Spin-Cooling of the Motion of a Trapped Diamond

TL;DR

This paper demonstrates spin-dependent torque and cooling of a trapped microdiamond's motion via nitrogen-vacancy centers, enabling control of macroscopic quantum states and potential applications in high-precision sensing and quantum information.

Contribution

It reports the first observation of spin-cooling and spin-dependent torque on a macroscopic diamond, combining microwave and laser excitation to control and cool the diamond's motion.

Findings

Observation of spin-dependent torque on a trapped diamond

Demonstration of spin cooling of the diamond's libration

Evidence of bistability and self-sustained oscillations due to spin-mechanical coupling

Abstract

Observing and controlling macroscopic quantum systems has long been a driving force in research on quantum physics. In this endeavor, strong coupling between individual quantum systems and mechanical oscillators is being actively pursued. While both read-out of mechanical motion using coherent control of spin systems and single spin read-out using pristine oscillators have been demonstrated, temperature control of the motion of a macroscopic object using long-lived electronic spins has not been reported. Here, we observe both a spin-dependent torque and spin-cooling of the motion of a trapped microdiamond. Using a combination of microwave and laser excitation enables the spin of nitrogen-vacancy centers to act on the diamond orientation and to cool the diamond libration via a dynamical back-action. Further, driving the system in the non-linear regime, we demonstrate bistability and self-sustained coherent oscillations stimulated by the spin-mechanical coupling, which offers prospects for spin-driven generation of non-classical states of motion. Such a levitating diamond operated as a compass with controlled dissipation has implications in high-precision torque sensing, emulation of the spin-boson problem and probing of quantum phase transitions. In the single spin limit and employing ultra-pure nano-diamonds, it will allow quantum non-demolition read-out of the spin of nitrogen-vacancy centers under ambient conditions, deterministic entanglement between distant individual spins and matter-wave interferometry.

Equations

$$S_{\varphi }(\omega )$$

$$S(t)$$

$$I\ddot{\varphi }$$

$$\varphi (\omega )=\chi (\omega ){\Gamma }_T(\omega )$$

$$\chi (\omega )=\frac{1}{I({\omega }_{\varphi }^2-{\omega }^2+i\omega \gamma )}.$$

$$⟨{\Gamma }_T(\omega ){\Gamma }_T({\omega }^{\prime })⟩=2\pi \delta (\omega +{\omega }^{\prime })S_T(\omega )$$

$$S_{T}(\omega)=-\frac{2kT}{\omega}{\rm Im}\Big{[}\frac{1}{\chi(\omega)}\Big{]}$$

$$S_T(\omega )=2kT\gamma I.$$

$$S_{\varphi }(\omega )$$

$$\frac{1}{2}I{\omega }_{\varphi }^2⟨{\varphi }^2⟩=\frac{1}{2}kT$$

$$⟨{\varphi }^2⟩=\int S_{\varphi }(\omega )d\omega ,$$

$$H_{\mathrm{meca}}=\frac{1}{2}I{\omega }_{\varphi }^2{\varphi }^2+\frac{L^2}{2I}.$$

$${\hat{S}}_{\alpha }=\frac{1}{N}\sum _{i=1}^N{\sigma }_{\alpha }^i,$$

$${\hat{H}}_{\mathrm{NV}}=\hslash ND{\hat{S}}_z^2+\hslash {\gamma }_{e}N\mathbf{B}\cdot \hat{\mathbf{S}}$$

$$E=\frac{1}{2}I{\omega }_{\varphi }^2{\varphi }^2+\frac{L^2}{2I}-\hslash N{\gamma }_{e}B\varphi +\mathrm{cte}$$

$${\varphi }_0=\frac{\hslash N{\gamma }_{e}B}{I{\omega }_{\varphi }^2}.$$

$$\frac{\partial S}{\partial t}$$

$$\frac{\partial S_z^{+1}}{\partial t}$$

$$\frac{\partial S_z^{-1}}{\partial t}$$

$$\frac{\partial S_z^0}{\partial t}$$

$$\frac{{\partial }^2\varphi }{\partial t^2}$$

$$\frac{\partial S_z^{+1}}{\partial t}$$

$$\frac{\partial S_z^{-1}}{\partial t}$$

$$\frac{\partial S_z^0}{\partial t}$$

$$P(\Delta ,\varphi )$$

$$\varphi (\omega )$$

$$\varphi (\omega )={\chi }_{\mathrm{eff}}{\Gamma }_T(\omega ).$$

$$\delta S_z=\xi (\omega )\varphi .$$

$$\varphi =\chi (\omega )\delta {\Gamma }_T+\chi \hslash {\gamma }_{e}BN\xi (\omega )\varphi .$$

