Published papers:
Buckyball quantum computer: realization of a quantum gate , M.S. Garelli and F.V. Kusmartsev, European Physical Journal B, Vol. 48, No. 2, p. 199, (2005)
Fast Quantum Computing with Buckyballs, M.S. Garelli and F.V. Kusmartsev, Proceedings of SPIE, Vol. 6264, 62640A (2006)
Theoretical Realization of Quantum Gates Using Interacting Endohedral Fullerene Molecules:
We have studied a physical system composed of two interacting endohedral fullerene molecules for quantum computational purposes. The mutual interaction between these two molecules is determined by their spin dipolar interaction. The action of static magnetic fields on the whole system allow to encode the qubit in the electron spin of the encased atom.
We herein present a theoretical model which enables us to realize single-qubit and two-qubit gates through the system under consideration. Single-qubit operations can be achieved by applying to the system resonant time-dependent microwave fields. Since the dipolar spin interaction couples the two qubit-encoding spins, two-qubit gates are naturally performed by allowing the system to evolve freely. This theoretical model is applied to two realistic architectures of two interacting endohedrals. In the first realistic system the two molecules are placed at a distance of $1.14 nm$. In the second design the two molecules are separated by a distance of $7 nm$. In the latter case the condition $\Delta\omega_p>>g(r)$ is satisfied, i.e. the difference between the precession frequencies of the two spins is much greater than the dipolar coupling strength. This allows us to adopt a simplified theoretical model for the realization of quantum gates.
The realization of quantum gates for these realistic systems is provided by studying the dynamics of the system. In this extent we have numerically solved a set of Schr{\"o}dinger equations needed for reproducing the respective gate, i.e. phase-gate, $\pi$-gate and CNOT-gate. For each quantum gate reproduced through the realistic system, we have estimated their reliability by calculating their related fidelity.
Finally, we present new ideas regarding architectures of systems composed of endohedral fullerenes, which could allow these systems to become reliable building blocks for the realization of quantum computers.
This document provides an outline for a colloquium on searches for spin-dependent short-range forces. It discusses the motivation for searching for such forces from theories of baryon asymmetry in the universe and the strong CP problem. It then describes several experimental techniques using polarized 3He to search for these forces, including measurements of precession frequency shifts. The document outlines improvements that could be made to experimental configurations, cells, masses, and magnetic fields to increase experimental sensitivity.
- The document discusses the history and technique of measuring the electric dipole moment (EDM) of the neutron, which would violate parity and time reversal symmetry.
- It outlines the key aspects of the new neutron EDM experiment being conducted at the Spallation Neutron Source at Oak Ridge National Laboratory, including using ultracold neutrons stored in superfluid helium-4 and measuring their precession with polarized helium-3 as a comagnetometer.
- It also describes experiments measuring the "dressed spin" effect where the effective g-factors of neutrons and helium-3 are made identical through application of a resonant radiofrequency dressing field, which could improve the sensitivity of the neutron E
This document provides an overview of nanomagnetism and the key developments in the field over time. It discusses early experiments and models from Einstein, de Haas, Heisenberg, and others. Key concepts covered include magnetic anisotropy, superparamagnetism, quantum tunneling of magnetization, and magnetic deflagration. Experimental work is highlighted from various researchers, including observations of quantum steps in microwave absorption and avalanches in Mn-12 acetate.
This document discusses electron spin resonance (ESR), which is similar in principle to nuclear magnetic resonance (NMR). It derives the magnetic moment of an electron's spin and shows that a spin placed in a constant magnetic field will precess around the field at the Larmor frequency. When an additional oscillating magnetic field is applied at the Larmor frequency, the spin will rotate at the Rabi frequency in the rotating frame. This rotation appears as oscillations between the spin states on the Bloch sphere in the lab frame.
The document investigates heat dissipation in a fuel drop containing magnetic nanoparticles when subjected to a circularly polarized magnetic field and fluid vorticity. It presents a theoretical model describing the magnetization of ferrofluids under these conditions. The model accounts for Brownian and Néel relaxation mechanisms and viscous and magnetic torques. It determines an expression for the heating rate of the drop as a function of field frequency, fluid vorticity, and model parameters. Simulations show the heating rate increases with greater difference between field and vorticity frequencies, reaching a constant saturation value determined by low vorticities.
This document summarizes key concepts about sound from a physics textbook chapter:
1) It provides formulas and sample calculations for determining the speed of sound in various materials like steel, copper, and air using properties like density, temperature, and molecular mass.
2) It also discusses the fundamental frequencies and overtones of vibrating air columns in open and closed pipes of different lengths, using the speed of sound and formulas involving wavelength and frequency.
3) The final sections cover sound intensity and decibel calculations, defining the reference intensity and showing examples of computing intensity from a given decibel level or vice versa.
This document provides 3 key points about angular impulse and momentum:
1) It defines angular momentum as the moment of linear momentum about a point, and derives equations relating angular momentum, moment of forces, and rate of change of angular momentum.
2) It discusses examples of applying the principle of conservation of angular momentum, including a ball on a cylinder and a ballistic pendulum.
3) It introduces the principle of angular impulse, which states that the angular impulse on a particle equals its change in angular momentum, and can be used to analyze impulsive forces.
The quantum bounce of neutrons has been observed at the peV energy level. An application of Ramsey's method of oscillating fields allows high-precision spectroscopy of neutrons bouncing on a surface. This technique could improve the sensitivity for testing neutron couplings to hypothetical short-range forces and influences on gravity. Future experiments aim to reach sensitivities needed to probe certain axion dark matter models and non-Newtonian gravity potentials.
This document provides an outline for a colloquium on searches for spin-dependent short-range forces. It discusses the motivation for searching for such forces from theories of baryon asymmetry in the universe and the strong CP problem. It then describes several experimental techniques using polarized 3He to search for these forces, including measurements of precession frequency shifts. The document outlines improvements that could be made to experimental configurations, cells, masses, and magnetic fields to increase experimental sensitivity.
- The document discusses the history and technique of measuring the electric dipole moment (EDM) of the neutron, which would violate parity and time reversal symmetry.
- It outlines the key aspects of the new neutron EDM experiment being conducted at the Spallation Neutron Source at Oak Ridge National Laboratory, including using ultracold neutrons stored in superfluid helium-4 and measuring their precession with polarized helium-3 as a comagnetometer.
- It also describes experiments measuring the "dressed spin" effect where the effective g-factors of neutrons and helium-3 are made identical through application of a resonant radiofrequency dressing field, which could improve the sensitivity of the neutron E
This document provides an overview of nanomagnetism and the key developments in the field over time. It discusses early experiments and models from Einstein, de Haas, Heisenberg, and others. Key concepts covered include magnetic anisotropy, superparamagnetism, quantum tunneling of magnetization, and magnetic deflagration. Experimental work is highlighted from various researchers, including observations of quantum steps in microwave absorption and avalanches in Mn-12 acetate.
