with the 0.7 cm wire. We confirmed that these resonance frequencies reasonably agreed with estimations made from the moments of inertia for the wheel and the cell, as well as the lengths and the Young's modulus of the W wires. The inset of Fig. 2 (b) shows a magnified resonance peak at 29 mHz, which was the focus of the following measurements. From fitting with a Lorentz function, we inferred the Q-factor of this torsion balance to be about 1,000.
To investigate cell rotation induced by spin-polarized Cs atoms, we employed a forced oscillation method at a resonance frequency of 29 mHz. We changed the helicity of the pumping light continuously between right-and left-circular polarization with the LCVR at 29 mHz, and measured the amplitude of the resulting oscillation at the same frequency.
The cell temperature was 50 • C. Figure 3 shows the typical time evolution of the oscillation amplitude under several conditions. LD1 and LD2 that produced the pumping light were tuned to the F = 3 → F ′ = 4 transition and F = 4 → F ′ = 4 transition, respectively, with the frequency difference equal to the Cs hyperfine splitting 9.19 GHz. The pumping light was incident to the cell before the measurement started. The amplitude measured before period (a) was a typical background level (a few µrad). During period (a), the LCVR  were effects that were unrelated to optical absorption by Cs atoms, presumably radiation pressure effects [1,12]. We roughly estimated the expected amplitudes of the oscillation from the radiation torque originating from photon spins (2 × 10 −17 N m) and the Q-factor of the torsion balance (∼ 1, 000), and obtained amplitudes ∼ 2 µrad, which was smaller than the observed amplitudes. Further studies are required to quantitatively evaluate the process of spin transfer from atoms to solids. This study can be regarded as a combination of the Einstein-de Haas (EdH) [13] and Beth experiments [1]. In the EdH experiment, the rotation of a magnetic material was shown to be induced by flipping its magnetization. This result, obtained before the development of quantum mechanics, demonstrated that magnetization was related to angular momentum.
It can now be explained as compensation of the change in direction of atomic spins in a solid, a source of magnetization, by the rotation of the solid to fulfill the conservation of angular momentum. In our experiment, with regard to angular momentum transfer from photons to matter, microscopic processes can be well understood as optical pumping, and were much more controllable than in the Beth experiment because atom-photon interactions can be precisely adjusted. With regard to angular momentum transfer from atomic spin to solid rotation, the experiment described here allows control of a wide variety of parameters; spins can be manipulated both optically and magnetically during the flight of atoms to the cell surface, the properties of which can also be modified using surface coating techniques.