The transient behavior of a two-level spin system coupled to an electric circuit is investigated by using the equations of Bloembergen and Pound. The equations are solved, in the limit where the circuit ringing time is very short compared with all other characteristic times, for two cases: 1) the spin-lattice and spin-spin relaxation times both infinite, with an externally applied driving field, and 2) the spin-lattice relaxation time infinite but the spin-spin relaxation time finite, in the absence of an external field. In case 1), it is shown that the motion of an initially inverted magnetization under the action of an applied signal consists roughly of two stages: in the first stage, the effect of radiation damping is unimportant and the motion of the system is determined principally by the applied signal via the ordinary Bloch equations, whereas in the second stage, the motion is essentially the same as if the applied signal had been turned off and only radiation damping were present. In case 2), it is shown that a delayed peak in the emitted radiation should be observed under certain conditions. The delayed peak condition is identical with that derived by Bloom. Curves are presented showing the peak power and the time at which the delayed peak occurs as functions of the relevant parameters. In connection with the ordinary maser behavior of a two-level spin system, it is shown that for values of the parameters typical of steady-state maser amplification, the effects of radiation damping should be unimportant. Finally, systems are examined for which the radiation damping time is much shorter than all other characteristic times (super-regenerative systems). It is indicated how such systems might be operated as one-shot multivibrators or as linear amplifiers. For the latter type of operation, an expression for the gain is derived which is found to be similar to that encountered in ordinary circuit theory.
We numerically solve the Liouville equation for the Tavis Cummings model of multiple spins coupled to a lossless single mode cavity, starting from an initial condition with small numbers of fully polarized spins tipped by a specified angle, and the cavity in its ground Fock state. Time evolution of the magnetizations and cavity states, following small to medium nutation by a classical field, yields a microscopic quantum mechanical picture of radiation damping in magnetic resonance, and the formation of the free induction signal, that is, the transfer of Zeeman energy, via spin coherence, to cavity coherence. Although the motion of the Bloch vector is nonclassical, our quantum description is related to the macroscopic picture of NMR reception, by showing the close relationship between the usual radiation damping constant, and the quantum mechanical Rabi nutation frequency (as enhanced by cavity coupling and stimulated emission.) That is, each is the product, of a nutation rate per oscillator current, and a current. Although the current in the damping constant is explicitly limited by cavity losses, which do not enter the formula for the Rabi frequency, we nonetheless show (in an appendix) how these losses can be introduced into our problem by means of a master equation. Numerical solution of the classical Bloch-Kirchhoff equations reinforces the conclusion that the strength of the free induction
We apply the methods of cavity quantum electrodynamics (CQED), to obtain a microscopic and fully quantum-mechanical picture of radiation damping in magnetic resonance, and the nascent formation of the free induction signal. Numerical solution of the Tavis-Cummings model - i.e., multiple spins 1∕2 coupled to a lossless single-mode cavity - shows in fine detail the transfer of Zeeman energy, via spin coherence, to excite the cavity - represented here by a quantized LC resonator. The case of a single spin is also solved analytically. Although the motion of the Bloch vector is non-classical, we nonetheless show that the quantum mechanical Rabi nutation frequency (as enhanced by cavity coupling and stimulated emission) gives realistic estimates of macroscopic signal strength and the radiation damping constant in nuclear magnetic resonance. We also show how to introduce dissipation: cavity losses by means of a master equation, and relaxation by the phenomenological method of Bloch. The failure to obtain the full Bloch equations (unless semi-classical conditions are imposed on the cavity) is discussed in light of similar issues arising in CQED (and in earlier work in magnetic resonance as well), as are certain problems relative to quantization of the electromagnetic near-field.
We study the radiation damping and the spin noise of a magnetic material coupled with a resonant circuit.Radiation damping in magnetic resonance is a dissipation phenomenon, where magnetization prepared after a Rabi pulse decays toward its equilibrium state.The magnetic sample loses its energy by the coupling with resonant circuit, that must be tuned in Larmor frequency of the sample spins.Even though this phenomenon had been studied many years ago, no full quantum description was done.We present a quantum Hamiltonian model, that explains the radiation damping.We use quantum Langevin equation method for this task.Beyond radiation damping, we show the magnetization acquires an unusual intrincate motion, if the circuit initial state is coherent.Using the same Langevin equation, we study the sample inuence on the resonant circuit noise.We calculate the current spectral density in the case of thermal equilibrium of whole system.We can verify the method eectiveness, comparing former papers.Moreover we study modications in the circuit noise, if an external oscillating tension is applied.In this situation, other two peaks emerge in the central peak sidebands of the sample absorption spectrum.It leads to appear three dips in circuit current spectrum.This eect is due to the splitting of the spin energy states.We comment about the analogy between this phenomenon and the resonance uorescence in Quantum Optics.
