Levitation of micro-particles in a Paul trap

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Nuclear Mangetic Resonance

Rabi’s discovery was followed by its implementation in a solid state system: Bloch ex-plained [85] and observed -simultaneously with Purcell- that this precession induced in solid could be observed macroscopically. Contrary to previous experiments where atoms could not be retained or thermalized, Bloch and Purcell were able to apply a uniform mag-netic field strong enough to polarize spin ensembles within a solid. Then, they applied a perpendicular rotating magnetic field for a short time and could observe the magnetic field generated by the precessing spins in a nearby pick-up coil, after the oscillating field was turned off. Indeed, after the excitation the magnetic moment of the nuclear spins are tilted away from their equilibrium point, closer to the transverse plane and they rotate around the permanent magnetic field at their Larmor frequency. This oscillation occurs over a finite timescale due to:
1. thermal fluctuations (ie relaxation)
2. inhomogeneities of the permanent magnetic field, which cause the spins not to precess at the same Larmor frequency and dephase from one another.
The nascent field of Nuclear Magnetic Resonance (NMR) was refined in 1950 by Erwin Hahn with the spin echo (or Hahn echo) technique [86]. Instead of a single pulse, two pulses were applied. A first precession can be observed after the first pulse but the signal is lost due to dephasing. However when the second pulse is applied after a free precession time τ , the nuclear spins precession are rephased after a time of 2τ from the first pulse. As we will see later, this technique can be used with NV spins and allows more complex investi-gations of inhomogeneities and dephasing as well as protecting the NV spins against them.
Since then, NRM has considerably evolved, nowadays applications include imaging (for example, medical) and analysis of molecule to obtain chemical/structural information. Although sensitivity has been improved, the amplitude of the magnetic field generated by the spins scales down with the size of the observed ensemble and constitutes the main limitation to increase resolution of imaging or observation of smaller samples.

Optically detected magnetic resonance

The weak magnetic field that a spin ensemble generates is not the only way to detect it: for example, the discovery of the spin was made through mechanical means. Almost simul-taneously to NMR, Electron Paramagnetic Resonance (EPR) was discovered by Yevgeny Zavoisky looking at the absorption of a microwave field by salts.
Its sensitivity dramatically improved when it was combined with optical spectroscopy. In the early days of NMR it was shown that due to spin-dependent lifetime of optically excited states in crystals, spins can be both polarized and read-out through optical means [87]. The high sensitivity of Optically Detected Magnetic Resonance (ODMR) lead, in particular, to the observation of single spins in molecules [88]. Interessingly, single spins can be used as point-like magnetometer [89], or as a quantum memory to store an arbi-trary quantum state.
The different systems of single spins in the solid state can be rated on account of their lifetime, decoherence/dephasing rate, read-out fidelity and operating temperature. The Nitrogen-Vacancy (NV) center is an atomic defect in diamond, which stands out due to its stable photoluminescence at room temperature. It further enables optical spin initial-ization and read-out, while microwave excitation allows transitions from one spin state to another. This led to the observation of a single NV spin at room-temperature in 1997 [90]. NV centers have since been used for nanoscale magnetometry [89, 91, 92] and to perform the first loophole-free Bell inequality test [19].

Magnetic Resonance Force Microscopy

Magnetic Resonance Force Microscopy (MRFM) is an important field to understand the advances in spin-mechanics. Mechanical detection of spins in the solid state (other than ferromagnets) was first realised in 1955 [93]: if one applies not only a strong magnetic field but also a gradient, a force is exerted on the solid when the spins are polarized. However it was only with advancements in Atomic Force Microscopy (AFM) that it sparked interest to increase the spatial resolution of magnetic resonance imaging.
The basis of Magnetic Resonance Force Microscopy (MRFM) operation is described in figure 1.2.b). A similar cantilever to the one used for AFMs is used as a mechanical oscillator: a mirror (eg constituted of a simple metallic coating) is placed on one side of it so its position can be measured through optical interferences of a reflected laser beam, and a ferromagnet tip is added at its extremity. The cantilever is displaced so as to sweep a plane close to the surface of a bulk material containing spins close to its surface. A microwave is then used to flip the spin at the same rate than the resonance of the mechanical oscillator. When the magnetic tip is close to the spin, it exerts a peri-odic force on the magnetic tip that resonantly excites the cantilever. The displacement of the oscillator can then be optically measured by taking advantage of its high quality factor.
The first MRFM experiment was realized in 1993 [94], it showed nm-scale resolution in 2003 [95] and in 2004 single spin detection was achieved [41]. This last experiment was highly promising: by reproducing this experience with a long lifetime spin (like the spin of an NV center), one could envision using it to both actuate and measure the position of a mechanical resonator (the cantilever). Such coupling can be used to first cool down the resonator’s motion and -if it is strong enough- to bring it in a quantum state. Practical limitations however make this experiment particularly challenging: first, in order to obtain a strong enough coupling, the magnetized cantilever must be at a distance of a few tens of nanometers from a single spin[43]. The mechanical oscillator must also be placed at cryogenic temperatures to reduce its heating rate and must have a micron-scale size for a single spin to be able to displace it.