$$S_{\varphi }(\omega )$$

$$=$$

Full text

Since the celebrated Einstein and de Haas’ experiment in 1915 Einstein and de Haas (1915), much work has been carried out in the detection of atomic spins through mechanical motion Alzetta et al. (1967), culminating in the observation of a magnetic force from single spins Rugar et al. (2004); J. et al. (2007) and magnetometry at the nanoscale Burgess et al. (2013). Conversely, single spins and qubits have also been utilized to sense the motion of objects close to the quantum ground state. Single qubit thermometry of mechanical oscillators at the quantum level was realized using superconducting qubit coupled to membranes LaHaye et al. (2009); O’Connell et al. (2010) and nitrogen-vacancy (NV) centers coupled to cantilevers Arcizet et al. (2011); Kolkowitz et al. (2012); Lee et al. (2017). A crucial next step is to reach strong coupling between long-lived spins and mechanical oscillators, which will enable ground state cooling, as in tethered quantum opto-mechanical platforms Arcizet et al. (2006); Gigan et al. (2006); Schliesser et al. (2008), and the observation of quantum superpositions of macroscopic systems Rabl et al. (2009). One further prospect is the entanglement between multiple spins Rabl et al. (2010), with far reaching implications in quantum information science and metrology Barson et al. (2017). Obtaining coupling rates that surpass both the decoherence of the spin and of the mechanical system is however still a challenge for most state-of-the-art platforms. Recently, there has been renewed focus on levitating objects Chang et al. (2010); Romero-Isart et al. (2010) motivated by the low mass and high Q-factors that they offer, together with the possibility to cool their motion using embedded spins Yin et al. (2013). There is a strong analogy between this platform where spins move a levitating crystal and laser-cooled atoms where electrons move atomic nuclei. It may thus be forecast that a levitating particle containing a few long-lived spins will ultimately reach similar level of control as trapped ions Leibfried et al. (2003) with bright prospects for the aforementioned applications.

In this work, we report on a controllable torque induced by the spins of atoms embedded into a macro-object. Specifically, we couple the spin of many nitrogen-vacancy (NV) centers to the orientation of a trapped diamond particle. This coupling then enables us to show read-out of the NV centers spin resonance together with cooling and lasing of the diamond motion.

The crystallographic structure of NV centers is depicted in Fig.1-a). The spin-spin interaction between the two electrons in the NV center ground state lifts the degeneracy of the spin triplet eigenstates by $D=2.87$ GHz at room temperature. Such an interaction implies that the NV center has a preferential quantization axis that is along one of the four crystal axes $\langle 111\rangle$. In the presence of a magnetic field ${\bf B}$ at an angle $\phi$ with respect to the NV axis, the energy difference between the two energy eigenstates $|m_{s}^{\prime}=\pm 1\rangle$ is about $\gamma_{e}B\phi$, where $\gamma_{e}$ is the gyromagnetic ratio of the electron. Spin control can then be performed using optical and microwave excitation and the angular dependence of the NV spin energy eigenstates is expected to allow rotation and cooling of the diamond angular motion. Once in a magnetic state via a resonant microwave excitation, the NV center will tend to align the corresponding diamond crystalline axis to the magnetic field, as illustrated in Fig.1-a-i). Further, laser triggered relaxation from the excited state can then extract the work exchanged between the spin magnetic energy and the librational motion (see Fig.1-a-ii)).

In our experiment, harmonic librational (sometimes called torsional, pendular or rotational) confinement is provided both by the Paul trap and the particle asymmetry. We measure the diamond libration using the laser reflection off the diamond surface. The $\mu$m size roughness on our 15 $\mu$m particle enables a specular pattern to be detected at the particle image plane, which after mode-matching one of the many bright spots to an optical fiber yields an angular sensitivity of about 0.3 mrad$/\sqrt{\rm Hz}$ and a resolution of about 10 mrad/Mcounts/s (see Methods). Under vacuum conditions ($\approx$ 1 mbar), the signal power spectrum plotted in Fig.1-b) shows harmonic motion of the three librational modes with frequencies $\omega_{\phi}/2\pi$ ranging from 200 Hz to 1 kHz and with a damping rate of about 15 Hz. Fig.1-b) also shows an optically detected magnetic resonance (ODMR) spectrum for a diamond outside the trap, in the presence of a magnetic field $B\approx 30$ G. Eight transitions, corresponding to the projections of the B field onto the four NV orientations are observed, with typical spin decoherence rates $1/T_{2}^{*}\approx 7$ MHz.

