If NMR systems are to be used as practical quantum computers, the number of coupled spins will need to be so large that it is not feasible to rely on purely heteronuclear spin systems. The implementation of a quantum logic gate imposes certain constraints on the motion of those spins not directly involved in that gate, the so-called "spectator" spins; they must be returned to their initial states at the end of the sequence. As a result, a homonuclear spin system where there is appreciable coupling between every pair of spins would seem to require a refocusing scheme that doubles in complexity and duration for every additional spectator spin. Fortunately, for the more realistic practical case where long-range spin-spin couplings can be neglected, simpler refocusing schemes can be devised where the overall duration of the sequence remains constant and the number of soft pulses increases only linearly with the number of spectator spins. These ideas are tested experimentally on a six qubit system: the six coupled protons of inosine.
A three-qubit 13C solid-state nuclear magnetic resonance (NMR) system for quantum information processing, based on the malonic acid molecule, is used to demonstrate high-fidelity universal quantum control via strongly-modulating radio-frequency pulses. This control is achieved in the strong-coupling regime, in which the timescales of selective qubit addressing and of two-qubit interactions are comparable.
We present a new concept for homonuclear dipolar recoupling in magic-angle-spinning (MAS) solid-state NMR experiments which avoids the problem of dipolar truncation. This is accomplished through the introduction of a new NMR pulse sequence design principle: the triple oscillating field technique. We demonstrate this technique as an efficient means to accomplish broadband dipolar recoupling of homonuclear spins, while decoupling heteronuclear dipolar couplings and anisotropic chemicals shifts and retaining influence from isotropic chemical shifts. In this manner, it possible to synthesize Ising…
In NMR (Nuclear Magnetic Resonance) quantum computation, the selective control of multiple homonuclear spins is usually slow because their resonance frequencies are very close to each other. To quickly implement controls against decoherence effects, this paper presents an efficient numerical algorithm fordesigning minimum-time local transformations in two homonuclear spins. We obtain an accurate minimum-time estimation via geometric analysis on the two-timescale decomposition of the dynamics. Such estimation narrows down the range of search for the minimum-time control with a gradient-type opt…
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.
The one-dimensional homonuclear periodic array of nuclear spins I = 1/2, owing to hyperfine interaction of nuclear spins with electronic magnetic moments in antiferromagnetic structure, is considered. The neighbor nuclear spins in such array are opposite oriented and have resonant frequencies determined by hyperfine interaction constant, applied magnetic field value and interaction with the left and right nuclear neighbor spins. The resonant frequencies difference of nuclear spins, when the neighbor spins have different and the same states, is used to control the spin dynamics by means of sele…
We investigate the dynamics of entanglement and quantum discord of two qubits in liquid state homonuclear NMR. Applying a phenomenological description for NMR under relaxation process, and taking a group of typical parameters of NMR, we show that when a zero initial state $|00> $ experiences a relaxation process, its entanglement disappears completely after a sequence of so-called sudden deaths and revivals, while the quantum discord retains remarkable values after a sequence of oscillations. That is to say, the quantum discord is more robust than entanglement.
