Levitation of micro-particles in a Paul trap 

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The Stern-Gerlach experiment

The quantum nature of the spin is most striking in the Stern Gelarch experiment [80]: even if discrete quanta were observed before [81, 82] it evidenced for the first time two distinct quantum states for the magnetic moment of atoms ie the spin states.
The Stern Gerlach experiment, developed in 1922 is described in figure 1.1.a). It showed the discretization of the magnetic moment of atoms by observing the deviation of a beam of silver atoms going through a magnetic gradient. Would the magnetic moment of the atoms be a classical quantity, one would expect a continuous range of deviation as a function of its amplitude. The process actually taking place for each atom is an entangle-ment between its deviated position and its spin projection along the magnetic field. When its position is measured, only one of the two position-spin state can be obtained.
The Stern Gerlach experiment was later modified by Isidor Isaac Rabi to observe Nu-clear Magnetic Resonance (NMR) of LiCl molecules [83]. This experiment is described in figure 1.1.b): a beam of molecule is spread by a magnetic gradient in a first region and recombined by an inverse magnetic gradient in a second region. In-between an os-cillating magnetic field is applied. When its frequency is close to the Larmor frequency ν = γeB0 where B0 is the uniform magnetic field between the gradients, the spin -and so the magnetic moment- of each molecule can be inverted. At resonance, this results in a lower recombination of the molecules beams as the deviations due to the gradients add up instead of cancelling. The inversion of the magnetic moment is explained by its precession about the permanent magnetic field and the probability to inverse it can be calculated and was shown to follow the so-called Rabi oscillations [84].

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 experiment where atoms could not be retained or thermalized, Bloch and Purcell were able to apply a uniform magnetic field strong enough to polarize spin ensembles within a solid. They then 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. These 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 Er-win 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 inves-tigations of inhomogeneities and dephasing as well as protecting the NV spin 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 constitute the main limitation to increase resolution of imaging or observation of smaller samples.

Optically detected magnetic resonance

The weak magnetic field which a spins ensemble generates is not the only way to detect it: in particular, the discovery of the spin was made through mechanical means. Al-most simultaneously to NMR, Electron Paramagnetic Resonance (EPR) was discovered by Yevgeny Zavoisky looking at the absorption of a microwave 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 state, spins can be both polarized and read-out through optical means [87]. The high sensitivity of Optically Detected Magnetic Resonance (ODMR) in particular lead to observation of single spins in molecules [88]. Single spins can be used as point-like mag-netometer [89], or as a quantum memory to store an arbitrary 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 that stands out due to its stable photoluminescence at room temperature. It further enables spin initialization and read-out while microwave excitation allows transitions from one spin state to another. This led to observation of a single NV spin at room-temperature in 1997 [90]. It has since been used for nanoscale magnetometry [89, 91, 92] and to perform the first loophole-free Bell inequality test [19].

Magnetic Resonance Force Microscopy

Mechanical detection of spins in the solid state was first realised in 1955 [93]: if one ap-plies 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 advancement 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 interference 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 mechanical oscillator. When the magnetic tip is close to the spin, it exerts a periodic 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 motion and -if it is strong enough- to bring it in a quantum state. Practical limitations however make this experience 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 temperature to reduce its heating rate and must have a micron-scale size for a single spin to be able to displace it.

Trapped ion: an example of quantum harmonic mechan-ical 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 mechanical oscillator: an object confined in a harmonic oscillator. Under the right con-dition, 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 an ion.
A Harmonic Oscillator (HO) can be used to describe any minimum of the energy – 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 some challenge that trapped ions were able to leverage. We will first present the formalism we use to described the HO in the quantum regime.

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The emergence of trapped ions

Most Mechanical Oscillator (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/isolated from the environment and a cooling mechanism to displace it from thermal equilibrium. Once in the ground state one can use the cooling mechanism but reversed 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 the motion can be manipu-lated 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 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, using laser light and the Doppler effect [98] is more common. Coupling to the spin as we present it was only recently proposed [99] and realized [100]. Still, this method uses a similar formalism and we will later use the same coupling to control the motion of levitating micro-diamonds.
The spin-mechanical coupling is now splitted in two terms we have named “longitudi-nal” and “transverse”. The longitudinal term corresponds to a spin-dependent force on par with a position dependent Zeeman effect. The transverse term corresponds to an energy exchange between the spin and the MO, it will in practice cause a cooling or heating of the mechanical oscillator.
These two effects will play an important role in the latter work of this thesis but for now we will focus on the transverse term which enables coherent manipulation of the MO corresponds to an energy exchange between the spin and the mechanical oscillator: the spin can flip while emitting or absorbing a phonon of the mechanical oscillator. This will in practice result in a cooling or heating of the mechanical oscillator.
These two effects will play an important role in the latter work of this thesis but for now we will focus on the cooling/heating mechanism which enables coherent manipulation of the MO.

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 led 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 adja-cent Fock state. 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 it has a long lifetime and coherence time can fully be 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.

Table of contents :

Introduction 
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 ion: 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 degree of freedom
2.2 Trap set-up
2.2.1 Diamond visualization
2.2.2 Trapping electrode(s)
2.2.3 Injection of micro-particle 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 diamond 
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
3.2 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 measurement
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 hybrid trap
5.1.1 Hard ferromagnet
5.1.2 Soft ferromagnet
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-diamond
General conclusion 
A Ring electrode
B Calculation of the cooling rate

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