We now measure the diamond rotation induced by the $N\approx 10^{9}$ NV electronic spins inside the diamond, with the same optical read-out as for the librational mode detection, as depicted in Fig. 2-a)-i). The expected magnitude of the spin-torque is $\Gamma_{s}=\hbar N\gamma_{e}BS_{z}\approx 10^{-19}$N.m. Here $S_{z}$ is population in one of the magnetic states, determined by the competition between the microwave and laser polarization (both at rates in the 100 kHz range). This torque gives a displacement of the particle angle in the trap $\delta\phi=\Gamma_{s}/I\omega_{\phi}^{2}\approx 10$ mrad, where $I\approx 10^{-22}$ kg.m2 is the particle moment of inertia. As can be seen in Fig. 2-a)-ii), sweeping a microwave around the spin resonances indeed enables conspicuous features to appear. Once in the magnetic state $|m_{s}^{\prime}=-1\rangle$ or $|m_{s}^{\prime}=+1\rangle$, the NV centers tend to align or anti-align the diamond orientation to the magnetic field, which is manifest in the anti-correlation between the detected intensity levels for all pairs of transitions. A standard ODMR is also measured under the same magnetic field amplitude and measurement time (see Fig. 2-a)-iii)) demonstrating perfect correlation in the frequency of the peaks in the two measurements.

This spin-mechanical effect is in fact much richer than a static spin-dependent torque. As shown in Fig.1-a), the NV centers are magnetized through a microwave tone whose detuning from the NV resonances changes as the diamond rotates. To first order, such a torque will increase (resp. decrease) the confinement of the Paul trap if the microwave is blue (resp. red) detuned from the spin-resonance at the equilibrium angular position. Further, since the spin lifetime is on the order of the libration period, it enables dynamical back-action. A delay between the NV magnetization and the angular oscillation, observed in Huillery et al. (2019), indeed induces a torque that depends on the velocity, in close analogy with opto-mechanical schemes Arcizet et al. (2006); Gigan et al. (2006); Schliesser et al. (2008) and with Sisyphus cooling of cold atoms. The net result is a pronounced cooling (resp. heating) of the diamond motion when the microwave is red (resp. blue) detuned from the spin resonance as sketched in Fig. 2-b). In order to observe such spin-spring and spin-cooling effects, we monitor the librational power spectrum as a function of the microwave detuning from the electronic spin resonance. Fig. 2-b) shows the result of measurements taken for three different microwave frequencies with a Rabi frequency of 10 kHz. A strongly modified spring and damping of the mechanical mode are observed. Assuming that the initial temperature is 300 K (see Methods), the resulting temperature after spin-cooling is here 80 K. Fig. 2-c) shows measurements of the damping rate and spring effects as a function of microwave frequency in good agreement with a theoretical model (see Methods). Cooling is at present limited by heating from the microwave excitation of the motion on the blue side. This could be eliminated by increasing the trapping frequency $\omega_{\phi}/2\pi$ above the NV spin transition linewidth.

We now make a step into a regime where the spin-mechanical interaction induces non-linear effects on the librational mode. With a stronger spin-torque (see Methods), Fig. 3-a)-i) displays the expected bistable behavior for the angular degree of freedom when the microwave is scanned across the spin resonance. The angle can be found at two metastable positions A or B depending on the history of the angular trajectory (see SI). The hysteresis behavior is indeed observed in the experiment, and shown in Fig. 3-a)-ii). The evolution of the particle orientation over time at a fixed microwave tone is also plotted in Fig. 3-a)-iii). Interestingly, the particle orientation jumps from site A to B in a seemingly unpredictable manner due to random kicks given to the particle. The average population at the angular position B can also be studied as a function of different microwave detunings and increases as the microwave is tuned towards resonance (see extended data and SI).

Let us now set the microwave frequency to the blue side of the spin-resonance in this strong spin-torque regime. Fig. 3-b)-i) shows the power spectral density as a function of the microwave pump power, where a transition from Brownian motion to a self-sustained oscillation is observed. Such a lasing-like action of a mechanical oscillator was observed in the first radiation pressure cooling experiments Arcizet et al. (2011) with proposed applications in metrology. The spin-mechanical gain that enables such lasing action here is provided by blue detuned microwave excitation which amplifies the angular motion up to a point where losses are compensated by the magnetic gain. The oscillator energy as a function of the microwave power is shown in Fig. 3-b)-ii). Lasing threshold is observed at 6 mW of microwave excitation. Another signature of mechanical lasing is shown in Fig. 3-b)-iii), which displays the probability distribution (PD) of the angular degree of freedom with and without microwave. Under blue detuned microwave excitation, the probability distribution departs from the Gaussian process (red curve) for Brownian motion, and turns to the characteristic PD of a coherent oscillation (blue curve). This effect shows that the librational mode can operate stably, deep in the non-linear regime and highlights further the analogy between the present spin-mechanical platform and opto-mechanical systems.