Swept-frequency pulses have found applications in a wide range of areas including spectroscopic techniques where efficient control of spins is required. For many of these applications, a good understanding of the evolution of spin systems during these pulses plays a vital role, not only in describing the mechanism of techniques, but also in enabling new methodologies. In magnetic resonance spectroscopy, broadband inversion, refocusing, and excitation using these pulses are among the most used applications in NMR, ESR, MRI, and $in$ $vivo$ MRS. In the present survey, a general expression for ch…
We report the implementation of the central building block of the Schulman-Vazirani procedure for fully polarizing a subset of two-level quantum systems which are initially only partially polarized. This procedure consists of a sequence of unitary operations and incurs only a quasi-linear overhead in the number of quantum systems and operations required. The key building block involves three quantum systems and was implemented on a homonuclear three-spin system using room temperature liquid state nuclear magnetic resonance (NMR) techniques. This work was inspired by the state initialization ch…
In the strong coupling regime with J-coupling much larger than chemical shift differences, J-coupling spectroscopy enables spectral identification of molecules even when conventional NMR fails. While this classically required the presence of a heteronucleus, we recently showed that J-coupling spectra can be acquired in many homonuclear systems using spin-lock induced crossing (SLIC). Here, we present an alternative method using a spin echo train in lieu of a spin-locking SLIC pulse, which has a number of advantages. In particular, spin echo acquisition within the pulse train enables simultaneo…
Mapping the physical dipolar Hamiltonian of a solid-state network of nuclear spins onto a system of nearest-neighbor couplings would be extremely useful for a variety of quantum information processing applications, as well as NMR structural studies. We demonstrate such a mapping for a system consisting of an ensemble of spin pairs, where the coupling between spins in the same pair is significantly stronger than the coupling between spins on different pairs. An amplitude modulated RF field is applied on resonance with the Larmor frequency of the spins, with the frequency of the modulation match…
Nuclear magnetic resonance (NMR) spectroscopy usually requires high magnetic fields to create spectral resolution among different proton species. At low fields, chemical shift dispersion is insufficient to separate the species, and the spectrum exhibits just a single line. In this work, we demonstrate that spectra can nevertheless be acquired at low field using a novel pulse sequence called spin-lock induced crossing (SLIC). This probes energy level crossings induced by a weak spin-locking pulse and produces a unique J-coupling spectrum for most organic molecules. Unlike other forms of low-fie…
We demonstrate experimentally the usefulness of selective pulses in NMR to perform quantum computation. Three different techniques based on selective pulse excitations have been proposed to prepare a spin system in a pseudo-pure state. We describe the design of novel ``portmanteau'' gates using the selective manipulation of level populations. A selective pulse implementation of the Deutsch-Jozsa algorithm for a two-qubit and a three-qubit quantum computer is demonstrated.
Quantum Information processing by NMR with small number of qubits is well established. Scaling to higher number of qubits is hindered by two major requirements (i) mutual coupling among qubits and (ii) qubit addressability. It has been demonstrated that mutual coupling can be increased by using residual dipolar couplings among spins by orienting the spin system in a liquid crystalline matrix. In such a case, the heteronuclear spins are weakly coupled but the homonuclear spins become strongly coupled. In such circumstances, the strongly coupled spins can no longer be treated as qubits. However,…
We demonstrate how NMR can in principle be used to implement all the elements required to build quantum computers, and briefly discuss the potential applications of insights from quantum logic to the development of novel pulse sequences with applications in more conventional NMR experiments.
Development of Neural Network-Based Optimal Control Pulse Generator for Quantum Logic Gates Using the GRAPE Algorithm in NMR Quantum Computer
Sharif University of Technology（谢里夫理工大学）；Centre for Quantum Engineering and photonics Technology, Sharif University of Technology（谢里夫理工大学量子工程与光子技术中心）；Koc University（科奇大学）
Geometric phases have been used in NMR, to implement controlled phase shift gates for quantum information processing, only in weakly coupled systems in which the individual spins can be identified as qubits. In this work, we implement controlled phase shift gates in strongly coupled systems, by using non-adiabatic geometric phases, obtained by evolving the magnetization of fictitious spin-1/2 subspaces, over a closed loop on the Bloch sphere. The dynamical phase accumulated during the evolution of the subspaces, is refocused by a spin echo pulse sequence and by setting the delay of transition …
I describe the use of techniques based on composite rotations to combat systematic errors in quantum logic gates. Although developed and described within the context of Nuclear Magnetic Resonance (NMR) quantum computing these sequences should be applicable to other implementations of quantum computation.
In this work, we present an implementation of quantum logic gates and algorithms in a three effective qubits system, represented by a (I = 7/2) NMR quadrupolar nuclei. To implement these protocols we have used the strong modulating pulses (SMP). The various stages of each implementation were verified by quantum state tomography (QST). It is presented here the results for the computational base states, Toffolli logic gates, and Deutsch-Jozsa and Grover algorithms. Also, we discuss the difficulties and advantages of implementing such protocols using the SMP technique in quadrupolar systems.
We propose a nanostructure switch based on nuclear magnetic resonance (NMR) which offers reliable quantum gate operation, an essential ingredient for building a quantum computer. The nuclear resonance is controlled by the magic number transitions of a few-electron quantum dot in an external magnetic field.
