Single-Spin Magnetomechanics with Levitated Micromagnets

J. Gieseler,1 A. Kabcenell,1 E. Rosenfeld,1 J. D. Schaefer,1 A. Safira,1 M. J. A. Schuetz,1 C. Gonzalez-Ballestero,2, 3 C. C. Rusconi,4 O. Romero-Isart,2, 3 and M. D. Lukin1 1Physics Department, Harvard University, Cambridge, MA 02318, USA. 2Institute for Quantum Optics and Quantum Information of the Austrian Academy of sciences, A-6020 Innsbruck, Austria. 3Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria. 4Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany. (Dated: December 24, 2019)

Realizing coherent coupling between individual spin degrees of freedom and massive mechanical modes is an outstanding challenge in quantum science and engineering. Such a coupling could be used to create mechanical systems with a strong quantum non-linearity, allowing preparation of macroscopic quantum states of motion [1]. In addition, mechanical systems can be used to mediate effective spin-spin interactions between distant spinqubits [2], enabling applications ranging from quantum information processing [3] and sensing [4-6] to tests of fundamental physics [7,8]. One particularly promising approach is to engineer a strong spin-mechanical coupling via magnetic field gradients [9][10][11][12][13]. Achieving strong spin-resonator coupling requires a combination of high quality mechanical resonators, strong magnetic field gradients, and spin qubits with very long spin coherence times.
In this Letter, we propose and demonstrate a new platform for strong spin-mechanical coupling based on levitated microscopic magnets coupled to the electronic ground state manifold of a single nitrogen vacancy (NV) center in diamond (Fig.1). The key idea is to utilize a levitated magnet that is localized in free space by electromagnetic fields. In such a system, dissipation is minimized since there is no direct contact with the environment. Specifically, we make use of a levitating mechanical resonator based on magnetostatic fields. This approach not only avoids clamping losses, but also circumvents photon recoil and heating associated with optical levitation [14][15][16] and is therefore predicted to yield large mechanical quality factors [17,18].
In addition, the levitated magnet naturally generates the strong magnetic field gradient that is required for spin-mechanical coupling. We specifically demonstrate the coupling to an individual NV-center, one of the most studied color centers in diamond [19]. Besides optical initialization and readout, the NV-center features long coherence times even at room temperature, which makes it an attractive candidate for scalable quantum networks in the solid state [3], quantum sensing [20][21][22] and quantum communication [23].
Before proceeding, we note that low dissipation mechanical resonators based on magnetic levitation have been explored previously [24,25]. However, experiments with superconducting levitation have so far been limited to millimeter-scale magnets [24,[26][27][28]. A recent experiment [29] demonstrated superconducting levitation of micro-magnets, but without spin-mechanical coupling and with much lower frequencies and Q-factors than shown here. Levitated spin-mechanical systems in which the spin is hosted inside the resonator have been implemented with nano-diamonds containing NV-center defects trapped in optical tweezers [30,31], Paul traps [32][33][34] and magneto-gravitational traps [35,36]. Nonetheless, the challenge remains to integrate these systems with strong magnetic field gradients, long coherence NVcenters and operation under ultra-high vacuum conditions. Our approach fulfills all these criteria simultaneously ( Fig.1).
Levitation of micromagnets We levitate single hard micro-magnets with a thin film of the type-II superconductor (sc), yttrium barium copper oxide (YBCO) (c.f. Fig.1). Since we do not apply additional magnetic fields, the micromagnet is the only magnetic field source. Thus, the magnetic flux through the YBCO film is determined only by the distance h cool between the magnet and the YBCO film, the orientation of the magnet θ cool , the magnetization of the magnet, and its radius a. After cooling the YBCO film below its critical temperature T c ≈ 90K, it becomes superconducting and magnetic flux that penetrates the film is frozen in. As a consequence, below T c , motion of the magnet induces currents in the superconductor that counteract changes in the magnetic field. This allows for stable 3D trapping using a procedure illustrated in (Fig.2a). The levitation height (h lev ) and trapping frequencies ω j (j = x, y, z) depend on the conditions during cooldown, which we can control by adjusting the relative distance between the superconductor and The coupling λ depends on the relative orientation of the NVcenter, ns, the magnetic moment nµ, the direction of motion of the magnet nm, the distance between the magnet and the NV-center r , the magnet radius a and the frequencies of the motion.