Coupling individual spins to the motion of a macroscopic oscillator will have far reaching applications in fundamental science, quantum information and metrology. The present spin-dependent torque itself may be employed for detecting atomic defects with electrons spins that cannot be efficiently detected through ODMR. Further, the approach may also be applied to other torsional nano-mechanical platforms Wu et al. (2016); Huillery et al. (2019), which can exploit the long NV spin-lattice relaxation at low temperatures for longer interrogation times and efficient cooling. Last, operating in the sideband resolved regime where $\omega_{\phi}/2\pi\gg 1/T_{2}^{*}$ can be realized after modest improvements to the present set-up. We estimated that using a 1 $\mu$m diameter pure diamond grown by chemical vapor deposition (CVD) attached to a 1 $\mu$m diameter ferromagnet, would enable entering the sideband resolved regime for this hybrid structure. Librational frequencies $\omega_{\phi}/2\pi$ above 200 kHz have indeed been observed recently Huillery et al. (2019) and NV centers with $1/T_{2}^{*}=50$ kHz electron-spin decoherence rates can readily be obtained in CVD grown microdiamonds enriched in 12C. Entering this regime would offer immediate prospects for ground state cooling the diamond libration and for multipartite spin-entanglement and would provide strong impetus to bridge the gap between trapped particles and trapped atoms.

Acknowledgments We would like to thank R. Blatt, C. Voisin, Y. Chassagneux, E. Baudin and S. Deléglise for discussions.

Author Contributions T. D. and P. H. contributed equally to this work. T. D, P. H., L. N. and G. H. performed the spin-torque experiments; T. D, P. H., and G. H. analyzed the data and performed the modeling with assistance from L. N., and G. H., T. D and P.H. wrote the manuscript. All authors contributed to the interpretation of the data and commented on the manuscript.

.1 The mechanical oscillator

The considered mechanical oscillator is the librational mode of a microdiamond levitating in a Paul trap. It was shown in Delord et al. (2017b, c) that the Paul trap induces a confinement of the angle due both the trap and diamond particle asymmetry. In the manuscript, we directly observe the three librational modes under vacuum using the speckle detection detailed in the Method section. We fit the power spectral density of the librational modes in Fig. 1-b) of the main text using the equations that are derived from the following linear response theory.

In the following we consider the simple case of a single excited angle, assuming that the other angles are not coupled to it. The librational motion of the diamond in the Paul trap is ruled by the equation of motion for the angle between the main axis of the diamond and the main axis of the trap Delord et al. (2017c). Noting this angle $\phi$, we get

[TABLE]

where $I$ is the moment of inertia, $\omega_{\phi}$ is the angular frequency, $\gamma$ is the damping rate due to collisions with the background gas and $\Gamma_{T}(t)$ is the associated Langevin torque.

Fourier Transforming this equation yields

[TABLE]

where

[TABLE]

The Langevin torque obeys the relation

[TABLE]

where

[TABLE]

when the number of phonon excitations is larger than $1$. One finds

[TABLE]

The librational spectrum is then found to be

[TABLE]

This formula describes very well the observed librational motion shown in Fig. 1 b) of the main text. Integrating this expression over $\omega$, we obtain

[TABLE]

where

[TABLE]

in agreement with the equipartition theorem. In principle, the area below the Lorentzian curves observed in Fig. 1 b) gives us direct access to the temperature (see Methods, for a discussion on temperature calibration).

.2 Rotating a levitating diamond using the spin of nitrogen-vacancy centers

We now discuss the theory behind the spin-induced torque from the nitrogen-vacancy centers. In the secular approximation of the Paul trap potential, the Hamiltonian describing the angular motion takes the form

[TABLE]

In our experiment, the angular position of the particle in the trap is shifted due to a torque induced by spins of NV centers in the presence of a magnetic field, which will add a magnetic potential energy.

The NV centers is a spin triplet system due to the presence of two paired electrons. The singlet state lies at much higher energies and only contributes to the triplet state polarization via the green laser. The dipolar interaction between the two electron spins lifts the degeneracy between the $|m_{s}^{\prime}=0\rangle$ and $|m_{s}^{\prime}=\pm 1\rangle$ states, which sets a preferential quantization axis that is not given by the external magnetic field as it would normally be for free electrons Doherty et al. (2011). The degeneracy is already lifted by $D=2.87$ GHz at room temperature in the absence of magnetic field. This property is at the core of the proposals put forward in Delord et al. (2017c); Ma et al. (2017) to let the NV centers act on the angle of a levitating diamond.

Here, we consider a large ensemble of NV centers (about $10^{9}$ NV centers), which are all along one of the 4 $<111>$ crystalline axes. Under an external magnetic field, the orientation dependent Zeeman effect lifts the degeneracy between these NV spin energies. Therefore when the microwave frequency is close to the resonance corresponding to only one axis, only those spins that are along this axis are in a magnetic state.

To estimate the maximal torque exerted by $N$ magnetized NV centers onto the diamond, one must search for the steepest variation of the Zeeman splitting with the NV angle with respect to the B field. Let us define the average spin operators as

[TABLE]

where $\alpha$ denotes the three spin directions $x$, $y$ and $z$ and $\sigma_{\alpha}^{i}$ are the Pauli matrices of the NV number $i$ corresponding to the direction $\alpha$.

The hamiltonian for the NV ensemble reads

[TABLE]

where $\gamma_{e}$ is the electron gyromagnetic factor, $\hbar$ the reduced Planck constant and $\bf B$ the external magnetic field. The first term describes the aforementioned spin-spin interaction which is the dominant energy contribution and the second term, the Zeeman energy.