While Nuclear Magnetic Resonance (NMR) techniques are unlikely to lead to a large scale quantum computer they are well suited to investigating basic phenomena and developing new techniques. Indeed it is likely that many existing NMR techniques will find uses in quantum information processing. Here I describe how the composite rotation (composite pulse) method can be used to develop quantum logic gates which are robust against systematic errors.
Characterizing and understanding noise affecting quantum states has immense benefits in spectroscopy as well as in realizing quantum devices. Transverse relaxation times under a set of dynamical decoupling (DD) sequences with varying interpulse delays were earlier used for obtaining the noise spectral densities of single-qubit coherences. In this work, using a pair of homonuclear spins and NMR techniques, we experimentally characterize noise in certain decoherence-free subspaces. We also explore the noise of similar states in a heteronuclear spin pair. Further, using a 10-qubit system, we inve…
The use of nuclear magnetic resonance (NMR) to carry out quantum information processing (QIP) often requires the preparation, transformation, and detection of pseudopure states. In our previous work, it was shown that the use of pairs of pseudopure states (POPS) as a basis for QIP is very convenient because of the simplicity in experimental execution. It is now further demonstrated that the product of the NMR spectra corresponding to two sets of POPS that share a common pseudopure state has the same peak frequencies as those of the common (single) pseudopure state. Examples of applying two dif…
Quantum computing using two-dimensional NMR has recently been described using scalar coupling evolution technique [J. Chem. Phys.,109,10603 (1998)]. In the present paper, we describe two-dimensional NMR quantum computing with the help of selective pulses. A number of logic gates are implemented using two and three qubits with one extra observer spin. Some many-in-one gates (or Portmanteau gates) are implemented. Toffoli gate (or AND/NAND gate) and OR/NOR gates are implemented on three qubits. Deutsch-Jozsa quantum algorithm for one and two qubits, using one extra work qubit, has also been impl…
We present a genetic algorithm for finding a set of pulse sequences, or rotations, for a given quantum logic gate, as implemented by NMR. We demonstrate the utility of the method by showing that shorter sequences than have been previously published can be found for both a CNOT and for the central part of Shor's algorithm (for N=15.) Artificial intelligence techniques like the genetic algorithm here presented have an enormous potential for simplifying the implementation of working quantum computers.
NMR is emerging as a valuable testbed for the investigation of foundational questions in quantum mechanics. The present paper outlines the preparation of a class of mixed states, called pseudo-pure states, that emulate pure quantum states in the highly mixed environment typically used to describe solution-state NMR samples. It also describes the NMR observation of spinor behavior in spin 1/2 nuclei, the simulation of wave function collapse using a magnetic field gradient, the creation of entangled (or Bell) pseudo-pure states, and a brief discussion of quantum computing logic gates, including …
In a lattice ${\cal L}$ of nuclear spins with ABCABCABC... type periodic structure embedded in a single-crystal solid, each ABC-unit can be used to store quantum information and the information can be moved around via some cellular shifting mechanism. Impurity doping marks a special site D$\not\in{\cal L}$ which together with the local spin lattice constitute a quantum automaton where the D site serves as the input/output port and universal quantum logic is done through two-body interactions between two spins at D and a nearby site. The novel NMR quantum computer can be easily scaled up and ma…
Electron spin echo envelope modulation (ESEEM) has been observed for the first time from a coupled hetero-spin pair of electron and nucleus in liquid solution. Previously, modulation effects in spin echo experiments have only been described in liquid solutions for a coupled pair of homonuclear spins in NMR or a pair of resonant electron spins in EPR. We observe low-frequency ESEEM (26 and 52 kHz) due to a new mechanism present for any electron spin with S>1/2 that is hyperfine coupled to a nuclear spin. In our case these are electron spin (S=3/2) and nuclear spin (I=1) in the endohedral fuller…
We give a general method of construting quantum circuit for random \QTR{it}{satisfiability} (SAT) problems with the basic logic gates such as multi-qubit controlled-NOT and NOT gates. The sizes of these circuits are almost the same as the sizes of the SAT formulas. Further, a parallelization scheme is described to solve random SAT problems efficiently through these quantum circuits in \QTR{it}{nuclear magnetic resonance} (NMR) ensemble quantum computing. This scheme exploits truly mixed states as input states rather than pseudo-pure states, and combines with the topological nanture of the NMR …
Quantum process tomography is a procedure by which the unknown dynamical evolution of an open quantum system can be fully experimentally characterized. We demonstrate explicitly how this procedure can be implemented with a nuclear magnetic resonance quantum computer. This allows us to measure the fidelity of a controlled-not logic gate and to experimentally investigate the error model for our computer. Based on the latter analysis, we test an important assumption underlying nearly all models of quantum error correction, the independence of errors on different qubits.