This document discusses electron spin resonance (ESR), which is similar in principle to nuclear magnetic resonance (NMR). It derives the magnetic moment of an electron's spin and shows that a spin placed in a constant magnetic field will precess around the field at the Larmor frequency. When an additional oscillating magnetic field is applied at the Larmor frequency, the spin will rotate at the Rabi frequency in the rotating frame. This rotation appears as oscillations between the spin states on the Bloch sphere in the lab frame.
The document investigates heat dissipation in a fuel drop containing magnetic nanoparticles when subjected to a circularly polarized magnetic field and fluid vorticity. It presents a theoretical model describing the magnetization of ferrofluids under these conditions. The model accounts for Brownian and Néel relaxation mechanisms and viscous and magnetic torques. It determines an expression for the heating rate of the drop as a function of field frequency, fluid vorticity, and model parameters. Simulations show the heating rate increases with greater difference between field and vorticity frequencies, reaching a constant saturation value determined by low vorticities.
This document summarizes key concepts about sound from a physics textbook chapter:
1) It provides formulas and sample calculations for determining the speed of sound in various materials like steel, copper, and air using properties like density, temperature, and molecular mass.
2) It also discusses the fundamental frequencies and overtones of vibrating air columns in open and closed pipes of different lengths, using the speed of sound and formulas involving wavelength and frequency.
3) The final sections cover sound intensity and decibel calculations, defining the reference intensity and showing examples of computing intensity from a given decibel level or vice versa.
This document provides 3 key points about angular impulse and momentum:
1) It defines angular momentum as the moment of linear momentum about a point, and derives equations relating angular momentum, moment of forces, and rate of change of angular momentum.
2) It discusses examples of applying the principle of conservation of angular momentum, including a ball on a cylinder and a ballistic pendulum.
3) It introduces the principle of angular impulse, which states that the angular impulse on a particle equals its change in angular momentum, and can be used to analyze impulsive forces.
The quantum bounce of neutrons has been observed at the peV energy level. An application of Ramsey's method of oscillating fields allows high-precision spectroscopy of neutrons bouncing on a surface. This technique could improve the sensitivity for testing neutron couplings to hypothetical short-range forces and influences on gravity. Future experiments aim to reach sensitivities needed to probe certain axion dark matter models and non-Newtonian gravity potentials.
Cluster aggregation with complete collisional fragmentationColm Connaughton
The document summarizes research on cluster-cluster aggregation (CCA) models where particles stick together upon contact. It discusses mean-field kinetic equations to model CCA with sources and sinks of particles. For the case of complete fragmentation, it presents an exact solution to the kinetic equations. It finds that nonlocal cascades where larger clusters interact mostly with smaller ones can be unstable, leading to oscillatory behavior over time rather than a stationary state. The document outlines approaches to model the nonlocal case using approximations to the Smoluchowski kinetic equation.
Nonequilibrium statistical mechanics of cluster-cluster aggregation, School o...Colm Connaughton
Colm Connaughton presented on nonequilibrium statistical mechanics models of cluster-cluster aggregation. He discussed simple models where particles move randomly and merge upon contact. More sophisticated models track the size distribution of clusters as they aggregate. The Smoluchowski equation describes this process. For certain collision kernels, clusters of arbitrarily large size can form in finite time, known as gelation. While some kernels mathematically describe instantaneous gelation, physical models avoid this with a cluster size cutoff. Stationary states can be reached with a particle source.
The document provides an introduction to dynamics and vibration in engineering. It discusses the Millennium Bridge incident as a motivating example of undesirable vibration. The course will cover modeling structural systems with multiple degrees of freedom and understanding vibration phenomena. Simple examples are presented, such as a pendulum and spring-mass system, to derive the equations of motion and introduce concepts like natural frequency, stiffness, mass, and harmonic motion. Initial conditions and their role in determining vibration responses are also discussed.
What happens when the Kolmogorov-Zakharov spectrum is nonlocal?Colm Connaughton
This document summarizes research on the behavior of the Kolmogorov-Zakharov (KZ) spectrum when it is nonlocal. It examines a model of cluster-cluster aggregation described by the Smoluchowski equation, which can be viewed as a model of 3-wave turbulence without backscatter. The research finds that when the exponents in the interaction term satisfy certain conditions, the KZ spectrum is nonlocal. In this case, the stationary state has a novel functional form and can become unstable, leading to oscillatory behavior in the cascade dynamics at long times. Open questions remain about whether physical systems exhibit this behavior and how the results are affected by including backscatter terms.
This document discusses the basic principles of seismic waves. It introduces longitudinal (P) waves and shear (S) waves, and derives the one-dimensional wave equation. It discusses wave phenomena like reflection, transmission, and refraction based on Snell's law at boundaries between layers. It also discusses the different arrivals of direct, reflected, and refracted/head waves that can be measured at the surface for seismic exploration purposes.
This document discusses base excitation in vibration analysis and provides examples. It begins by introducing base excitation as an important class of vibration analysis that involves preventing vibrations from passing through a vibrating base into a structure. Examples of base excitation include vibrations in cars, satellites, and buildings during earthquakes. The document then provides mathematical models and equations to analyze single degree of freedom base excitation systems. Graphs of transmissibility ratios are presented and examples are worked through, such as calculating car vibration amplitude at different speeds. Rotating unbalance is also covered as another source of vibration excitation.
Talk given at the workshop "Multiphase turbulent flows in the atmosphere and ocean", National Centre for Atmospheric REsearch, Boulder CO, August 15 2012
This document provides information about the dynamics of machinery course for several mechanical engineering students. It includes the learning objectives, symbols and definitions, response of a damped system under harmonic motion, an example problem, and key concepts about magnification factor, phase angle, and total response of a system. The example calculates the total response of a single-degree-of-freedom system subjected to an external harmonic force and free vibration.
The Inverse Smoluchowski Problem, Particles In Turbulence 2011, Potsdam, Marc...Colm Connaughton
This document summarizes Colm Connaughton's presentation on solving the inverse Smoluchowski problem to determine particle collision kernels from observed cluster size distributions. It describes how the forward problem maps kernels to distributions but the inverse problem is ill-posed. Tikhonov regularization is used to obtain approximate kernel reconstructions from numerical solutions with known test kernels, demonstrating partial success in reconstructing kernel features despite ill-posedness. Future work aims to address limitations and applicability to real problems.
Instantaneous Gelation in Smoluchwski's Coagulation Equation Revisited, Confe...Colm Connaughton
Invited talk given at "Boltzmann equation:
mathematics, modeling and simulations
In memory of Carlo Cercignani", Institut Henri Poincare, Paris, February 11, 2011.
