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International Journal of Modern Physics B

Magnetic resonance with number states versus coherent states

Navin Khaneja

Published Dec 3, 2018• 0 citations
Publisher

Abstract

In this paper, we study the interaction of quantized radio-frequency (rf)/microwave-field with nuclear spin in Nuclear Magnetic Resonance (NMR) or electron spin in Electron Paramagnetic Resonance (EPR). In magnetic resonance experiments, interaction of quantized rf-field leads to entanglement of spin with the electromagnetic field. In an entangled state, the spins are depolarized with no net transverse magnetization, which cannot give a detectable signal in inductive detection (or Q detection) that detects transverse magnetization. We show that when the electromagnetic field is in coherent state, inductive detection becomes possible. We use the mathematics of quantum optics 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.

DOI
10.1142/s0217979218503368

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In this paper, we study the interaction of quantized radio-frequency (rf)/microwave-field with nuclear spin in Nuclear Magnetic Resonance (NMR) or electron spin in Electron Paramagnetic Resonance (EPR). In magnetic resonance experiments, interaction of quantized rf-field leads to entanglement of spin with the electromagnetic field. In an entangled state, the spins are depolarized with no net transverse magnetization, which cannot give a detectable signal in inductive detection (or Q detection) that detects transverse magnetization. We show that when the electromagnetic field is in coherent state, inductive detection becomes possible. We use the mathematics of quantum optics 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.
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.
Using electron paramagnetic resonance (EPR) at Rabi frequencies, we study the Bloch-Siegert effect in the coherent dynamics of electron spin states driven by a bichromatic radiation consisting of a transverse microwave (MW) and longitudinal radio frequency (RF) fields. The Rabi oscillations on the electron spin states dressed by the MW field are observed at the double resonance, when the MW frequency equals the Larmor frequency of the spin system and the frequency ωrf of the RF field is close to the Rabi frequency ω1 in the MW field. The dispersive (|ωrf - ω1| ~ ωrf, ω1) and near-resonant (ωrf ≈ ω1) regimes are investigated. We demonstrate that the Bloch-Siegert shift increases with increas- ing RF amplitude and at the fixed RF amplitude it has the largest values in the low-frequency (ωrf < ω1) range of the dispersive regime. Experimental results were obtained in the time-resolved EPR for E' centres in quartz.
We show that there is excellent quantitative agreement between the usual semiclassical predictions for nuclear magnetic resonance experiments (quantized spins interacting with a classical radio frequency magnetic field, including radiation damping), and the predictions of a fully quantized description of the same experiment. We also propose a pictorial description of the relevant dynamical quantum states, which helps to lead the intuition towards the same conclusion. Our model ignores spin relaxation, and simulates the equivalent damping resistor of the sample circuit by a very long piece of lossless coaxial cable. The classical rf field (pulses and free induction decay) is described quantum mechanically by means of Glauber’s “coherent” or “quasiclassical” states.
Modern experimental techniques can generate magnetic fields of the form H(t) = H0 z-hat + H1 [x-hat cos(ωt) + y-hat sin(ωt)], at frequencies within an order of magnitude of the nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR) frequencies, ωn0 and ωe0, respectively, when acting on atoms or molecules. We derive simple closed-form expressions for the exact nuclear- and electronic-spin wave functions that enable controlled transitions between entangled states, allowing an atom or molecule to function as a quantum computer. These solutions also enable precise NMR or EPR measurements of nuclear moments in atoms and molecules. We present examples relevant to measurements of the nuclear moments of 14N, 7Li, and 133Cs. Because existing hyperfine measurements of the lowest three nuclear moments of 133Cs are mutually inconsistent, the proposed NMR/EPR experiments provide a route to measuring all seven of its nuclear moments with high precision.
The dynamics of a two-level spin system dressed by bichromatic radiation is studied under the conditions of double resonance when the frequency of one (microwave) field is equal to the Larmor frequency of the spin system and the frequency of the other (radio-frequency) field ω rf is close to the Rabi frequency ω 1 in a microwave field. It is shown theoretically that Rabi oscillations between dressed-spin states with the frequency ɛ are accompanied by higher-frequency oscillations at frequencies n ω rf and n ω rf ± ɛ, where n = 1, 2,.... The most intense among these are the signals corresponding to n = 1. The counter-rotating (antiresonance) components of the RF field give rise to a shift of the dressed-state energy, i.e., to a frequency shift similar to the Bloch-Siegert shift. In particular, this shift is manifested as the dependence of the Rabi-oscillation frequency ɛ on the sign of the detuning ω 1 − ω rf from resonance. In the case of double resonance, the oscillation amplitude is asymmetric; i.e., the amplitude at the sum frequency ω rf + ɛ increases, while the amplitude at the difference frequency ω rf − ɛ decreases. The predicted effects are confirmed by observations of the nutation signals of the electron paramagnetic resonance (EPR) of E ′ 1 centers in quartz and should be taken into account to realize qubits with a low Rabi frequency in solids.
We propose a new approach to the measurement of a single spin state, based on nuclear magnetic resonance (NMR) techniques and inspired by the coherent control over many-body systems envisaged by quantum information processing. A single target spin is coupled via the magnetic dipolar interaction to a large ensemble of spins. Applying radio frequency pulses, we can control the evolution so that the spin ensemble reaches one of two orthogonal states whose collective properties differ depending on the state of the target spin and are easily measured. We first describe this measurement process using quantum gates; then we show how equivalent schemes can be defined in terms of the Hamiltonian and thus implemented under conditions of real control, using well established NMR techniques. We demonstrate this method with a proof of principle experiment in ensemble liquid state NMR and simulations for small spin systems.