Trapped ions: an example of quantum harmonic me-chanical oscillator

In the Stern-Gerlach experiment, magnetic coupling displaces a free-falling beam of atoms depending on the orientation of their spin. Similarly, one can couple a single spin to a harmonic mechanical oscillator: an object, the position of which is confined in a harmonic potential. Under the right conditions, this coupling can actually be used to generate a quantum state of the mechanical oscillator. Here we will first describe this coupling in the case of a trapped ion.
A Harmonic Oscillator (HO) can be used to describe any energy minimum -at the first order approximation- and is therefore very pervasive in physics. According to quantum mechanics, the energy states of a HO can be quantized. Observing such quantized state for a mechanical HO however presents challenges, which trapped ions were able to leverage. We will first present the formalism that we use to described the HO in the quantum regime.

The emergence of trapped ions

Most Mechanical Oscillators (MO) have a high average phonon number at room temper-ature and do not lend themselves easily to quantum manipulation. Indeed 300kB/~ ∼7 THz, which means any oscillator of lower frequency will be in a thermal state at room temperature, that is in a non-coherent superposition of many Fock states. Observing a MO in the quantum regime therefore requires either to cool down the environment so that kBT < ~ω or to have the MO decoupled or isolated from the environment and a cooling mechanism to displace it from the thermal equilibrium. Once in the ground state one can use reverse the cooling mechanism to create an arbitrary state of higher energy (eg Fock state, superposition state) [34].
Trapped ions were the first system that met this criteria. At high vacuum the center of mass of a trapped ion constitutes a well isolated HO. Then, its motion can be manipulated by using laser or microwave fields to couple its motion to its internal degrees of freedom such as electron orbitals or spins. Single ions were first isolated in a Paul trap in 1980 [97]. The Paul trap was proposed by W. Paul [9]: it uses a dynamical electric potential to confine the ion and eventually earned him the Nobel prize in 1989. Typical electrodes that generate the electric potential are shown in figure 1.3.a). Cooling of ion ensembles was first showed in 1978 with an oscillator cooled to lower than 40K using laser light [10]. The limitation for the achieved temperature was found to be the linewidth of the optical transition compared to the frequency of the MO: the latter must be higher than the former to enable efficient cooling. Such regime is called the Resolved Sideband (RSB) regime and was reached ten years later thereafter enabling ground state cooling of a mercury ion [11]. In the RSB regime one can perform a Rabi oscillation on the sidebands and can not only cool down the MO to its ground state, but also map any superposition state from the electronic states unto a superposition of adjacent phonon states or entangle the MO state with the electronic state [15].

Spin-mechanical coupling

We now take a look at the manipulation of the motion of trapped ion in the quantum regime. We will here only describe a method using magnetic coupling to an electron spin. It should be noted that for trapped ions, the use of laser light and the Doppler effect [98] is actually more common. Coupling to the spin as presented here was proposed later [99] and only recently realized [ 100]. Still, this method uses a similar formalism and allows us to introduce the coupling we will use to control the motion of levitating micro-diamonds.
We consider the quantized energy of the Center of Mass (CoM) of a trapped ion containing a one halve electron spin. The energy states of the two degrees of freedom are depicted in figure 1.3.b). One can use a magnetic field gradient to couple the spin and the motion of the ion: the energies of the spin states will depends on the position of the ion because of the varying Zeeman effect.

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Center of mass spin-mechanics with NV spins

The same spin-mechanical coupling that allows control of an ion’s motion can actually be used with a massive mechanical oscillator coupled to a well controlled two level system. The field of opto-mechanics has already achieved impressive results regarding the control of a mechanical oscillator in the quantum regime [35, 37, 38]. However, the use of a two level system offers interesting prospects. In particular, one could transfer the high degree of control, which is now achieved for certain two-level systems in the solid-state, unto the mechanical oscillator.
Figure 1.4.a) depicts the states of a two level system (here, a spin) dressed by the Fock states of a mechanical oscillator. As explained for trapped ions, a strong spin-mechanical coupling allows coherent diagonal transitions between two different spin states and ad-jacent Fock states. The ability to generate an arbitrary state for the two level system can then be transferred to the state of the mechanical oscillator, if its decoherence and heating rate are slower than the coupling. The NV spin is an attractive system for such scheme because of its long lifetime and coherence time and as it can be fully controlled using optical and microwave fields [90]. Here, we will specifically describe schemes that have been proposed to couple the motion of a mechanical oscillator to an NV spin.

