arXiv subjects

Navin Khaneja

Publications and source records attributed to Navin Khaneja.

At least 19 recordsLinked to original sources

arXiv2024

Feedback Pulses

We have a new paradigm to design NMR pulses. Pulses, we call feedback pulses. We want broadband inversion and excitation. We have many offsets, start evolving them all starting from the north pole. Monitor them on the Bloch sphere, see which offset is worst (most away from south pole). Change the rf-phase to the offset ($π/2$ ahead of offset transverse magnetization phase) and irradiate at that offset frequency and evolve for some time and monitor and repeat, looking for worst offset. When we are on resonance to a offset, we are doing well, inverting it and when we are off resonant, we don't hurt much (even if hurt little, we will come back to the offset in good time). By the process of monitoring, and setting phase we eventually push everything to the south pole and bingo, we have an inversion pulse. Feedback is done in simulation, but what results in end is a broadband inversion pulse. For broadband excitation, start with all offsets (symmetric around origin) on y axis. By feedback push them to the south pole. When we run the resulting sequence backward with phases, $π$ incremented, we will get an excitation pulse. For band-selective excitation pulse put offsets in pass band on the $y$ axis and in the stop band on the south pole. Use feedback to push everything to the south pole. Again, run backwards with $π$ incremented phases, to get band selective excitation. Suddenly, we have it all, simple and easy. The paper, introduces the feedback pulse algorithm, simulations and experiments.

quant-ph↗

arXiv2023

Critique of Feynman Propagator, the $\E \cdot x$ gauge

Consider Møller scattering. Electrons with momentum $p$ and $-p$ scatter by exchange of photon say in $z$ direction to $p+q$ and $-(p+q)$. The scattering amplitude is well known, given as Feynman propagator $ \M = \frac{(e \hbar c)^2}{ε_0 V} \frac{\bar{u}(p+q) γ^μ u(p) \ \bar{u}(-(p+q)) γ_μ u(-p)}{q^2}$, where $V$ is the volume of the scattering electrons, $e$ elementary charge and $ε_0$ permitivity of vacuum. But this is not completely correct. Since we exchange photon momentum in $z$ direction, we have two photon polarization $x,y$ and hence the true scattering amplitude should be $$ \M_1 = \frac{(e \hbar c)^2}{ε_0 V} \frac{ \bar{u}(p+q) γ^{x} u(p) \ \bar{u}(-(p+q)) γ_{x} u(-p)\ \ + \bar{u}(p+q) γ^{y} u(p) \ \bar{u}(-(p+q)) γ_{y} u(-p) \ }{q^2}. $$ But when electrons are non-relativistic, $\M_1 \sim 0$. This is disturbing, how will we ever get the coulomb potential, where $\M \sim \frac{(e \hbar c)^2}{ε_0 V q^2}$. Where is the problem ? The problem is with the gauge in Dirac equation. For a plane wave along $z$ direction, with electric field $E_x \sin (kz - ωt)$, the Lorentz gauge is $$ (A_0, A_x, A_y, A_z) = \frac{E_x}ω \cos(kz-ωt)(0, 1, 0, 0)$$. But this gauge is not suited for calculating optical transitions, because we don't recover the Rabi frequency $q E_x d$ ($d$ electric dipole moment). What we find is something orders of magnitude smaller. Nor is it suitable for calculating electron electron scattering because we don't recover Coulomb potential. What we find is something orders of magnitude smaller. Instead, we work with $\E \cdot x$ gauge $$ (A_0, A_x, A_y, A_z) = \frac{-E_x}{2} ( x\ \sin(kz-ωt), -\frac{\cos(kz-ωt)}ω, 0, \frac{x}{c} \sin(kz-ωt) ) $$ ($c$ light velocity) to find everything correct. What we get is new propagator.