The behavior of a nonlinear oscillator (NO) coupled to a radiation field is investigated. The NO considered is an angular momentum oscillator, of energy $\ensuremath{\hbar}\ensuremath{\omega}{L}_{3}$, that describes the collective effect of a number of identical two-level (or spin-\textonehalf{}) systems under given idealized conditions, a large total angular momentum quantum number
L0 corresponding to a large number of two-level systems. The field is described by a set of modes. Free decay---with the NO initially excited and the radiation field in the ground state---and forced oscillation---with the NO subject to a prescribed resonant driving field---are studied. The analysis is performed both classically and quantum mechanically, using the classical and Heisenberg equations of motion, respectively, so as to display explicitly the difference in the results. Interest in the comparison between the two formalisms is motivated by the expectation that for large
L0, the NO should behave essentially classically, except near its highest-energy state in the absence of a driving field. The general equations of motion are reduced to equations for the NO variables only. In the classical analysis of free decay, expressions for the energy and oscillating coordinates are derived. The decay time is shown to approach infinity as
L3(0) approaches
L0 (the limiting condition being that of unstable equilibrium) and the radiative frequency shift is shown to be approximately proportional to
L3(t). It is also shown that use of the rotating-wave approximation alters qualitatively the expression for the frequency shift. In the classical analysis of forced oscillation, approximate results are obtained for a weak driving field and a strong driving field, the NO being initially in the ground state. The weakfield results exhibit a monotonic approach of
L3(t) to a constant (negative) value---or steady state---at which the power absorbed equals the power radiated; in the strong-field case,
L3(t) oscillates periodically between the limits $\ifmmode\pm\else\textpm\fi{}{L}_{0}$, the coupling to the radiation field having negligible effect on the frequency of this oscillation. In the quantum-mechanical analysis, the equations of motion become simplified for
L0=21, and this case is treated first. The free-decay results are essentially similar to those of the Weisskopf-Wigner theory, exhibiting an exponential decay of
〈L3(t)〉 and a radiative frequency shift in the oscillating coordinates. Under forced oscillation, with $〈{L}_{3}(0)〉=\ensuremath{-}{L}_{0}, 〈{L}_{3}(t)〉$ approaches a constant value either monotonically, if the driving field is sufficiently weak, or by means of a damped oscillation, if the driving field is strong. For
L0>21, the free decay is treated by a method that involves the derivation of a set of expressions for the
kth derivative of
〈L3(t)〉 as an expectation value of a polynomial in
L3 of order
k+1. This set, together with the eigenvalue equation for
L3, is shown to lead to a solution for all the moments of
L3. The method is used to obtain complete solutions for several low values of
L0, and also to obtain initial derivatives of
〈L3(t)〉 as polynomials in
L0 for
〈L3(0)〉=L0. For large
L0, a comparison of classical and quantum-mechanical equations shows that only the condition $〈{L}_{3}(0)〉\ensuremath{\approx}{L}_{0}$ requires quantum-mechanical treatment, and that for a short time only. The free-decay problem with initial condition
〈L3(0)〉=L0 solved quantum mechanically up to such a time
t1 by means of the initial derivatives previously derived, and then this solution if used to provide initial conditions determining a classical solution for $t\ensuremath{\ge}{t}_{1}$. The statistical aspects introduced by the quantum mechanics are preserved in the classical solution, and their significance is discussed, with several examples, comparison is also made with a nonstatistical approximation. In the case of forced oscillation, the behavior of
〈L3(t)〉---for arbitrary
L0---is examined in detail for a strong field, and turns out to be described by a damped oscillation, the existence of the damping being independent of
L0. The apparent inconsistency of this result with the expectation that a large system subject to strong forces should behave classically is discussed, and
〈L23(t)〉 is examined. It is concluded that in a single experiment, the NO energy oscillates approximately like the classical energy, without damping; quantum mechanics introduces, however, a slight randomness (or unpredictability) in the frequency of this oscillation, because of the coupling with the radiation field. Since
〈L3(t)〉 describes the average over an ensemble of which each member consists of a NO coupled to a radiation field, the random frequency variation among the members accounts for the damping of the average. The usefulness of the combination of classical and quantum-mechanical analyses in achieving and interpreting theoretical results for a class of phenomena involving the collective interaction between a number of atoms and a radiation field is pointed out.
The phenomenon of overdamping, when frictional forces overwhelm the restoring force of an harmonic oscillator, is well known in classical mechanics. Analogous phenomena occur in time-dependent quantum mechanics when probability loss rates (e.g. photoionization rates) become larger than the characteristic inverse times for coherent excitation (the Rabi frequencies). These results often seem surprising, even for simple two-state quantum systems. We discuss examples of this overdamping phenomenon—when an increase of the loss rate actually produces a slower rate of probability loss and even freezes population—for a number of quantum systems: steady and pulsed two-state excitation, time evolution with adiabatic passage, simple multistate chains, and branched chains of excitation linkage. In each situation there occurs for very large damping, often unexpectedly, phenomena akin to the overdamping of the classical harmonic oscillator.