This document summarizes key concepts and equations related to wave motion from a physics textbook. It discusses transverse and longitudinal waves, and defines terms like amplitude, wavelength, frequency, period, and speed. It also covers topics like standing waves, harmonics, wave energy, and calculating the fundamental frequency and overtones of vibrating strings and ropes based on their tension, mass, and length. Sample problems are provided and worked through applying these concepts and equations to different scenarios.
This summary provides the key information from the document in 3 sentences:
The document discusses nuclear physics concepts such as nuclear structure, binding energy, mass defect, radioactive decay, and half-life. It includes examples calculating physical properties like mass, radius, binding energy, and activity for various nuclei. The examples analyze nuclear reactions and decays, and solve related problems involving time, mass, energy, and radioactive decay calculations.
Feedback of zonal flows on Rossby-wave turbulence driven by small scale inst...Colm Connaughton
The document summarizes research on the interaction between large-scale zonal flows and small-scale Rossby wave turbulence. It describes how modulational instability can generate large-scale zonal jets from small-scale Rossby waves through an inverse cascade. The generated jets then provide negative feedback on the small-scale waves by distorting them and inducing spectral diffusion through a nonlocal turbulence theory. Numerical simulations demonstrate this generation of jets and spectral transport between scales.
This document provides an introduction to the concept of wave turbulence. It discusses how waves interact nonlinearly at finite amplitudes to produce a statistical, non-equilibrium dynamics. Key points:
- Wave turbulence involves dispersive waves that are excited and damped by external processes, leading to interactions between many degrees of freedom.
- The nonlinear interactions can be modeled using a Hamiltonian approach by including higher-order terms that couple different Fourier modes.
- A central goal is developing a statistical description of the system using correlation functions and obtaining a closed kinetic equation for the wave spectrum.
- In the weak turbulence regime, this kinetic equation can be solved perturbatively to obtain scaling laws for the wave spectrum in both physical and
This chapter discusses refraction of light, including:
- The index of refraction, which is the ratio of the speed of light in a vacuum to its speed in a medium.
- Snell's law, which relates the angles of incidence and refraction based on the indices of refraction of the media.
- Total internal reflection, which occurs when light travels from an optically dense medium to a less dense one at an angle greater than the critical angle.
- Wavelength changes when light moves between media due to the index of refraction.
Several example problems are worked through applying these concepts to compute angles, indices of refraction, wavelengths and speeds of light in various materials.
We propose a possible experimental realization of a quantum analogue
of Newton's cradle using a configuration which starts from a
Bose-Einstein condensate. The system consists of atoms with two
internal states trapped in a one dimensional tube with a longitudinal
optical lattice and maintained in a strong Tonks-Girardeau regime at
maximal filling. In each site the wave function is a superposition of
the two atomic states and a disturbance of the wave function
propagates along the chain in analogy with the propagation of
momentum in the classical Newton's cradle. The quantum travelling
signal is generally deteriorated by dispersion, which is large for a
uniform chain and is known to be zero for a suitably engineered
chain, but the latter is hardly realizable in practice. Starting from
these opposite situations we show how the coherent behaviour can be
enhanced with minimal experimental effort.
Voici un pdf non travaillé de ma soutenance de thèse. Je m'excuse pour les déformations des équations et les animations qui se chevauchent. Je n'ai pas trouver le temps de régler cela.
Optical control of resonant light transmission for an atom-cavity system_Arij...Arijit Sharma
This document summarizes an experiment that demonstrated optical control of light transmission through a Fabry-Perot cavity containing rubidium vapor. A probe laser tuned to the rubidium D2 transition frequency was coupled into the cavity. When a second, control laser beam intersected the cavity mode and was tuned to a different rubidium transition frequency, it could either suppress or enhance the transmission of the probe laser through the cavity, depending on the transition frequencies used. This provided a way to switch the cavity transmission on and off optically. Both steady-state and transient responses were investigated experimentally and qualitatively explained by the coupling of atomic states induced by the probe and control beams.
This document discusses various concepts and technologies related to interstellar travel, including:
1) Current rocket technologies are insufficient to reach other stars due to limitations of chemical fuels; multistage rockets and new fuels like nuclear or antimatter could help.
2) The nearest star is over 4 light years away, so travel at current speeds would take tens of thousands of years; new propulsion methods like ion drives, solar sails, and ramjets could enable much faster travel.
3) However, special relativity shows that nothing can travel faster than light speed, posing a fundamental limit to interstellar travel; speculative ideas like wormholes or generation ships may be needed to overcome this.
Cluster aggregation with complete collisional fragmentationColm Connaughton
The document summarizes research on cluster-cluster aggregation (CCA) models where particles stick together upon contact. It discusses mean-field kinetic equations to model CCA with sources and sinks of particles. For the case of complete fragmentation, it presents an exact solution to the kinetic equations. It finds that nonlocal cascades where larger clusters interact mostly with smaller ones can be unstable, leading to oscillatory behavior over time rather than a stationary state. The document outlines approaches to model the nonlocal case using approximations to the Smoluchowski kinetic equation.
Nonequilibrium statistical mechanics of cluster-cluster aggregation, School o...Colm Connaughton
Colm Connaughton presented on nonequilibrium statistical mechanics models of cluster-cluster aggregation. He discussed simple models where particles move randomly and merge upon contact. More sophisticated models track the size distribution of clusters as they aggregate. The Smoluchowski equation describes this process. For certain collision kernels, clusters of arbitrarily large size can form in finite time, known as gelation. While some kernels mathematically describe instantaneous gelation, physical models avoid this with a cluster size cutoff. Stationary states can be reached with a particle source.
The document provides an introduction to dynamics and vibration in engineering. It discusses the Millennium Bridge incident as a motivating example of undesirable vibration. The course will cover modeling structural systems with multiple degrees of freedom and understanding vibration phenomena. Simple examples are presented, such as a pendulum and spring-mass system, to derive the equations of motion and introduce concepts like natural frequency, stiffness, mass, and harmonic motion. Initial conditions and their role in determining vibration responses are also discussed.
What happens when the Kolmogorov-Zakharov spectrum is nonlocal?Colm Connaughton
This document summarizes research on the behavior of the Kolmogorov-Zakharov (KZ) spectrum when it is nonlocal. It examines a model of cluster-cluster aggregation described by the Smoluchowski equation, which can be viewed as a model of 3-wave turbulence without backscatter. The research finds that when the exponents in the interaction term satisfy certain conditions, the KZ spectrum is nonlocal. In this case, the stationary state has a novel functional form and can become unstable, leading to oscillatory behavior in the cascade dynamics at long times. Open questions remain about whether physical systems exhibit this behavior and how the results are affected by including backscatter terms.