quant-ph↗

arXiv2023

Aspects of electron scattering, the elastic, and the inelastic

A electron of mass $m$, when electrically scatters of nucleus, of mass $M$, transfers momentum $q$ to the nucleus. The energy lost by electron is more than the energy gained by the nucleus. The resulting energy goes in exciting the atom to a higher energy state as in Frank Hertz experiment and sodium, neon, mercury vapor lamps, or ionization of atom as in bubble and cloud chamber experiments, or just production of X-rays as in Bremsstraulung. In this paper, we study these phenomenon. These experiments are inelastic scattering experiments. We remark, why neutrinos donot scatter and can penetrate earth, why muons travel further than electrons in materials and why a material like lead plate can slow down electrons and positrons efficiently. We look at the elastic scattering of electrons as in electron diffraction and electron microscopes. We look at scattering of electrons in the condensed matter, these phenomenon range from scattering of electrons of periodic potential, to give Bloch waves, scattering of electrons of phonons and impurities to give resistance, scattering of electrons of lattice to give cooper pairs and superconductivity. We study electron scattering from exchange potential as in Fermi liquid theory and resulting $T^2$ resistance at low temperatures. Electron scattering of exchange potential resulting in chemical reactions. We turn our attention to electron-proton scattering both eleastic and inelastic, as in deep inelastic scattering experiments and understand the independence of ineleastic cross-section of with respect to transferred momentum. We see, why we can just say that there are three quarks in proton from elastic cross-section. Our main contribution in this article is we are detailed at places, we find literature terse.

quant-ph↗

arXiv2022

Extension of a Linear Controller Scheme to Non-Linear Systems and its Application on Inverted Pendulum

This paper presents the control and stabilization of the rotary inverted pendulum based on a general controller scheme. The proposed scheme has its foundation in classical control theory, and the importance of an integrator in disturbance rejection is emphasized. The system's dynamics are obtained by the Euler Lagrange method and are approximated for small-angle as balancing the pendulum is the objective. Experimental results demonstrate that the proposed control scheme can achieve the stabilization of a non-linear system. Also, the boundedness and convergence of the non-linear system with the controller subjected to the initial condition are validated.

eess.SY↗

arXiv2021

A General Controller Scheme for Stabilization & Disturbance Rejection with Application to Non-Linear Systems and its Implementation on 2 DOF Helicopter

A general controller scheme for stabilizing a non-linear system, which has its origin from the linear system theory, is proposed in this paper. The proposed controller can stabilize the non-linear system subjected to initial conditions. An effective way to obtain the controller parameters is presented with the knowledge of the system model. The controller is designed for the linear time-invariant (LTI) system, which can reject any disturbance acting on it. Paper emphasis the idea of an integrator controller in disturbance rejection. The concept is extended to the application to non-linear systems where the non-linearities are assessed as the disturbance to the refined linear part of the system. Boundedness and convergence of the non-linear system with the controller are proved to justify system stabilization. Hardware implementation of the controller on the 2 dof helicopter model is presented with experimental results, which validates the proposed control scheme.

eess.SY↗

arXiv2021

Refined Transformation Approach for Stabilization of MIMO System by Pole Placement

The paper presents a distinctive and straightforward technique for stabilization of multi-variable systems. The idea is to decouple the system state matrix depending on different inputs and outputs. Refined special canonical transformations are described for the design of controller and observer for a single-input and single-output (SISO) case and are extended to multi-input multi-output (MIMO) systems. These transformations help in the stabilization of the error dynamics of the observer and in placing the closed loop poles of the system. The idea is not only in the transformations taken but also how the gain matrices are selected which simplifies the computation.

eess.SY↗

arXiv2018

Electron conduction in solid state via time varying wavevectors

In this paper, we study electron wavepacket dynamics in electric and magnetic fields. We rigorously derive the semiclassical equations of electron dynamics in electric and magnetic fields. We do it both for free electron and electron in a periodic potential. We do this by introducing time varying wavevectors $k(t)$. In the presence of magnetic field, our wavepacket reproduces the classical cyclotron orbits once the origin of the Schröedinger equation is correctly chosen to be center of cyclotron orbit. In the presence of both electric and magnetic fields, our equations for wavepacket dynamics differ from classical Lorentz force equations. We show that in a periodic potential, on application of electric field, the electron wave function adiabatically follows the wavefunction of a time varying Bloch wavevector $k(t)$, with its energies suitably shifted with time. We derive the effective mass equation and discuss conduction in conductors and insulators.