The three-qubit conditional swap gate(Fredkin gate) is a universal gate that can be used to create any logic circuit and has many direct usages. In this paper, we experimentally realized Fredkin gate with only three transition pulses in solution of alanine. It appears that no experimental realization of Fredkin gate with fewer pulses has yet been reported up to now. In addition, the simple structure of our scheme makes it easy to be implemented in experiments.
A proposal for a scalable, solid-state implementation of a quantum computer is presented. Qubits are fluorine nuclear spins in a solid crystal of fluorapatite [Ca_5 F(PO_4)_3] with resonant frequencies separated by a large field gradient. Quantum logic is accomplished using nuclear-nuclear dipolar couplings with decoupling and selective recoupling RF pulse sequences. Magnetic resonance force microscopy is used for readout. As many as 300 qubits can be implemented in the laboratory extremes of T=10 mK and B_0=20 T with the existing sensitivity of force microscopy.
Liquid phase NMR is a general purpose test-bed for developing methods of coherent control relevant to quantum information processing. Here we extend these studies to the coherent control of logical qubits and in particular to the unitary gates necessary to create entanglement between logical qubits. We report an experimental implementation of a conditional logical gate between two logical qubits that are each in decoherence free subspaces that protect the quantum information from fully correlated dephasing.
We propose an experimentally feasible scheme to achieve quantum computation based on a pair of orthogonal cyclic states. In this scheme, quantum gates can be implemented based on the total phase accumulated in cyclic evolutions. In particular, geometric quantum computation may be achieved by eliminating the dynamic phase accumulated in the whole evolution. Therefore, both dynamic and geometric operations for quantum computation are workable in the present theory. Physical implementation of this set of gates is designed for NMR systems. Also interestingly, we show that a set of universal geomet…
We briefly describe the use of GRAPE pulses to implement quantum logic gates in NMR quantum computers, and discuss a range of simple extensions to the core technique. We then consider a range of difficulties which can arise in practical implementations of GRAPE sequences, reflecting non-idealities in the experimental systems used.
We describe the use of composite rotations to combat systematic errors in single qubit quantum logic gates and discuss three families of composite rotations which can be used to correct off-resonance and pulse length errors. Although developed and described within the context of NMR quantum computing these sequences should be applicable to any implementation of quantum computation.
The experimental realisation of the basic constituents of quantum information processing devices, namely fault-tolerant quantum logic gates, requires conditional quantum dynamics, in which one subsystem undergoes a coherent evolution that depends on the quantum state of another subsystem. In particular, the subsystem may acquire a conditional phase shift. Here we consider a novel scenario in which this phase is of geometric rather than dynamical origin. As the conditional geometric (Berry) phase depends only on the geometry of the path executed it is resilient to certain types of errors, and o…
This letter presents a two-dimensional nuclear magnetic resonance(NMR) approach for constructing a two-logical-qubit decoherence-free subspace (DFS) based on the fact that the three protons in a CH3 spin system can not be resolved in one-dimension NMR spectroscopy, but to a certain extent, can be distinguished by two-dimensional multiple-quantum NMR. We used four noisy physical nuclear spins, including three protons and one carbon in the CH3 spin system, to generate two decoherence-free logical quantum bits. It made full use of the unaddressed spins which could not be used in one-dimensional s…
We propose a method of controlling a quantum logic gate in a solid-state NMR quantum computer. A switchable inter-qubit coupling can be generated by using the longitudinal component of the Suhl-Nakamura interaction induced by a local singlet-triplet excitation in a 1-D antiferromagnet with a spin gap.
Internal states of different ions in an electrodynamic trap are coupled when a static magnetic field is applied -- analogous to spin-spin coupling in molecules used for NMR. This spin-spin interaction can be used, for example, to implement quantum logic operations in ion traps using NMR methods. The collection of trapped ions can be viewed as a $N$-qubit molecule with adjustable coupling constants.