The greatest angular variation of the three eigenenergies with angle can be found between the ground and lowest excited state at approximately $\phi_{\rm opt}=\pi/4$ under moderate magnetic fields (less than 100 G)Ma et al. (2017). The derivative of the energy as a function of the angle can also be estimated close to this angle and can be found to be around $\gamma_{e}B$.

We can then write the Hamiltonian in the new eigenstate basis, and consider the NV electronic spins as being a simple two-level system in the $\{|m_{s}^{\prime}=0\rangle,|m_{s}^{\prime}=-1\rangle\}$ basis. Assuming that the spins are promoted to the excited $|m_{s}^{\prime}=-1\rangle$ spin state by a resonant microwave, and linearizing the energy dependance with respect to $\phi$ at $\pi/4$, the total energy is changed to

[TABLE]

The center of the angular potential is thus shifted to

[TABLE]

The number of NV centers that are magnetized during a typical microwave scan is at most one fourth of the total number of NV centers due to the 4 involved NV orientations. Including the detrimental influence of the transverse component of the magnetic field on the spin polarization, only $50\%$ of the population is transferred to the $|0\rangle$ state by the laser before being excited by the microwave, so in total, only about $10^{8}$ NV centers are in a magnetized state. Taking a $15\mu$m diameter diamond, using typical librational mode frequency of $\omega_{\phi}/2\pi=500$ Hz and an external magnetic field of 100 G, we obtain an angle shift of around $\phi_{0}=5$ mrad.

Let us note that the spin-induced angular displacement will significantly increase when decreasing the particle size. Indeed, the moment of inertia scales like the fifth power of the particle diameter. Magnetizing a single spin inside a 500 nm diameter diamond with similar trapping conditions than the presented experiments would lead to an angular displacement of 0.5 mrad. Our sensitive detection technique (via the speckle) imposes the minimum size of the employed particles to be the illumination wavelength, but detecting such angular displacement within the spin lifetime does not seems out of reach for future experiments.

The above analysis however neglects an important effect that is at the core of the spin-mechanical coupling, that is the back-action of the oscillator onto the spin state. The root of this effect is that the magnetic state population also depends on $\phi$. The reason for this is that when the diamond rotates, the B field projection onto the NV axis changes. The microwave will then be brought in or out of resonance, which will in turn change the total magnetization. This mechanism is truly analogous to the radiation pressure back-action force onto moveable mirrors in a high-finesse optical cavity. There, when the light enters the cavity, it can push the mirror and displace it enough so that the cavity resonance condition is changed. This change in the resonance condition can in turn reduce or enhance the intra-cavity light intensity Aspelmeyer et al. (2014), which can increase again, or reduce, the radiation pressure force. The analogy with the present system is very strong since the interpretation of our experiment can be carried our by essentially replacing the NV spin degree of freedom with the cavity field and the mirror position by the particle angle.

Let us analyse the mechanisms at play including such, potentially delayed, interaction between the spin and librational degree of freedom.

.3 The linear regime : Spin-spring effect and spin-cooling

The above mentioned spin-mechanical coupling mechanism was discussed in Delord et al. (2017c); Ge and Zhao (2018).

The analysis of Delord et al. (2017c) was carried out for diamonds in a Paul trap in the sideband resolved regime, where the motional frequency is larger than the spin decoherence rate. Reducing the mean phonon number in the librational mode can be done in a very similar way than with single ions/atoms by tuning the microwave frequency close to the motional red sideband transition. Fast cooling can be realized by using a green laser that polarises the magnetized spins back to the ground state. In a regime where the spin-excitation rate on the red-sideband is faster than any heating mechanism of the angular degree of freedom (typically collisions with the background gas), such a combination of microwave and laser excitation can lead to ground state cooling of the librational mode.

In our diamonds, many paramagnetic impurities are present, which induces inhomogeneous broadening of the NV spin resonance transitions by about 7 MHz, making it difficult to reach the sideband resolved regime. Another difficulty is that the particles that we trap have a large moment of inertia (in the $10^{-22}{\rm kg/m^{2}}$ range). Technical limitations on the trap parameters (size and voltage) then impose an upper limit to the librational frequency.

Using a large particle however means that we can use many NV centers so the spin-dependent torque can be very large. Further, although we are far from the sideband resolved regime, since $T_{2}^{*}=50$ns and the librational mode period is in the millisecond range, significant cooling of the oscillation can be obtained because of a significant delay between the magnetization of the NV centers (measurements of the magnetization rate are shown in the extended data) and the librational motion. Such spin-cooling mechanism is sketched in Fig. 2-b). We now quantify the cooling efficiency using a numerical model.