quant-ph↗

arXiv2017

Coherent states in Magnetic Resonance

In NMR experiments, interaction of quantized radio-frequency (rf) field leads to entanglement of nuclear spin with the electromagnetic field. In an entangled state, the nuclear spins are depolarized with no net magnetization, which cannot give a detectable signal in inductive detection. We show that when the electromagnetic field is in coherent state, inductive detection is just true. We develop the mathematics to study the evolution of a coherent rf-field with a sample of all polarized spins. We show that evolution can be solved in closed form as a separable state of rf-field and spin ensemble, where spin ensemble evolves according to Bloch equations in an rf field. We extend the analysis and results to a spin ensemble with Boltzmann polarization. The rabi frequency and coupling strength of spins to rf-field depends on number state of the rf-field. We show that in interaction with a coherent rf-field, this variation in coupling strength, introduces negligible error.

quant-ph↗

arXiv2017

Conservation of energy, density of states and spin lattice relaxation

The starting point of all NMR experiments is a spin polarization which develops when we place the sample in static magnetic field $B_0$. There are excess of spins aligned along $B_0$ (spin up with lower energy) than spins aligned opposite (spin down with higher energy) to the field $B_0$. A natural question is what is the source of this excess spin polarization because relaxation mechanisms can flip a up spin to a down spin and vice-versa. The answer lies in the density of states. When a molecule with spin down flips to spin up it loses energy. This energy goes into increasing the kinetic energy of the molecule in the gas/solution phase. At this increased kinetic energy, there are more rotational-translational states accessible to the molecule than at lower energy. This increases the probability the molecule will spend in spin up state (higher kinetic energy state). This is the source of excess polarization. In this paper, we use an argument based on equipartition of energy to explicitly count the excess states that become accessible to the molecule when its spin is flipped from down to up. Using this counting, we derive the familiar Boltzmann distribution of the ratio of up vs down spins. Although prima facie, there is nothing new in this paper, we find the mode counting argument for excess states interesting. Furthermore, the paper stresses the fact that spin polarization arises from higher density of states at increased kinetic energy of molecules.

cond-mat.mes-hall↗

arXiv2017

Chirp Mixing

In this paper, we develop the theory of chirp mixing. The working principle is simple, given coupled homonuclear spins with offsets in range [-B, B], we adiabatically sweep through the resonances. This achieves cross polarization between the z magnetization of the coupled spins. We repeat this basic operation many times with a supercycle to achieve appropriate mixing time. When we sweep through the resonances, midway between the resonances of the coupled spin I and S, the effective field seen by two spins is the same and hence they precess at same frequency around their effective fields. This means the coupling, which normally gets averaged due to the chemical shift difference is no more averaged for a short time and we get mixing. In this paper, we develop these basic ideas. By virtue of its design, the chirp mixing is much more broadband compared to state of the art methods. The proposed methodology is demonstrated on 13C mixing in a sample of Alanine.

quant-ph↗

arXiv2017

Time optimal control in coupled spin systems: a second order analysis

In this paper, we study some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation. Time optimal control helps to minimize relaxation losses. The ability to synthesize, local unitaries, much more rapidly than evolution of couplings, gives a natural time scale separation in these problems. The generators of evolution, $\g$, are decomposed into fast generators $\k$ (local Hamiltonians) and slow generators $\p$ (couplings) as a Cartan decomposition $\g = \p \oplus \k$. Using this decomposition, we exploit some convexity ideas to completely characterize the reachable set and time optimal control for these problems. In this paper, we carry out a second order analysis of time optimality.