Conditional geometric phase shift gate, which is fault tolerate to certain errors due to its geometric property, is made by NMR technique recently under adiabatic condition. By the adiabatic requirement, the result is inexact unless the Hamiltonian changes extremely slowly in the process. However, in quantum computation, everything has to be completed within the decoherence time. High running speed of every gate in quantum computation is demanded because the power of quantum computer can be exponentially proportional to the maximum number of logic gate operation that can be taken sequentially …
We report the first use of "logical labeling" to perform a quantum computation with a room-temperature bulk system. This method entails the selection of a subsystem which behaves as if it were at zero temperature - except for a decrease in signal strength - conditioned upon the state of the remaining system. No averaging over differently prepared molecules is required. In order to test this concept, we execute a quantum search algorithm in a subspace of two nuclear spins, labeled by a third spin, using solution nuclear magnetic resonance (NMR), and employing a novel choice of reference frame t…
The enormous theoretical potential of Quantum Information Processing (QIP) is driving the pursuit for its practical realization by various physical techniques. Currently Nuclear Magnetic Resonance (NMR) has been the forerunner by demonstrating a majority of quantum algorithms. In NMR, spin systems consisting of coupled nuclear spins are utilized as qubits. In order to carry out QIP, a spin system has to meet two major requirements: (i) qubit addressability and (ii) mutual coupling among the qubits. It has been demonstrated that the magnitude of the mutual coupling among qubits can be increased…
I describe the use of techniques based on composite rotations to combat systematic errors in controlled phase gates, which form the basis of two qubit quantum logic gates. Although developed and described within the context of Nuclear Magnetic Resonanace (NMR) quantum computing these sequences should be applicable to any implementation of quantum computation based on Ising couplings. In combination with existing single qubit gates this provides a universal set of robust quantum logic gates.
We describe a method for improving coherent control through the use of detailed knowledge of the system's Hamiltonian. Precise unitary transformations were obtained by strongly modulating the system's dynamics to average out unwanted evolution. With the aid of numerical search methods, pulsed irradiation schemes are obtained that perform accurate, arbitrary, selective gates on multi-qubit systems. Compared to low power selective pulses, which cannot average out all unwanted evolution, these pulses are substantially shorter in time, thereby reducing the effects of relaxation. Liquid-state NMR t…
The composite rotation approach has been used to develop a range of robust quantum logic gates, including single qubit gates and two qubit gates, which are resistant to systematic errors in their implementation. Single qubit gates based on the BB1 family of composite rotations have been experimentally demonstrated in a variety of systems, but little study has been made of their application in extended computations, and there has been no experimental study of the corresponding robust two qubit gates to date. Here we describe an application of robust gates to Nuclear Magnetic Resonance (NMR) stu…
Liquid crystals offer several advantages as solvents for molecules used for nuclear magnetic resonance quantum computing (NMRQC). The dipolar coupling between nuclear spins manifest in the NMR spectra of molecules oriented by a liquid crystal permits a significant increase in clock frequency, while short spin-lattice relaxation times permit fast recycling of algorithms, and save time in calibration and signal-enhancement experiments. Furthermore, the use of liquid crystal solvents offers scalability in the form of an expanded library of spin-bearing molecules suitable for NMRQC. These ideas ar…
Quantum superposition and entanglement of physical states can be harnessed to solve some problems which are intractable on a classical computer implementing binary logic. Several algorithms have been proposed to utilize the quantum nature of physical states and solve important problems. For example, Shor's quantum algorithm is extremely important in the field of cryptography since it factors large numbers exponentially faster than any known classical algorithm. Another celebrated example is the Grovers quantum algorithm. These algorithms can only be implemented on a quantum computer which oper…
Coherent oscillations between any two levels from four nuclear spin states of I=3/2 have been demonstrated in a nanometre-scale NMR semiconductor device, where nuclear spins are all-electrically controlled. Using this device, we discuss quantum logic operations on two fictitious qubits of the I=3/2 system, and propose a quantum state tomography scheme based on the measurement of longitudinal magnetization, $M_z$.