.3.1 Numerical model

To fit the experimental data Fig.2-c), we study the dynamics of the coupled angle and spin degrees of freedom using Bloch equations. The green laser induces decay from the $|m_{s}=\pm 1\rangle$ levels to the ground state $|m_{s}=0\rangle$ at a rate $\gamma_{\rm las}$ of about tens of kHz. We note $\Omega/2\pi$, the Rabi frequency of the microwave signal and $\Delta/2\pi$ the frequency detuning with respect to the $|m_{s}=0\rangle$ to the $|m_{s}=-1\rangle$ transition at equilibrium.

The dynamics of the paramagnetic spin bath ($\mu$s) is faster than the timescales of the motion and spin-spring shifts so we can treat the paramagnetic impurities as a general Markovian reservoir that limits the spin coherence time $T_{2}^{*}$ to 50 ns Delord et al. (2018), much below the $T_{1}$ time of the spin populations ($\approx ms$ in our experiment). We write $S$ the mean expectation value of the Pauli operator $\hat{\sigma}^{-}_{i}$ of the individual spins on the subspace {$|0\rangle$,$|-1\rangle$}. We also write $\hat{S}_{z}^{\beta}=|\beta\rangle\langle\beta|$, the population in the three spin states $|\beta\rangle$ and $S_{z}^{\beta}$ its expectation value. The evolution of these expectation values is ruled by the following set of equations

[TABLE]

The last equation ensures that the spin population in this manifold is preserved. The important ingredient here is the angular dependent detuning that appears in the first equation. These coupled equations of motion are then coupled to the angular motion via

[TABLE]

where $\Gamma=\hbar N\gamma_{e}B/I$. This equation describes the angle evolution under the NV spin torque. We note that if the spin population in the two magnetic excited states is the same, no torque is applied to the diamond. $\Gamma_{T}(t)$ term is the delta-correlated Langevin noise term.

In the experiment, the polarization dynamics and librational motion evolve on $ms$ timescales, much longer than the $T_{2}^{*}$ time ($ns$). We can thus adiabatically eliminate the evolution of $S$ and use rate equations that describe the spin populations only. They read

[TABLE]

where $P(\Delta,\phi)$ is given by

[TABLE]

where $\sigma=1/T_{2}^{*}$. $P(\Delta,\phi)$ is a Lorentzian function quantifying the change in the polarization rate of the NV centers in the magnetic state. Using this model, the spin resonance lineshape would not accurately model our experiment, where the coupling of the NV centers to a paramagnetic spin bath gives rise to a Gaussian lineshape. Here, since the spin bath dynamics is fast compared the measurement time, it does not play any other role than changing the actual lineshape, we can thus instead write $P$ as a Gaussian function

[TABLE]

with a width given by $\sigma\sqrt{{\rm ln}(2)}/2\pi$.

The solutions for $\phi$ obtained from this numerical model can be used to estimate the effective damping and spin-spring effects and hence the temperature.

.3.2 Relation between the spin population fluctuations and the final temperature

To understand how to relate the results from our numerical simulations to the effective temperature, we can again write the equation of motion for $\phi$ in the Fourier domain

[TABLE]

As was done in Arcizet et al. (2006b), in the linear regime, one can also introduce an effective susceptibility $\chi_{\rm eff}$ such that

[TABLE]

In order to find $\chi_{\rm eff}$, one needs a relationship between the angle and the spin population. To do this, we decompose the spin and angles as the sum of a mean value and a fluctuating component, as $S_{z}=\overline{S_{z}}+\delta S_{z}$ for instance. Linearizing the Bloch equations will then allow to get a relationship between $\delta S_{z}$ and $\phi$

[TABLE]

where $\xi$ is a complex number that depends on all the parameters of the spin-mechanical interaction. Injecting this relation back into Eq. (13), we get

[TABLE]

The power spectral density of the librational motion $\phi$ can finally be found to be

[TABLE]

It was shown in Arcizet et al. (2006b), that the real part of $\xi$ gives either an extra binding or anti-binding confinement (depending on its sign), also know as a spring effect. The imaginary part of $\xi$, is related to delayed action of the spin onto the angle and provides the cooling/heating mechanism. Provided that the dynamics is still that of a damped harmonic oscillator, effective damping and spring effects can be found.

We checked this by solving the full set of numerical simulations. The set of experiments shown in Fig. 2-d) where performed by exciting parametrically the librational motion using the internal spins inside the diamond. The ring down of the librational mode was then observed and was very well approximated by exponentially decaying curves (see Methods).