quant-ph↗

arXiv2017

Chirp Excitation

The paper describes the design of broadband chirp excitation pulses in NMR. We first develop a three stage model for understanding chirp excitation in NMR. We then show how a chirp $π$ pulse can be used to refocus the phase of the chirp excitation pulse. The resulting magnetization still has some phase dispersion in it. We show how a combination of two chirp $π$ pulses instead of one can be used to eliminate this dispersion, leaving behind a small residual phase dispersion. The excitation pulse sequence presented here allow exciting arbitrary large bandwidths without increasing the peak rf-amplitude. Although methods presented in this paper have appeared elsewhere, we present complete analytical treatment that elucidates the working of these methods.

quant-ph↗

arXiv2017

Broadband Homonuclear Decoupling

We present a solution to the problem of broadband decoupling of a coupled homonuclear two-spin system. We describe a pulse sequence that creates an effective field perpendicular to the coupling interaction with a magnitude propotional to the chemical shift of the spins over a broad range of chemical shifts. When the chemical shift difference as imprinted on the perpendicular field is greater than the coupling between the spins, we get effective decoupling. The pulse sequence may be useful in various NMR applications.

quant-ph↗

arXiv2017

Double swept band selective excitation

The paper describes the design of band selective excitation and rotation pulses in high resolution NMR by method of double sweep. We first show the design of a pulse sequence that produces band selective excitation to the equator of Bloch sphere with phase linearly dispersed as frequency. We show how this linear dispersion can then be refocused by nesting free evolution between two adiabatic inversions (sweeps). We then show how this construction can be generalized to give a band selective $x$ rotation over desired frequency band. Experimental excitation profiles for the residual HDO signal in a sample of $99.5\%$ D$_2$O are obtained as a function of resonance offset.

quant-ph↗

arXiv2016

Broadband excitation by method of double sweep

The paper describes the design of broadband excitation pulses in high resolution NMR by method of double sweep. We first show the design of a pulse sequence that produces broadband excitation to the equator of Bloch sphere with phase linearly dispersed as frequency. We show how this linear dispersion can then be refocused by nesting free evolution between two adiabatic inversions (sweeps). We then show how this construction can be generalized to exciting arbitrary large bandwidths without increasing the peak rf-amplitude and by incorporating more adiabatic sweeps. Finally, we show how the basic design can then be modified to give a broadband $x$ rotation over arbitrary large bandwidth and with limited rf-amplitude. Experimental excitation profiles for the residual HDO signal in a sample of $99.5\%$ D$_2$0 are displayed as a function of resonance offset. Application of the excitation is shown for $^{13}$C excitation in a labelled sample of Alanine.

quant-ph↗

arXiv2015

Time-optimal polarization transfer from an electron spin to a nuclear spin

Polarization transfers from an electron spin to a nuclear spin are essential for various physical tasks, such as dynamic nuclear polarization in nuclear magnetic resonance and quantum state transformations on hybrid electron-nuclear spin systems. We present time-optimal schemes for electron-nuclear polarization transfers which improve on conventional approaches and will have wide applications.

quant-ph↗

arXiv2015

Reduced coupling with global pulses in quantum registers

Decoupling is an important tool to prolong the coherence time of quantum systems. Most decoupling schemes have been assuming selective controls on the system and it is believed that with global pulses one can only decouple systems with certain coupling terms like secular dipole-dipole coupling. In this article we show that with global pulses it is possible to reduce the coupling strength of other types of coupling, which we demonstrate with Ising coupling. The complexity of such pulses is independent of the size of system.

quant-ph↗

arXiv2013

Efficient synthesis of quantum gates on indirectly coupled spins

Experiments in coherent nuclear and electron magnetic resonance,and quantum computing in general correspond to control of quantum mechanical systems, guiding them from initial to final target states by unitary transformations. The control inputs (pulse sequences) that accomplish these unitary transformations should take as little time as possible so as to minimize the effects of relaxation and decoherence and to optimize the sensitivity of the experiments. Here, we derive a time-optimal sequences as fundamental building blocks for synthesize unitary transformations. Such sequences can be widely implemented on various physical systems, including the simulation of effective Hamiltonians for topological quantum computing on spin lattices. Experimental demonstrations are provided for a system consisting of three nuclear spins.

quant-ph↗