This document discusses the basic principles of seismic waves. It introduces longitudinal (P) waves and shear (S) waves, and derives the one-dimensional wave equation. It discusses wave phenomena like reflection, transmission, and refraction based on Snell's law at boundaries between layers. It also discusses the different arrivals of direct, reflected, and refracted/head waves that can be measured at the surface for seismic exploration purposes.
This document discusses base excitation in vibration analysis and provides examples. It begins by introducing base excitation as an important class of vibration analysis that involves preventing vibrations from passing through a vibrating base into a structure. Examples of base excitation include vibrations in cars, satellites, and buildings during earthquakes. The document then provides mathematical models and equations to analyze single degree of freedom base excitation systems. Graphs of transmissibility ratios are presented and examples are worked through, such as calculating car vibration amplitude at different speeds. Rotating unbalance is also covered as another source of vibration excitation.
Talk given at the workshop "Multiphase turbulent flows in the atmosphere and ocean", National Centre for Atmospheric REsearch, Boulder CO, August 15 2012
This document provides information about the dynamics of machinery course for several mechanical engineering students. It includes the learning objectives, symbols and definitions, response of a damped system under harmonic motion, an example problem, and key concepts about magnification factor, phase angle, and total response of a system. The example calculates the total response of a single-degree-of-freedom system subjected to an external harmonic force and free vibration.
The Inverse Smoluchowski Problem, Particles In Turbulence 2011, Potsdam, Marc...Colm Connaughton
This document summarizes Colm Connaughton's presentation on solving the inverse Smoluchowski problem to determine particle collision kernels from observed cluster size distributions. It describes how the forward problem maps kernels to distributions but the inverse problem is ill-posed. Tikhonov regularization is used to obtain approximate kernel reconstructions from numerical solutions with known test kernels, demonstrating partial success in reconstructing kernel features despite ill-posedness. Future work aims to address limitations and applicability to real problems.
Instantaneous Gelation in Smoluchwski's Coagulation Equation Revisited, Confe...Colm Connaughton
Invited talk given at "Boltzmann equation:
mathematics, modeling and simulations
In memory of Carlo Cercignani", Institut Henri Poincare, Paris, February 11, 2011.
This document summarizes key concepts and equations related to wave motion from a physics textbook. It discusses transverse and longitudinal waves, and defines terms like amplitude, wavelength, frequency, period, and speed. It also covers topics like standing waves, harmonics, wave energy, and calculating the fundamental frequency and overtones of vibrating strings and ropes based on their tension, mass, and length. Sample problems are provided and worked through applying these concepts and equations to different scenarios.
This summary provides the key information from the document in 3 sentences:
The document discusses nuclear physics concepts such as nuclear structure, binding energy, mass defect, radioactive decay, and half-life. It includes examples calculating physical properties like mass, radius, binding energy, and activity for various nuclei. The examples analyze nuclear reactions and decays, and solve related problems involving time, mass, energy, and radioactive decay calculations.
Feedback of zonal flows on Rossby-wave turbulence driven by small scale inst...Colm Connaughton
The document summarizes research on the interaction between large-scale zonal flows and small-scale Rossby wave turbulence. It describes how modulational instability can generate large-scale zonal jets from small-scale Rossby waves through an inverse cascade. The generated jets then provide negative feedback on the small-scale waves by distorting them and inducing spectral diffusion through a nonlocal turbulence theory. Numerical simulations demonstrate this generation of jets and spectral transport between scales.
This document provides an introduction to the concept of wave turbulence. It discusses how waves interact nonlinearly at finite amplitudes to produce a statistical, non-equilibrium dynamics. Key points:
- Wave turbulence involves dispersive waves that are excited and damped by external processes, leading to interactions between many degrees of freedom.
- The nonlinear interactions can be modeled using a Hamiltonian approach by including higher-order terms that couple different Fourier modes.
- A central goal is developing a statistical description of the system using correlation functions and obtaining a closed kinetic equation for the wave spectrum.
- In the weak turbulence regime, this kinetic equation can be solved perturbatively to obtain scaling laws for the wave spectrum in both physical and
This chapter discusses refraction of light, including:
- The index of refraction, which is the ratio of the speed of light in a vacuum to its speed in a medium.
- Snell's law, which relates the angles of incidence and refraction based on the indices of refraction of the media.
- Total internal reflection, which occurs when light travels from an optically dense medium to a less dense one at an angle greater than the critical angle.
- Wavelength changes when light moves between media due to the index of refraction.
Several example problems are worked through applying these concepts to compute angles, indices of refraction, wavelengths and speeds of light in various materials.
We propose a possible experimental realization of a quantum analogue
of Newton's cradle using a configuration which starts from a
Bose-Einstein condensate. The system consists of atoms with two
internal states trapped in a one dimensional tube with a longitudinal
optical lattice and maintained in a strong Tonks-Girardeau regime at
maximal filling. In each site the wave function is a superposition of
the two atomic states and a disturbance of the wave function
propagates along the chain in analogy with the propagation of
momentum in the classical Newton's cradle. The quantum travelling
signal is generally deteriorated by dispersion, which is large for a
uniform chain and is known to be zero for a suitably engineered
chain, but the latter is hardly realizable in practice. Starting from
these opposite situations we show how the coherent behaviour can be
enhanced with minimal experimental effort.
Voici un pdf non travaillé de ma soutenance de thèse. Je m'excuse pour les déformations des équations et les animations qui se chevauchent. Je n'ai pas trouver le temps de régler cela.
Optical control of resonant light transmission for an atom-cavity system_Arij...Arijit Sharma
This document summarizes an experiment that demonstrated optical control of light transmission through a Fabry-Perot cavity containing rubidium vapor. A probe laser tuned to the rubidium D2 transition frequency was coupled into the cavity. When a second, control laser beam intersected the cavity mode and was tuned to a different rubidium transition frequency, it could either suppress or enhance the transmission of the probe laser through the cavity, depending on the transition frequencies used. This provided a way to switch the cavity transmission on and off optically. Both steady-state and transient responses were investigated experimentally and qualitatively explained by the coupling of atomic states induced by the probe and control beams.
This document discusses various concepts and technologies related to interstellar travel, including:
1) Current rocket technologies are insufficient to reach other stars due to limitations of chemical fuels; multistage rockets and new fuels like nuclear or antimatter could help.
2) The nearest star is over 4 light years away, so travel at current speeds would take tens of thousands of years; new propulsion methods like ion drives, solar sails, and ramjets could enable much faster travel.
3) However, special relativity shows that nothing can travel faster than light speed, posing a fundamental limit to interstellar travel; speculative ideas like wormholes or generation ships may be needed to overcome this.
Cosmogenese et communication interstellaire est un cours d'Astrophysique et de gravité quantique acessible pour tous et dans le domaine d'application de l'ingénierie tels que le voyage interstellaire.
