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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. laser optical detection of position microwave magnetic antenna force cantilever S magnetic ωr N tip spin defect(s), periodically bulk material hlipped at ωr+δ.
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.
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.
Quantum harmonic oscillator with the ladder operators method
The Schrodinger equation for the wavefunction |Ψi of a particle moving in a harmonic potential in a single dimension reads: ∂Ψ ˆ (1.1) i~ ∂t = H |Ψi .
with the following Hamiltonian ˆ ~2 2 1 2 2 (1.2) H = pˆ + mωr xˆ , 2m 2. where m is the mass of the particle, ωr the angular frequency of the HO, xˆ and pˆ the operators for the particle’s position and momentum respectively. Such equation can be solved with the ladder operator method developed by Paul Dirac [96]. In this method we use the creation aˆ† and annihilation aˆ operators to describe the motion of the oscillator: xˆ = a0 aˆ† + aˆ with a0 = ~ , p0 = pˆ = ip aˆ† aˆ 2mωr s 0 . Using these operators the Hamiltonian can be rewritten: Hˆ = ωr aˆ†aˆ + 1 . 2 Its eigenstates are the so-called Fock states or number states: aˆ† n |ni = √ |0i .
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. The Hamiltonian of this coupled system can be written using the ladder operators: H/~ = ωraˆ†aˆ + ωsSˆz + a0 aˆ + aˆ† GmγeSˆz (1.7) = ωraˆ†aˆ + ωsSˆz + λ(ˆa + aˆ†)Sˆz with λ = Gmγea0.
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.
Confinement of a charged dielectric particle in a Paul trap
Let us first describe the mechanism that allows Paul traps to confine charged particles. We will see that this mechanism, well-established for the CoM of an ion [9, 48, 98] can be extended to the angular degree of freedom of a particle with a macroscopic and anisotropic charge distribution [73].
Confinement of the CoM
Paul traps rely on electric forces to confine ions or charged particles. A simple harmonic electric potential is however not sufficient to confine a particle in all three directions of space. This is evidenced by the Laplace equation for the electric potential φ(x, y, y) ∂2φ + ∂2φ + ∂2φ = 0, (2.1) 2 2 2 ∂x ∂y ∂z.
that shows that if two directions are confined (eg ∂2φ/∂2x, ∂2φ/∂2y > 0), the third one is anti-confined (∂2φ/∂z2 < 0).
A Paul trap circumvents this issue by using an oscillating electric field. A voltage os-cillating at a frequency Ω is applied to electrodes in order to generate the electric potential φ in a certain region of space close to the center of the trap (x, y, z = 0): φ(x, y, z, t) = Vac cos(Ωt) + Vdc ηxx2 + ηyy2 + ηzz2 , (2.2) z02.
where VAC is the amplitude of the oscillating voltage, VDC a bias voltage, z0 a character-istic dimension of the electrodes and ηx, ηy > 0, ηz = −ηx − ηy geometric factors related to the shape of the electrodes. A typical electrodes configuration is a ring associated to endcap electrodes as shown in figure 1.3.a) but many variations are possible [111–113].
Table of contents :
Acknowledgements
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 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