For the most part of the paper, we model the experiment numerically using Monte-Carlo simulations and the XMDS package Dennis et al. (2013). For the spin-cooling and spin-spring effects shown in Fig. 2.c), we used the measured librational frequency $\omega_{\phi}=2\pi\times$ 480 Hz, the measured damping rate $\gamma/2\pi=16$ Hz and the magnetic field splitting $\gamma_{e}B=260$ MHz as parameters. The number of NV centers, polarization rate, Rabi frequency and the longitudinal spin lifetime $T_{1}$ in the excited state are not known with a high precision and are left as free parameters. The number of NV centers in particular can be deduced from the detected count rate, but this can only be an order of magnitude estimate since we do not know precisely the collection efficiency of the apparatus, how the laser light is coupled to the NV centers in the diamond and the PL is refracted by the diamond. Doing this we found around $10^{9}$ NV centers in total, but use it as a free parameter when determining the best value in the experiment. Similarly, the moment of inertia is not known precisely due to the imprecision in the quoted diamond diameter $d$ and shape, which translates into a large error in the moment of inertia (which scales like $d^{5}$). We thus set the parameter $\Gamma$ as a free parameter. Numerically, we turn on a microwave at a time t=20 ms, and record the librational mode ring down. When the microwave is quasi-resonant, it induces a spin torque which displaces the angle from the equilibrium position. To find the optimum values for the resulting cooling and spring-constants, and to obtain a fit to the experimental data, we analyze $S_{z}$, $\theta$, $\gamma_{\rm eff}$ and $\omega_{\rm eff}$ as a function of the microwave detuning from the spin transition. The numerical results of the Monte-Carlo simulation are adjusted to match the experimental parameters corresponding to the Fig.2-c) of the paper. Doing a Monte-Carlo simulation was important to describe damping of the oscillations on the blue-side. Without the Brownian motion of the angle included in the numerical simulations, strong phonon-lasing (described below) takes place in the regime where the damping on the red matches the experimental data. The Brownian motion induces phase diffusion of the laser, which effectively damps out our averaged signal.

Using these numerical data, we can then fit a cosine function multiplied by an exponentially decaying curve to deduce both the spin spring and damping rates. We observe an increase/decrease in the trapping frequency and a decrease/increase of the damping rate when the microwave is tuned to the blue/red. The analytical formula is in very good agreement with the ring down numerical data with $1/T_{1}=0.6$, $\gamma_{\rm las}=2$ and $\Omega/2\pi=30$ in kHz units, as free parameters.

We can then deduce effective damping and spring coefficients when the microwave is tuned to the red side of the spin resonance, and write the net susceptibility of the levitating diamond angle to the spin-torque as

[TABLE]

from which we can deduce the final temperature using the equipartition theorem.

The imaginary part of $\xi(\omega)$ determines the damping term in the mechanical susceptibility, so the delay between the librational motion and the depolarization is the cause of cooling/heating. This model reproduces qualitatively the experimental results. To optimize the cooling, the laser induced re-polarization rate should thus be on the order of the trapping frequency. Again, this simplified analysis neglects the $T_{1}$ decay, which tends to demagnetize the whole spin ensemble on millisecond times scales. The librational frequency must thus be on the order of the $T_{1}$ time, which is the main limitation to the cooling efficiency in the experiment. Increasing the microwave and laser powers makes the librational mode evolution become bistable when the microwave is tuned to the red, and motional lasing occurs on the blue as we describe next.

.4 The non-linear regime : Bistability and phonon-lasing

The above calculations were performed in the regime where the spins were weakly polarized by the laser and microwaves. Here, we use the simplified two-level model to express the effective spin-mechanical potential energy in the non-linear regime. We will explain Fig. 3 of the main text, where bistability and phonon-lasing of the librational modes are observed. Let us first discuss the regime where the microwave is tuned to the red.

.4.1 Bistable regime

The phenomenon of bistability has been analyzed and experimentally realized by many research groups and can be observed in a vast range of experimental settings. Bistability is associated with a wealth of interesting phenomena Lugiato (1984), one prominent example being the possibility to observe driven-dissipative phase transitions Minganti et al. (2018). One situation that has been extensively studied is when atoms are placed in an optical cavity, which was realized in the early days of cavity Quantum electrodynamics Rempe et al. (1991). The present system bears strong analogy with these experiments, since the NV spin degree of freedom plays the role of the cavity field in the experiments.

In order to describe this observed bistability, we can start from a simple two-level model. In the steady state limit, the Bloch equations give the following equation for $S_{z}$

[TABLE]

Using the equation of motion for the angle in the steady state limit gives

[TABLE]

where $\kappa=(1/T_{2}^{*})^{2}+4\Omega^{2}/(\gamma_{\rm las}T_{2}^{*})$ and $I=\frac{\Gamma}{\omega_{\phi}^{2}}\frac{2\Omega^{2}}{\gamma_{\rm las}T_{2}}$. This relation is a third order polynomial equation for $\phi$. We solve it numerically using the parameters of our experiment and found that when $\Delta<0$ it can have two real and one imaginary solutions. Physically, this effectively corresponds to a stable and unstable (saddle point) for the angle. The result of the numerical simulations is shown in Fig. 3-a) as a function of the detuning, where a characteristic S-shape curve is observed. When the microwave is tuned to the blue of the spin resonance transition, two stable solution for the angle (at positions A and B) are seen to co-exist.