the particle during cooldown (h cool ). We observe the levitated magnet through a long working distance microscope objective that is positioned outside the vacuum chamber, and record its motion with a fast camera (Fig.2b). From the video frames, we extract time-traces of the particle's center-of-mass position in the camera coordinate system (x c (t), y c (t)) and calculate their power spectral densities (PSD). The distinct peaks in the PSD shown correspond to the three centerof-mass modes ω j /2π (Fig.2c). Fig.2d displays the center-of-mass frequencies as a function of the normalized levitation heighth lev = h lev /a for two particles with radius a 1 = 23.2 ± 0.7µm and a 2 = 15.5 ± 0.3µm, respectively. The lines are a fit to a power law f (h lev ) = f maxh −β lev , where f max is the frequency in the limit when the gap between the particle and the superconductor goes to zero. In our experiment, we find f max = (2.3 ± 0.4, 2.4 ± 0.4, 5.6 ± 1.0)kHz and β = (1.9 ± 0.1, 2.1 ± 0.1, 2.0 ± 0.1) for particle 1 and f max = (8.8 ± 1.1, 9.5 ± 1.1, 25.2 ± 3.3)kHz and β = (2.1±0.1, 2.1±0.1, 2.3±0.1) for particle 2, which is in good agreement with the expected value of β = 2.5 from a simple dipole model [SI]. The measured center-of-mass frequencies are comparable to those achieved with optical levitation [37] and significantly exceed motional frequencies in Paul traps [38] and magneto-gravitational traps [35,36]. However, the observed dependence of the maximum frequency on the particle radius is stronger than the dipole model's prediction of f dp max ∝ 1/a. We attribute this to the multi-domain nature of our particles and note that spherical particles as large as a ∼ 1µm, which we expect to be achievable with this technique, can be single domain [39]. (a) To levitate the magnets, we adjust the magnet-sc distance above the critical temperature Tc. Then we cool the sc below Tc to freeze in the magnetic flux from the magnet. After cooldown, we reduce the distance until the magnet begins to levitate. (b) We observe the magnet motion through a microscope objective and record its motion with a video camera. (c) Power spectral density extracted from video analysis of the magnet motion in the vertical (horizontal) direction, corresponding to the data shown as circles in (e). The inset shows a typical ringdown measurement of the y-mode (ωy = 0.839kHz) from which we extract the Qfactor. (d) Frequencies of center-of-mass motion as a function of the levitation height normalized to magnet radius. Dashed (solid) lines show the three center-of-mass modes for magnet 1 (2). (e) Q-factor as a function of trap frequency for magnet 2. Each symbol corresponds to a different levitation experiment and the different colors corresponds to the three different center-of-mass modes. Fig.2e shows the Q-factors of magnet 1 for three different levitation heights for all three translational modes. We measure the dissipation with ring down measurements, exciting one mode with an AC magnetic field and observing its energy decay (Fig.2c inset). From an exponential fit of the energy decay we extract the decay time 1/γ j and the Q-factor Q j = ω j /γ j for each mode. The measured Q-factors are around one million and depend only weakly on the trapping frequencies and thus on the levitation height. Air damping can be ruled out at our experimental conditions with pressures below 10 −5 mBar and the most likely source of dissipation is the magnet-superconductor interaction. Note that, even though this Q-factor is somewhat lower than what has been demonstrated with non-magnetic optically levitated [16] and nano-fabricated mechanical resonators [40][41][42][43], it represents the state of the art for magnetized resonators [44,45] and the ultimate limit, in particular for magnets with a < µm, is still an open question.