Coupling schemes

Interaction between a two level system and a large mechanical oscillator can be achieved through several means [42]. Here, we focus on the case of the NV center, for which two types of interaction have been used.
Strain coupling
One can first use the intrinsic strain in the diamond crystal to mediate the coupling: figure 1.4.b) depicts a cantilever made of bulk diamond, with a single NV spin embedded in the cantilever. The strain within the crystal actually depends on the position of the cantilever, because the cantilever applies a stress on the diamond as it deforms the crystal.
The strength of this coupling is however small: it has only been observed while applying a strong drive to the mechanical oscillator so that the large amplitude of its oscillations compensates for the low zero-phonon coupling rate. Coherent control of an NV spin was achieved using this method [103, 104] but back-action of the NV spin on the mechanical oscillator has not been observed.
Magnetic coupling
The NV spins can also be coupled to a cantilever through magnetic coupling [43]. Fig-ure 1.4.c) illustrates how one obtains such coupling: a nano-diamond with a single NV spin is positioned at the tip of a cantilever, while a magnetic structure is brought in its close vicinity to generate a strong magnetic gradient Gm.
The first experiment to observe this coupling was performed with the scheme de-scribed in figure 1.4.c) [44, 105]. A magnetic gradient of up to 4.5 104 T/m was achieved by positioning a magnetic structure below a nano-diamond attached to a silicon carbide nano-wire. The spin-mechanical coupling allowed for a single NV spin to measure excited oscillations of the nano-wire.
In another experiment, a reversed set-up was used: a magnetized cantilever was ap-proached close to the surface of a bulk diamond where shallow single NV spins were implanted [45]. A magnetic gradient up to 105 T/m was achieved and the thermal fluctu-ation of the mechanical oscillator could be measure by the NV spin. In both experiments, the coupling rate λ did not exceed the Hz range and control of the mechanical oscillator could not be achieved.

Levitated diamonds

An alternative system, which reduces the mass of the mechanical oscillator and suppresses its heating would be to use a levitating nano-diamond as the mechanical oscillator. Such particles, under high vacuum have reached record-high quality factor [46]. The center of mass motion of these levitating particles is usually manipulated using optical [49, 51, 53] or electric forces [54, 55]. Optical measurement of the particle’s position enables cooling of its motion through a feed-back loop [ 53]. Recently, this method allowed the observation of the quantized motion of a levitating silica nano-sphere [56].
Optical tweezers [47] is the most established trapping method and could provide a high enough frequency (∼ 100 kHz) to reach the sideband resolved regime for spin-mechanics with levitating diamonds. Several experiments have therefore been performed where nano-diamonds are levitated in optical tweezers, and embedded NV spins are manipulated through electron spin resonance [62–64].
However, these first experiments were performed under atmospheric pressure and, under vacuum conditions, a strong heating of the diamond was observed [64, 66, 67]. This heat-ing is attributed to absorption of the trapping beam by impurities in the diamond crystal [64, 67, 106]. Although the use of ultra-pure diamonds could partially solve this issue [106], even a few impurities will eventually limit the pressure achievable, given the high optical power used to trap the particle [106]. It should be noted that heating of the inter-nal degrees of freedom was also observed with levitated silica nano-spheres [107], but at a much higher vacuum due to the low absorption of silica at the wavelength of the trapping laser.
Apart from reducing the absorption of the levitated particle, it is also possible to opti-cally cool down the internal degrees of freedom using anti-stokes emission in rare-earth crystals [108, 109]. An interesting development is the realization of such cooling in an optically levitated nano-crystal [110]: if such particle can be efficiently combined with a nano-diamond, it could prevent it from heating under high vacuum.
To prevent a levitating diamond from heating, another scattering-free trap could be employed: optical fields are then only required to control and observe the NV spin, with a much lower intensity than what is used in optical traps. Both Paul traps [65] and magneto-gravitational traps [49] have shown stable trapping of nano-diamonds. The frequency of the center of mass oscillations in those traps is however for now limited to the kHz range [49, 52], thereby preventing one from attaining the resolved sideband regime.
Spin-mechanics experiments are still possible outside of this regime. In particular, mat-ter wave interferometry experiments have been proposed with a levitating diamond when the trap is turned off or loosened [57, 60]. The general idea is similar to the Stern-Gerlach or Rabi experiments described in section 1.1: a magnetic gradient is used to spatially separate two spin-dependent paths, which are then recombined by applying a π pulse on the NV spin.
Finally, levitation experiments are prone to the use of the angular degrees of freedom. Its investigation was made possible following the levitation of non-spherical particles in Paul traps [68] and optical tweezers [69, 70], in particular, light-driven rotation at MHz frequencies have been observed [69, 70]. As we will see latter in this thesis, the NV spin is in fact well-suited for coupling to the angular degrees of freedom.