"Squeezed States in Bose-Einstein Condensate"Chad Orzel
1. The document discusses the formation of squeezed quantum states in Bose-Einstein condensates trapped in optical lattices. By slowly ramping up the depth of the optical lattice, the atoms can be prepared in a number-squeezed state.
2. Releasing the atoms from the lattice allows their wavefunctions to overlap and interfere, providing a way to probe the quantum phase state of the atoms. Number-squeezed states are observed to produce interference patterns with higher contrast than coherent states.
3. Variational calculations of the quantum state dynamics during lattice ramping and dephasing agree qualitatively with experimental observations of the transition between coherent and squeezed states.
1. A cyclotron uses a magnetic field to accelerate charged particles in a circular path between two "dees". As the particles accelerate, their frequency is synchronized with an oscillator to maintain resonance.
2. Ampere's Circuital Law states that the line integral of magnetic field B around any closed loop equals the permeability constant μ0 times the current passing through the enclosed area.
3. The magnetic field at the center of a straight solenoid is directly proportional to the number of turns per unit length and the current, according to the formula B=μ0nI.
1. A particle moving perpendicular to a magnetic field will follow a circular path. The radius of the path is determined by the particle's mass, charge, speed, and the magnetic field strength.
2. A velocity selector uses uniform, perpendicular electric and magnetic fields. Particles pass through undeflected if their speed equals the ratio of the field strengths.
3. A mass spectrometer accelerates ions and uses a magnetic field to cause circular orbits. Heavier ions have smaller orbit radii allowing separation based on mass.
1. The Stern-Gerlach experiment discovered that silver atoms split into two beams, indicating the presence of an intrinsic "spin" angular momentum of 1/2 beyond orbital angular momentum.
2. Elementary particles are classified as fermions, with half-integer spin, and bosons, with integer spin. The spin of the electron is represented by a two-component spinor.
3. In a magnetic field, the spin precesses around the field direction at the Larmor frequency, independent of initial spin orientation. This principle underlies paramagnetic resonance and nuclear magnetic resonance spectroscopy.
A Pedagogical Discussion on Neutrino Wave Packet EvolutionCheng-Hsien Li
This document discusses the time evolution of a neutrino wave packet. It presents a method to calculate higher-order corrections to the wave packet solution by expanding the momentum distribution and energy terms. The results show that including higher-order terms causes the wave packet to evolve into a spherical shape as expected by relativity. While the solution is limited to early times, it demonstrates that higher-order terms are needed to accurately model the wave packet's evolution into a spherical wavefront.
This document discusses magnetic materials and their properties. It begins by defining key terms like magnetic flux density (B), magnetic field intensity (H), magnetization (M), and permeability (μ). It then covers the classification of magnetic materials into diamagnetic, paramagnetic, and ferromagnetic types based on their magnetic susceptibility (χ). The document also provides microscopic explanations for magnetism based on orbital and spin motions of electrons and nuclei. It derives formulas for diamagnetic susceptibility using Langevin's model and for paramagnetic susceptibility using Curie's law.
Alfvén waves and their kinetic modifications like kinetic Alfvén waves play an important role in space weather and plasma energization processes. Kinetic Alfvén waves are able to dissipate energy and accelerate particles in various regions of space including the solar atmosphere, Earth's magnetosphere during substorms, and in forming features of the solar wind like proton beams. Open questions remain about the detailed theory and observations of kinetic Alfvén wave generation, propagation, and dissipation across different plasma environments.
Magnetic Effects Of Current Class 12 Part-3Self-employed
1. The document describes the working principles of a cyclotron, which uses a magnetic field to accelerate charged particles in a circular path between two "dees". As the particles accelerate, their frequency increases until it matches the frequency of an external oscillator, allowing for resonant acceleration.
2. Ampere's Circuital Law is explained, which relates the line integral of magnetic field around a closed loop to the current passing through the enclosed area. Magnetic fields from a straight solenoid and toroidal solenoid are also derived using this law.
3. The magnetic field is non-zero only within the winding area of a toroidal solenoid, and zero both inside and outside the solenoid ring.
1. A cyclotron uses a magnetic field to accelerate charged particles in a circular path between two "dees". The cyclotron frequency depends only on the mass to charge ratio of the particle and the magnetic field strength.
2. Ampere's Circuital Law states that the line integral of the magnetic field around any closed loop is equal to the permeability constant times the current passing through the enclosed area.
3. The magnetic field inside a straight solenoid is uniform and its strength depends on the number of turns per unit length, the current passing through, and the permeability constant.
4. For a toroidal solenoid or "toroid", the magnetic field exists only in the area enclosed by
This document provides an overview of plasma physics concepts. It defines an ionized gas and explains how the Saha equation describes ionization equilibrium. It also discusses how an ionized gas can become a plasma if it exhibits collective behavior and quasineutrality. Additionally, it introduces the Maxwellian velocity distribution and kinetic equations like the Boltzmann and Vlasov equations that govern plasma behavior.
1. A cyclotron uses a magnetic field to accelerate charged particles in a circular path between two "dees". As the particles accelerate, their frequency increases until it matches the oscillating frequency applied, allowing for resonance and maximum energy transfer.
2. Ampere's Circuital Law states that the line integral of the magnetic field around any closed loop is equal to the permeability constant times the current passing through the enclosed area.
3. The magnetic field at the center of a straight solenoid is directly proportional to the number of turns per unit length and the current passing through, and inversely proportional to the length of the solenoid.
This document contains a physics exam with 41 multiple choice questions covering topics like kinematics, forces, energy, oscillations, optics, electricity, magnetism, modern physics and thermodynamics. The questions are from an unsolved past paper from 2006 for AEEE, which appears to be an entrance exam of some kind. Each question provides 4 possible answers, but only one is correct.
This document discusses the magnetic properties of materials. Some key points:
1. Magnetic properties are studied using parameters like magnetic dipoles, magnetization, magnetic susceptibility, and permeability. Materials respond differently to external magnetic fields based on these properties.
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Phd thesis- Quantum Computation
1. Theoretical Realization of Quantum
Gates Using Interacting Endohedral
Fullerene Molecules
Maria Silvia Garelli
(M.S.Garelli@lboro.ac.uk)
Department of Physics, Loughborough University, LE11 3TU, U.K.
3. Properties of the N@C60 •The encapsulated Nitrogen
atom can be considered as an
independent atom, with all the
•Repulsive interaction properties of the free atom.
Between the Fullerene •Since the charge is completely
cage and the encapsulated screened, the Fullerene cage
atom. No charge transfer. does not take part in the
interaction process. It can just
•The atomic electrons of be considered as a trap for the
the encased atom are Nitrogen encased atom.
tighter bound than in
the free atom. The N
atom is stabilized in its
ground state.