Coupling to NV-center Next, we demonstrate coupling the motion of a levitated micromagnet to the electronic spin associated with an individual negatively charged NV-center. In our sample, NV-centers are hosted inside a diamond slab and implanted d impl ∼ 40 nm below the diamond surface. The diamond replaces the glass slide of the previous experiment and is placed across the pocket that contains the magnet. The pocket is ∼ 80µm deep and the magnet radius is a 3 = 15.1±0.1µm. We levitate the magnet with the superconductor z md = 44 ± 5µm below the diamond using the same method as before. The NV-center is located at (x d , y d ) = (83, 29) ± 5µm with respect to the magnet center such that the distance between the center of the magnet and the NVcenter is |r | = (z md + a 3 ) 2 + x 2 d + y 2 d = 99 ± 5µm. The NV-center's electronic ground state has spin S = 1 with the lower-energy |m s = 0 level separated from the |m s = ±1 levels by a zero-field splitting D zf /(2π) ≈ 2.87 GHz and its symmetry axes n s aligned along one of four crystallographic orientations set by the tetrahedral symmetry of the diamond lattice. A microwave (MW) signal at the transition frequency ω MW , drives the transition |m s = 0 → |m s = ±1 which results in a decrease of the PL signal. The spin-dependent PL of the NV defect is due to a non-radiative intersystem crossing decay pathway, which also allows for efficient spin-polarization in the |m s = 0 spin sublevel through optical pumping [46]. The magnetic field dependent PL can therefore be used to optically detect magnetic fields [47], which we will use to sense the motion of the magnet (Fig. 3b). Fig. 3c shows the optically detected magnetic resonance (ODMR) spectrum of the NV-center with a fit to a Lorentzian corresponding to the |+1 transition. The spectrum is measured with a home-built fluorescence microscope that we integrated with the cryostat, replacing the long working distance objective outside the chamber with a high NA objective inside the chamber to maximize the collection efficiency of the fluorescence photons (Fig. 3a). In the presence of a microwave tone, the spin mechanical coupling λ g causes a variation in PL, since a displacement x of the magnet shifts the electron spin resonance by δω NV = (λ g /x xp )x, where x zp = /2mω x is the zero point motion, in this case of 24 ± 1fm. To measure the magnet's motion with the NV, we excite one of its modes with a broadband fluctuating magnetic field, which drives it into a quasi-thermal state and allows us to observe it as a peak in the power spectral density of the NV PL counts (Fig.3f). We confirm that the peak is due to the moving magnet with the camera (Fig.3d). The camera and NV measurements are taken sequentially. Notably, we observe a small systematic frequency shift of ≈ 1 Hz between the NV and camera measurements, which is likely due to the laser field turned on during the NV measurement.
The mean spectral power in the NV peak is c 2 NV = s 2 δω 2 NV , where s is the slope of the ODMR signal at the microwave frequency, which we measure by applying a calibration tone to the microwave signal. The mean spectral power in the camera measurement, x 2 , allows us then to extract the spin-mechanical coupling as λ g = x zp δω 2 NV / x 2 . To measure the coupling and confirm the thermal character of the driven mode, we consider the area under the PSD integrated over a time interval much shorter than 1/γ, and construct its distribution over repeated measurements. For both the camera and the NV measurements, the distribution agrees with an exponential distribution P (E) = β exp(−βE), where the decay constant β is the inverse of the variances x 2 and δω NV (Fig.3e,g). The resulting coupling strength is 48 ± 2mHz, in satisfactory agreement with the theoretical value for the gradient coupling to a dipole λ g = γ e µ0ρµa 3 r 4 x zp f g (θ) = 2π × (18 ± 3)mHzf g (θ) (Fig.4b). Here f g (θ) is a function on the order of 1 that depends on the relative position and orientation of the NV-center and the magnet.
Discussion We now discuss the prospects of using this system to achieve strong coupling. The minimal separation d min q = |r | − a between magnet and NV-center is given by the NV implantation depth and the onset of strong attractive surface forces that will make the magnet stick to the diamond surface. For a given separation and assuming that the frequency scales as ω j /2π = αa −n , the radius a = (n + 3)/(5 − n)d min q yields the maximum gradient coupling for a dipolar particle. A conservative gap d min q = 0.25µm, α = 15kHz µm, corresponding to our observations in Fig.2, and n = 1 for the dipole model, results in a = 0.25µm and λ g /2π ∼ 2.6kHz. Since the motional frequency can be reduced by adjusting the levitation height, one can even reach the elusive ultra-strong coupling regime λ g > ω j [48].