Table of contents :

1 Basics of spin-mechanics 
1.1 Spin detection
1.1.1 The Stern-Gerlach experiment
1.1.2 Nuclear Mangetic Resonance
1.1.3 Optically detected magnetic resonance
1.1.4 Magnetic Resonance Force Microscopy
1.2 Trapped ions: an example of quantum harmonic mechanical oscillator
1.2.1 Quantum harmonic oscillator with the ladder operators method
1.2.2 The emergence of trapped ions
1.2.3 Spin-mechanical coupling
1.2.4 Coherent manipulation of the mechanical state
1.3 Center of mass spin-mechanics with NV spins
1.3.1 Coupling schemes
1.3.2 Levitated diamonds
2 Levitation of micro-particles in a Paul trap 
2.1 Confinement of a charged dielectric particle in a Paul trap
2.1.1 Confinement of the CoM
2.1.2 Confinement of the angular degrees of freedom
2.2 Trap set-up
2.2.1 Diamond visualization
2.2.2 Trapping electrode(s)
2.2.3 Injection of micro-particles in the Paul trap
2.2.4 Tuning the stability and confinement of the Paul trap
2.2.5 Vacuum conditions
2.3 Center of mass motion
2.4 Angular confinement: the librational modes
2.4.1 Origin of the confinement
2.4.2 Detection of the angular position
2.4.3 Librational modes in the underdamped regime
2.5 Limitations
2.5.1 Effect of the radiation pressure
2.5.2 Trap-driven rotations
2.6 Conclusion
3 Spin control in levitating diamonds 
3.1 The NV center in diamond
3.1.1 Atomic and electronic structure of the NV center
3.1.2 Orbital states and optical observation
3.1.3 Optically detected magnetic resonance
3.1.4 Impact of the magnetic field
3.1.5 Hyperfine coupling to nuclear spins
3.1.6 NV spins lifetime and coherence
3.1.7 Spin properties in diamond particles
3.1.8 Samples for the levitation experiment
3.2 Observation and control of NV centers in levitating diamonds
3.2.1 NV optical observation
3.2.2 External antenna
3.2.3 Integrated ring antenna with Bias T
3.3 NV spins to monitor the angular stability
3.3.1 Paul trap angular stability
3.3.2 ESR spectra in rotating diamonds
3.4 Coherent control and spin properties in levitating diamonds
3.5 NV thermometry
3.6 Conclusion
4 Spin-mechanical coupling 
4.1 Spin-induced torque
4.1.1 Theoretical description
4.1.2 Mechanically-detected Electron Spin Resonance
4.1.3 Calibration of the angular detection sensitivity
4.2 Linear back-action
4.2.1 Theoretical description
4.2.2 Ring-down measurements
4.2.3 Cooling of the thermal fluctuations
4.3 Non linear back-action
4.3.1 Bistability
4.3.2 Lasing of a librational mode
4.4 Spin-mechanics in the quantum regime
4.4.1 Spin-mechanical Hamiltonian
4.4.2 Coupling rate
4.4.3 Decoherence sources
4.4.4 Role of the geometry
4.4.5 Cooling efficiency
5 Levitating ferromagnets 
5.1 Magnet libration in a hybrid trap
5.1.1 Hard ferromagnet
5.1.2 Soft ferromagnets
5.2 Libration of iron rods
5.2.1 Levitation of asymmetric iron particles
5.2.2 Ring-down of the librational mode
5.2.3 Characterization of the mechanical properties
5.3 Hybrid diamond-ferromagnet particles
5.3.1 Nano-diamonds on iron micro-spheres
5.3.2 Nickel coating on micro-diamonds
General conclusion 
A Ring electrode
B Calculation of the cooling rate


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