The probability of being in either one of these two points depends on the history of the angle motion, which often implies a hysteretic behavior. In the experiment, we expect that if the microwave is scanned from the blue to the red, the particle will follow the upper angular trajectory (where the saddle point is B) up to the end, and finally drop to the small angle solution. On the other hand, if the microwave is scanned from the red to the blue, the particle will stay in the saddle point A much longer until the spin torque pulls it towards large angles. If the scan is performed on times scales of the librational period then hysteresis behavior is expected. This hysteresis behavior was observed in the experiment (Fig.3-b)) and numerical simulations of the coupled Bloch and torque equation very well describe the data.

If now the microwave is parked to the red of the spin resonance, there may be sudden stochastic jumps between the two metastable angular positions. External noise sources may indeed push the angle from one point to the other so that sharp jumps from the two stable positions A and B may occur on times scales of the motional period, and remain at the stable points for times scales dictated by the amplitude of the external driving noise. The effect was observed in the experiment, and is shown in Fig. 3-c) of the main text. The origin of this jump is likely to be dominated by collisions with the gas particles, but any other sources of noise, such as the microwave signal generator amplitude noise or laser noise, could also explain our observations.

We also measured the relative time spent in each of these stable dynamical potential wells in these condition (long measurement times) as a function of the microwave detuning. The evolution of the histograms showing the time spent in each of those wells is shown in the extended data Fig. 9.

These results point towards a naive interpretation that the two local potential minima are effectively separated by a potential barrier whose height depends upon the microwave detuning. Going further in this interpretation, one could estimate that the jump rate between two local minima from Kramer’s theory Ricci et al. (2017); Rondin et al. (2017). In the overdamped damped regime, the transition rate from the well A to B is :

[TABLE]

where $\omega_{i}^{A,B,C}$ correspond to the angular confinement frequencies of the mode $i$ at the point A,B or C. The reverse rate is obtained by swapping indices A and B. In our measurements the particle librational is underdamped so the transfer is slowed down by the slow transfer of energy between the librational mode and the bath, but the prefactor for the transition rates from A to C and C to A is the same, up to a difference in the librational mode frequencies in A and B Ricci et al. (2017); Rondin et al. (2017).

Here, we are interested in the ratio $R=P_{A}/(P_{A}+P_{B})$, which quantifies the normalized averaged population in the site A. This ratio is mainly governed by the difference in the potential height and effective temperature, including all noise sources, so we assume that the two frequencies at the points A and B are of the same order of magnitude for simplicity.

[TABLE]

Such a dependence of $P_{A}$ with the temperature of the bath is also confirmed by the Monte-Carlo simulations and offers a rich playground to studies of thermalization in out-of-equilibrium systems.

.4.2 Librational lasing

Phonon-lasing takes place when stimulated emission overcomes absorption of phonons from the mechanical mode. This effect has recently been observed using the mechanical modes of micro-toroids in Grudinin et al. (2010). Here the phonon laser is replenished via the dual pumping mechanism provided by the microwave (why magnetizes the NV states) and the green laser that polarises the NV in the ground state. In addition, delay also plays a crucial role in the phonon-lasing process. Indeed, the critical point where phonon-lasing occurs is to be found when the damping $\gamma_{\rm eff}$ becomes negative, that is when the delay makes the system unstable. Fig. 6-a) shows the result of numerical simulations in this regime. Numerical simulations show the damping and spin-spring shifts with a torque coefficient $\Gamma={0.7,0.9,1.2,1.5}$ kHz from traces i) to iv) and resonant microwave Rabi frequency of 100 kHz. On the blue side, the damping coefficient becomes negative and the spring effect stabilizes to a value that is slightly greater than the librational mode frequency defined solely by the Paul trap (here 480 Hz). This is observed experimentally in the Fig. 3.b)-i) of the main text. In this regime, the oscillator runs unstable and self sustained oscillations then take place. In the transient regime, the librational mode undergoes amplification, until it settles to a constant coherent oscillation. This is shown Fig. 6-b), where a microwave signal detuned by 7 MHz from the spin resonance spin transition is applied at a time t=20 ms. Here we set $\Gamma$ to 1.5 kHz. At this detuning, the signal is amplified for 120 ms, up to a point where a regime of self-sustained oscillation sets-in with an angular amplitude that settles at 0.05 rad.

Phonon-lasing occurs at a threshold value that depends on the ratio between losses and gain. The losses are related to residual gas damping and the longitudinal spin relaxation time $T_{1}$ limits the population inversion. Numerical simulations of the librational mode velocity are plotted as a function of microwave Rabi frequency in Fig. 6-c) for three different values of the spin-lattice relaxation rate $1/T_{1}=\{1,2,3\}$kHz for trace i)-ii) and iii) respectively. A threshold behavior is observed at microwave Rabi frequencies of $24,38$ and 59 kHz for the traces i),ii) and iii). This theory is in good agreement with the experimental observations shown in Fig. 3.b)-ii) in the main text.