•Nitrogen central site The only Physical quantity of interest
position inside the
is the spin of the encapsulated atom.
fullerene cage.
We suppose that the N atom is a ½-spin
particle
4. Decoherence times:
•T1 due to the interactions between
a spin and the surrounding environment
• T2 due to the dipolar interaction between
the qubit encoding spin and the surrounding
endohedral spins randomly distributed in the
sample
• T1 and T2 are both temperature dependent
• Their correlation T2 ≅ 2/3 T1 is
(N@C60 in CS2)
constant over a broad range of temperature
• below 160 K, CS2 solvent freezes, leaving regions
of high fullerene concentrations
⇒ dramatical increase of the local spin concentration T2=0.25ms
⇒ T2 becomes extremely short due to dipolar spin coupling
• temperature dependence due to Orbach processes
J. J. L. Morton, A. M. Tyryshkin, A. Ardavan, K. Porfyrakis, S. A. Lyon, G. A. Briggs, J. Chem. Phys. 124, 014508 (2006).
6. Physical system:
Two N@C60
Buckyballs
The mutual interaction between the two encased spins is dominated by the
dipole-dipole interaction , while the exchange interaction is negligible*
r r r r r r
ˆ ˆ ˆ ˆ
H = g (r )[σ 1 ⊗ σ 2 − 3(σ 1 ⋅ n ) ⊗ (σ 2 ⋅ n )]
µ0 µ B 2
where g (r ) = is the dipolar coupling constant
2πr 3
r Hamiltionian of the two-qubit system
By choosing n ˆ ˆ ˆ ˆ ˆ ˆ
parellel to the x-axis H = g (r )(σ z1 ⊗ σ z 2 + σ y1 ⊗ σ y2 − 2σ x1 ⊗ σ x2 )
*J. C. Greer,Chem. Phys, Lett. 326, 567 (2000); W. Harneit, Phys. Rev. A 65, 032322 (2002); M. Waiblinger, B. Goedde, K. Lips, W. Harneit, P. Jakes,
A. Weidinger, K. P. Dinse, AIP Conf. Proc. 544, 195 (2000).
7. Qubit-encoding
two-level system
If we apply a static magnetic field of amplitude B0
dierected along the z axis we obtain
a two-level system for each spin,
due to the splitting of the spin-z component
Hamiltonian of a two-qubit system subjected to the spin dipolar mutual
interaction and to the action of static magnetic field along the z direction
ˆ ˆ ˆ ˆ ˆ ˆ
H = g (r )(σ z1σ z 2 + σ y1σ y2 − 2σ x1σ x2 )
ω0 = µ B B0
1, 2 1, 2
ˆ ˆ
−ω01σ z1 − ω02 σ z2
whereω and ω are the precession frequencies of spin 1 and spin 2, respectively
01 02
8. Single addressing of each qubit
Current density > 107A/cm2 d = 1µm
ρ = 1µm
I = 0.3 A
With the use of atom chip technology*,
two parallel wires carrying a current
of the same intensity generate
a magnetic field gradient.
µ0 1 1
Bg =
x+ ρ +d /2 x−ρ −d /2
+
2π
ïthe two particles are characterized by different
resonance frequencies
*S. Groth, P. Kruger, S. Wildermuth, R. Folman, T. Fernholz, D. Mahalu, I. Bar-Joseph, J. Schmiedmayer, Appl. Phys. Lett. 85, 14 (2004)
10. Theoretical model borrowed from NMR quantum computation*
ESR techniques allow to induce transitions between the spin states
by applying microwave fields whose frequency is equal to the
precession frequency of the spin.
• Single-qubit
gates
on resonance spin-microwave field interaction
• Two-qubit gates
naturally existing spin dipolar interaction
* M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University press, 2000)
L. M. K. Vandersypen, I. L. Chuang, Rev. Mod. Phys. 74, 1037 (2005)
11. SINGLE-SPIN SYSTEM: single-qubit gates
The state of a ½ spin particle in a static magnetic field B0 directed along the z axis can
r
be manipulated by applying an on resonance MW field,Bm = Bm (cos ωmt + φ , sin ωmt + φ ,0)
which rotates in the x-y plane at a frequency wm =2w0 characterized by a phase f and
an amplitude Bm
H m = − µ0 B0σ z − µ B Bm [cos(ωmt + φ )σ x − sin(ωmt + φ )σ y ]
Total Hamiltonian 1 24 14444444 4444444
4 3
spin − static field
2 3
spin − MW field
Considering the Schrödinger equation and performing a change of coordinates to a frame
rot
rotating a frequency wr about the z axis defined by ψ = e −iωrσ z ψ , by choosing wr=w0
we obtain the Control Hamiltonian
H rot = −ωa [cos[(ωm − 2ω0 )t + φ ]σ x − sin[(ωm − 2ω0 )t + φ ]σ Y ]
ωa = µ B Bm
12. When the applied MW-field is resonant with the spin precession frequency, i.e. wm=2w0 ,
the Hamiltonian is time-independent, H = −ωa [cos(φ )σ x − sin(φ )σ Y ] , and its related
time evolution can be easily written as follows
− iHt iω a t [cos(φ )σ x −sin(φ )σ y ]
U (t ) = e =e
r r r
θσ ⋅n
Rotation of an angle q about n axis Rn (θ ) = e
r
−i
2
•U(t) is a rotation in the x-y plane of an angle q
proportional to wat, which is determined by phase f .
•Bm (angle of rotation) and f (axis of rotation) can be varied
with time.