The cooperativity C = λ 2 QT 2 /(2πk B T ) > 1 marks the onset of coherent quantum effects in a coupled spin-phonon system. With a mechanical Q-factor of 10 8 , which has been demonstrated in levitated systems [49], the coupling exceeds the thermal decoherence rate Γ th /2π = k B T /(2π Q) = 0.8kHz at T = 4K. NV-centers in bulk diamond, such as the sample used in our experiment, can have up to second long extended coherence timesT 2 at these temperatures using spin manipulation such as multi-pulse dynamical decoupling sequences limited only by pulse errors [50,51]. The minimum spin manipulation (and therefore sensing) frequency in such sequences is set by the power spectrum of the noise. It is typically a few kHz for bulk diamond NV-centers, which is within reach for the mechanical frequencies in our current geometry. Hence, this system can reach the high cooperativity (C > 1) and even the strong coupling regime (λ > 2π/T 2 , Γ th ). Such a strong coupling enables groundstate cooling, quantum-by-quantum generation of arbitrary states of motion [9], and spin-spin entanglement Besides the translational degrees of freedom, levitated particles are free to rotate. For the hard magnets used in our experiment, the anisotropy energy strongly couples the particle orientation to the magnetization axis. The coupling leads to hybrid magneto-rotational modes, which correspond to a librational mode at frequency ω l = √ ω L ω I that precesses around the local magnetic field B 0 at the Einstein-deHaas frequency ω I = ρ µ V mag /(I 0 γ 0 ) due to the intrinsic spin angular momentum of the polarized electrons in the magnet. Here, B 0 is the sum of the field due to the sc and additional external fields and I 0 = 2ρ m V mag a 2 /5 is the moment of inertia, V mag being the volume of the magent and ρ m (ρ µ ) its mass (spin) density. Since the Larmor frequency ω L = γ 0 B 0 , ω l is higher than the translational mode frequencies even without additional fields and can be tuned with moderate magnetic fields ∼ 10mT to MHz frequencies. The high frequencies of the librational modes make them inaccessible to our current detection based on video analysis and DC magnetometry. Future work will explore these modes using optical interferometry and SQUID [53] or NV-AC magnetometry [4,38]. The rotational modes couple to the NV-center with a dipole-dipole coupling λ dp = γ e µ 0 m zp a 3 r 3 f dp (θ, ϕ), where f dp (θ, ϕ) is a function ∼ 1 that depends on the relative position and orientation of the NV-center and the magnet and m zp = γ e ρ µ /2V mag is the zero point magnetization of the Kittel magnon [54]. The weaker distance dependence yields λ dp = 0.4kHz for a a = 5µm magnet at 5µm distance from the NV-center (Fig.4b). This parameter regime is readily accessible with the experimental approach presented here, and it is sufficient to probabilistically cool the librational mode near its ground state [55] (based on the requirements for a harmonic oscillator mode ω 0 /2π ∼ MHz, λ/2π ∼ 100Hz, Q > 10 6 ).  [52] for h lev /a = 3. The lowest (solid) lying modes correspond to the center-of-mass and the highest (dashed) lying modes to rotational motion. Data points are the experimental frequencies for particle 1 Fig.2d. MHz frequencies are predicted for sub-µm particles. (b) Gradient (solid) and dipole-dipole (dashed) couplings, respectively. Straight lines show couplings for constant gapparticle size ratiod = d/a = 5.5 (color) andd = 1 (black). Curved lines (gray) show coupling for constant gap d = 250nm with a maximum coupling at a = d. The black data point is the experimental gradient coupling for particle 3.
Outlook These considerations indicate that our approach is a promising platform for quantum nanomechanics. Our experimental technique also allows us to achieve levitation with the superconductor in the Meissner state, and thus presents a path forward to observe precession due to the intrinsic spin angular momentum of the magnet with applications in highly sensitive magnetometry [56].
Since our mechanical resonator is all magnetic, we maximize the spin to mass ratio ρ µ / √ ρ m for a given magnetic material, which maximizes the spin-mechanical coupling. This leads to strong spin-mechanical coupling even for moderate experimental parameters. In addition, this system features libration modes, which are expected to reach an unprecedented spin-mechanical parameter regime even for magnets with a ∼ 5µm. Magnets of this size can be levitated with the experimental technique introduced here. The combination of high mechanical Q-factor, strong spin-mechanical coupling, and long spin-coherence is key for a range of applications such as magnetometers, accelerometers and gyroscopes [53], where the magnet is the sensor which is read out through the NV-center [4]. Furthermore, it may enable the exploration of new phenomena, including dynamics between a levitated nanomagnet and a single flux vortex [57][58][59], precession of a non-rotating magnet due to its intrinsic spin angular momentum [60], preparation of non-Gaussian quantum states [9], mechanically mediated quantum networks [3,61], detection of dark matter [56], and measuring the magnets internal degrees of freedom [54]. This work was performed in part at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrastructure Network (NNCI), which is supported by the National Science Foundation under NSF award no. 1541959. CNS is part of Harvard University. We gratefully acknowledge Frank Zhao for assistance with sample magnetization.