The requirement of performing both single-qubit and two-qubit operations in the implementation of universal quantum logic often leads to very demanding constraints on quantum computer design. We show here how to eliminate the need for single-qubit operations in a large subset of quantum computer proposals: those governed by isotropic and XXZ,XY-type anisotropic exchange interactions. Our method employs an encoding of one logical qubit into two physical qubits, while logic operations are performed using an analogue of the NMR selective recoupling method.
The controlled-NOT gate and controlled square-root NOT gate play an important role in quantum algorithm. This article reports the experimental results of these two universal quantum logic gates (controlled square-root NOT gate and controlled-NOT gate) on a 7-qubit NMR quantum computer. Further, we propose a simple experimental method to measure and correct the error in the controlled phase-shift gate, which is helpful to construct a more perfect phase-shift gate experimentally and can also be used in more qubits discrete Fourier transformation.
Many coherence transfer experiments in Nuclear Magnetic Resonance Spectroscopy, involving network of coupled spins, use temporary spin-decoupling to produce desired effective Hamiltonians. In this paper, we show that significant time can be saved in producing an effective Hamiltonian, if spin-decoupling is avoided. We provide time optimal pulse sequences for producing an important class of effective Hamiltonians in three spin networks. These effective Hamiltonians are useful for coherence transfer experiments and implementation of quantum logic gates in NMR quantum computing. It is demonstrate…
Quantum computation is a subject of much theoretical promise, but has not been realized in large scale, despite the discovery of fault-tolerant procedures to overcome decoherence. Part of the reason is that the theoretically modest requirements still present daunting experimental challenges. The goal of this Dissertation is to reduce various resources required for robust quantum computation, focusing on quantum error correcting codes and solution NMR quantum computation. A variety of techniques have been developed, including high rate quantum codes for amplitude damping, relaxed criteria for q…
Most quantum error correcting codes are predicated on the assumption that there exists a reservoir of qubits in the state $\ket{0}$, which can be used as ancilla qubits to prepare multi-qubit logical states. In this report, we examine the consequences of relaxing this assumption, and propose a method to increase the fidelity produced by a given code when the ancilla qubits are initialized in mixed states, using the same number of qubits, at most doubling the number of gates. The procedure implemented consists of altering the encoding operator to include the inverse of the unitary operation use…
We study a novel class of open-loop control protocols constructed to perform arbitrary nontrivial single-qubit logic operations robust against time-dependent non-Markovian noise. Amplitude and phase modulation protocols are crafted leveraging insights from functional synthesis and the basis set of Walsh functions. We employ the experimentally validated generalized filter-transfer function formalism in order to find optimized control protocols for target operations in SU(2) by defining a cost function for the filter-transfer function to be minimized through the applied modulation. Our work deta…
Our work addresses the problem of generating maximally entangled two spin-1/2 (qubit) symmetric states using NMR, NQR, Lipkin-Meshkov-Glick Hamiltonians. Time evolution of such Hamiltonians provides various logic gates which can be used for quantum processing tasks. Pairs of spin-1/2's have modeled a wide range of problems in physics. Here we are interested in two spin-1/2 symmetric states which belong to a subspace spanned by the angular momentum basis {|j = 1, μ>; μ = +1, 0,-1}. Our technique relies on the decomposition of a Hamiltonian in terms of SU(3) generators. In this context, we defin…
We have designed efficient quantum circuits for the three-qubit Toffoli (controlled-controlled NOT) and the Fredkin (controlled-SWAP) gate, optimized via genetic programming methods. The gates thus obtained were experimentally implemented on a three-qubit NMR quantum information processor, with a high fidelity. Toffoli and Fredkin gates in conjunction with the single-qubit Hadamard gates form a universal gate set for quantum computing, and are an essential component of several quantum algorithms. Genetic algorithms are stochastic search algorithms based on the logic of natural selection and bi…
Physical implementation of Quantum Information Processing (QIP) by liquid-state Nuclear Magnetic Resonance (NMR), using weakly coupled spin-1/2 nuclei of a molecule, is well established. Nuclei with spin$>$1/2 oriented in liquid crystalline matrices is another possibility. Such systems have multiple qubits per nuclei and large quadrupolar couplings resulting in well separated lines in the spectrum. So far, creation of pseudopure states and logic gates have been demonstrated in such systems using transition selective radio-frequency pulses. In this paper we report two novel developments. First,…