•w0 cannot be varied with time because depends on the
amplitude B0 of the static magnetic field
π
Example: p/2 rotation about the y axis −i σ y
4
U =e
it can be realized by choosing f= p/2 and allowing the time evolution for a time
t=p/4wa= p /4mBBm
13. Two-Spin System
Single-qubit gates: can be performed Two-qubit gates: naturally accomplished
through the selective resonant interaction through the mutual spin dipolar interaction
between the MW-field and the spin
to be transformed
Since the dipolar interaction couples the two spins,
it naturally realizes two-qubit gates
To realize single-qubit gates we need to assume that the
spin-dipolar interaction is negligible in comparison
with the spin-MW field interaction term
ASSUMPTION − iHt − i ( H DD + H US ) t − iH US t •HDD dipolar interaction term
U (t ) = e =e ≈e •HUS is the interaction between
two uncoupled spins and the MW-field
The interaction terms between two uncoupled spins and a MW-field
dominate the time evolutionï the spin dipolar interaction is negligible ï
single-qubit rotation can be performed in good approximation
14. QUANTUM GATES
iπ
e 4 0 0 0
p/4-phase gate −i
π realizes a p gate up to a p/2 rotation
= 0 0 of both spins about the z axis and
4
U PG e 0
−i
π
4
up to a global phase
0 0 e 0
π
i
0 0 0 e4
1 0 0 0 1 0 0 0
0 1 0 0 CNOT-gate 0 1 0 0
p-gate U CNOT =
0 0 1
Gπ =
0 0 1 0
0
0
0
0 0 − 1
0 1 0
15. Refocusing: is a set of transformations which allow the removal of
the off-diagonal coupling terms of HDD
π π
− i σ z2 i σ Circuit representing U(t)
−iH DD t 2 −iH DD t 2 z2
U (t ) = e e e e
π π
− i σ z2 i σ z2
2 2
= U b (t )e U a (t )e
−i 4 g ( r )σ z 1σ z2 t
=e
π
m i σ z2
2
• e is a ±p rotation about the z axis of the second spin
• Ua(t) and Ub(t) represent the time evolution when the system is subjected
to a static field and to the mutual dipolar interaction only
ï they can be interpreted as two-qubit operations
by allowing evolution U(t) for a time t=p/16 g(r), a p/4-phase gate is realized
16. p-gate Circuit representing Gp
π π
−i σ z1
4
− i σ z2
4
π
Gπ = i e e U (t = )
16 g (r )
CNOT-gate Circuit representing CNOT
π π π
−i σ z1 i σ y2 − i σ y2
2 4 4
CNOT = ie e Gπ e
18. Realistic dynamics
reproduction of theoretical single-qubit and two-qubit quantum gates following the theory
previously presented
Assumption e − iHt ≅ e − iHUS t
in a realistic system in general is NOT satisfied
înumerical solution of the Schrödinger equation
The reliability of the realistic system as a candidate for performing quantum gates
will be checked from the comparison between the numerical results and
the theoretically predicted outcomes and through the study of the fidelity
of the quantum gate
19. Distant buckyballs: we assume that the distance between the centres of the two
buckyballs is r=7nm
This sut-up can be assembled by encasing buckyballs in a nanotube (peapod)
•Buckyball diameter: d@0.7nm
•distance between two buckyballs
in a nanotube: dist@0.3nm
(due to Van der-Waals forces)
} We need to place 9 empty buckyballs between
the two fullerenes in order to obtain r=7 nm
2
{
µ0 µ B
g (r ) = = 2.38 × 105 Hz dipolar coupling constant
2πr 3
r=7 nm î Bg1 = 1.87 ×10 −4 T
gradient field amplitudes
Bg 2 = −1.87 × 10 − 4 T
20. B01= B02 =(0.3+3.04x10-5)T, ν 0 = 2ω0 / 2π = 8.40 ×109 Hz
static magnetic field along resonance 1 1
the z direction frequencies ν 0 = 2ω0 / 2π = 8.39 ×109 Hz
2 2
î ∆ω p = ω p1 − ω p2 = 2ω01 − 2ω0 2 = 6.28 ×107 Hz
This condition allows us to omit the transverse coupling
î Dwp>>g(r) terms in the dipolar Hamiltonian
î The mutual dipolar interaction
Hamiltonian can be simplified as
H approx = g (r )(1 − 3 cos 2 θ )σ z1σ z2
q is the angle between the static magnetic field
and the line joining the centres of the buckys
H approx = −2 g (r )σ z1σ z2
q=0 î
21. •Hamiltonian of two distant buckys subjected to static fields along the z axis
H = H approx + H US
= −2 g (r )σ z1σ z2 − ω01σ z1 − ω0 2 σ z2
Energy-level diagram for two uncoupled spins (light lines)and for two spins described
by the Hamiltonian presented above (solid lines)
22. Total Hamiltonian (additional MW-field)
H = H approx + H US (t )
= −2 g (r )σ z1σ z2 − ω01σ z1 − ω02 σ z2
− ωa1 [cos(ωm1 t + φ )σ x1 − sin(ωm1 t + φ )σ y1 ]
− ωa2 [cos(ωm2 t + φ )σ x2 − sin(ωm2 t + φ )σ y2 ]
rot − iω01σ z1t − iω02σ z2 t
Total Hamiltonian in the rotating frame ψ =e e ψ
rot
H = H approx + H US
= −2 g (r )σ z1σ z2
− ωa1 [cos[(ωm1 − 2ω01 )t + φ ]σ x1 − sin[(ωm1 − 2ω01 )t + φ ]σ y1 ]
− ωa2 [cos[(ωm2 − 2ω0 2 )t + φ ]σ x2 − sin[(ωm2 − 2ω02 )t + φ ]σ y2 ]
23. • single-qubit gates: MW-field and the spin to be rotated are in resonance, i.e.
ωm = 2ω0 î first spin can be rotated
1 1
ωm = 2ω0
2 2 î second spin can be rotated
Typical experimental time exp θ î Bm@1.7mT
of a single-qubit rotation* t SQ = ≅ 32ns
gµ B Bm
• two-qubit gates: naturally realized by the mutual spin dipolar interaction Happrox
time-evolution operator if we allow this time-evolution for
2 ig ( r )σ z1σ z2 t
U (t ) = e î a time t=p/8g(r)=1.65ms we obtain
related to Happrox
a controlled p/4 phase gate
Happrox is already diagonal î the refocusing procedure is not needed
*J.J.L.Morton, A. M. Tyryshkin, A. Ardavan, K. Porfyrakis, S.A. Lyon, G.A. Briggs,Phys. Rev. A.71, 012332 (2005).
24. •Realization of a p-gate: we need to solve a Schrödinger equation for each of the
following transformations, which define a p-gate
1 0 0 0
π π
−i σ z1 −i σ z2 0 1 0 0
Gπ = i e 4
e 4
U (t = π / 8 g (r )) =
0 0 1 0
•Numerical output matrix
0 0 0 − 1
Up2=
Comments :
the dipolar interaction influences the perfect reproduction of single-quibit rotations
and subsequently of a p-gate but the time required for performing a single qubit rotation
is tSQ=32 ns. The time during which the system is influenced by the spin dipolar interaction
is T=2p/g(r)=2.6x10-5s îtSQ<<T during the completion of a single-qubit rotation
we can consider the system as being unaffected by the mutual spin dipolar interaction
îwhen performing Single-Qubit rotations, the spin-Mw field term dominates
25. • Realization of a CNOT-gate: we need to solve a Schrödinger equation for each of the
following transformations, which define a CNOT-gate
1 0 0 0
π π π
−i σ z1 i σ y2
2 4
− i σ y2
4 0 1 0 0
CNOT = ie e Gπ e =
0 0 0 1
0 0 1 0
•Numerical output matrix
UCNreal=
26. π π π π
tout = 3 +3 + = 1.85µs
4 µ B Bm
1
4 µ B Bm 8 g (r )
2
•Operational times: π π π π
CNOT
tout = 5 +5 + + = 2.05µs
4 µ B Bm1
4 µ B Bm 2 µ B Bm 8 g (r )
2 1
p/8g(r) determines the order of magnitude of tout
•Number of quantum operations T2 T2 n<104 î small number
n = π ≅ CNOT ≅ 10 2
allowed before relaxation: tout tout of operationîthe system
is not reliable
Possibility of increasing T2 two order of magnitude:
Proposal: investigation of experiments for the study of relaxation processes of
Buckyballs in a nanotube îreduction of dipolar interactions between
the encased spin and the randomly distributed spins in the sample
The nanotube represents a further shield for the
encased spin against the outer environment
27. Quantum gate fidelity
The fidelity quantifies the distance between the realistic evolved state σ ' = UσU †
and the ideal evolved state ψ
ideal
F(ψ ideal
,σ ' ) = ideal
ψ σ'ψ ideal
= ideal
ψ U ψ ψ U†ψ ideal
Since the starting state is not known in advance, we can consider the
minimum fidelity, which minimizes over all possible starting states
î F = min F ( ψ ideal
,σ ' )
c α
p-gate: F=0.998 F differs from its ideal value F=1
by of the order of 0.2%(0.8%)
CNOT-gate: F=0.991 ïThe realistic transformations are in
HIGH ACCORDANCE with the theoretical predictions and the system is
highly reliable for reproducing a p-gate through the study of its dynamics
28. Considerations on experimental limitations
•Single-qubit rotations: a rotation of spin 1 can be accomplished by centering a
selective MW-pulse at the precession frequency of spin 1,
i.e. wm1=2w01, and characterized by a frequency bandwidth
which has to cover the range of frequencies 2w01 ±4 g(r) but not
overlap the range 2w02 ±4 g(r), which corresponds to the range
of frequencies for the excitation of spin 2
Frequency bandwidth
difference between the upper and lower values
of the range which allow the swap of the selected spin
∆Ω = 2ω01 + 4 g (r ) − (2ω01 − 4 g (r )) = 8 g (r )
î the frequency bandwidth DW depends only on the dipolar coupling constant g(r)
29. ∆Ω = 8 g (r ) = 1.9 MHz and ∆t = t SQ = 32ns
î the bandwidth theorem DWDt@2p is not satisfied
Two options:
•If tSQ=32ns î DW=1.95x108 Hz The first is preferable because it
allows single-qubit rotations in
•If DW=1.9 MHz î tSQ=3.3 ms a shorter time
The frequency bandwidth depends on g(r). Since tSQ is given, the bandwidth
theorem allows us to put a constraint on g(r) and consequently on r, the distance
between the two encased particles
30. Conclusions:
Condition Dwp>>g(r) (1)
allows to know exactly the frequency bandwidth, i.e.
∆Ω = 8 g (r )
Since Dtª32ns, from the bandwidth theorem DWDtª1, we obtain
8
∆Ω = 8 g (r ) = 1.96 ×10 Hz
which implies g(r)=2.45x107Hz and rª1.5nm. This value of r can be
obtained by attaching functional groups between the two buckys.
In this case The system would be a good candidate
as a building block for quantum
π T
π /4
tout ≈ ≅ 1.6 ×10 s ⇒ n = π 24 ≥ 10 4
−8
/
computers and would allow the
8 g (r ) tout possibility of applying quantum
error correcting codes
31. From (1)îDwp>109HzîNew addressing scheme:
We need to investigate alternative designs for addressing each single qubit,
which can allow the achievement of the desirable value of Dwp
• Quantum Cellular Automaton with different species of encased particles
the two particles have to be characterized by a very different value
of the gyromagnetic ratio g
•New design for the magnetic field gradient more steep magnetic field gradient
Finally:
Is it exprimentally possible to
realize single-qubit rotations in
a time shorter than t=32 ns? T2
n= ≅ 10 4
If so î π(
toutCNOT )
32. Scalability: Buckyballs can be easily maneuvered:
• buckyballs embedded in a silicon substrate
• Peapod: buckyballs in a nanotube
proposal: improved T2 in a peapod
Readout: difficulty in the readout of single electron spins.
TNT(erbium-doped) fullerene promising candidates for the readout
Promising results of recent experiments:
•direct excitation of IONC STATES in TNT’sïopens the opportunity of identifying
useful readout transitions and coherently and selectively excite these transitions
•Application of suitable magnetic fields on TNT samplesïthe observed spectrum split
confirms that Er3+ ions are Kramer ions. They maintain the two-fold degeneracy in their
quantum states even under complete crystal-field splittingï ENCODING of a QUBIT
in this pseudo-1/2 spin and EXCITING selective luminecsent transitionsï COULD
ALLOW THE DETECTION OF INDIVIDUAL SPIN STATES
33. TWO-SPIN SYSTEM
TWO-QUBIT GATES: naturally accomplished through the mutual spin dipolar interaction
SINGLE-QUBIT GATES: can be performed through the selective resonant interaction
between the MW-field and the spin to be transformed
Total Hamiltonian of the two-spin system in the rotating frame
H (t ) = H DD + H US
= g (r )[cos(2ω01 − 2ω02 )t (σ x1σ x2 + σ y1σ y2 ) − 2σ z1σ z2 ]
− ωa1 [cos[(ωm1 − 2ω01 )t + φ ]σ x1 − sin[(ωm1 − 2ω01 )t + φ ]σ y1 ]
− ωa2 [cos[(ωm2 − 2ω0 2 )t + φ ]σ x2 − sin[(ωm2 − 2ω02 )t + φ ]σ y2 ]
where HDD is the dipolar interaction term and HUS is the interaction
between two uncoupled spins and the MW-field
34. Since H(t) is time-dependent î Unitary time-evolution
t
U (t , t0 ) = T exp[−i ∫ H (t ' )dt ']
t0
T is the time-ordering operator
In order to easily perform unitary transformations, the Hamiltonian has to be
time-independent, such that the unitary evolution can be written as U(t)=exp[-iHt].
To cancel the time-dependence in H(t) we chose:
• ω0 = ω0
1 2
the precession frequencies of the two spins are equal
• ωm1, 2 = 2ω01, 2 resonant MW-field
ASSUMPTION U (t ) = e − iHt = e − i ( H DD + HUS ) t ≈ e − iHUS t
The interaction terms between two uncoupled spins and a MW-field dominate
the time evolutionï the spin dipolar interaction is negligible ï single-qubit rotation
can be performed in good approximation
35. Since in the realistic case the dipolar interaction is always
present, we cannot reproduce single-qubit rotations
in perfect agreement with the theoretical predictions.
However, the dipolar interaction is essential for performing
two-qubit transformations
fl
Two-qubit gates:can be realized by allowing the system to
evolve freely under the action of the mutual
spin dipolar interaction.
Since the dipolar interaction couples the two spins, it naturally
realizes two-qubit gates