Rotational coherence

It is interesting to note that to actually implement a useful algorithm it is necessary to implement a certain number of quantum operations within the coherence time. Recently, we reported that it was possible to increase the number of coherent rotations by a factor of 10 by matching the Rabi frequency with the frequency of the proton in the polyoxometalate SIM GdW30. Under these conditions, it was possible to perform at least 80 such operations (Figure 2.14) [75]. 

Coherent rotation of magnetic dipoles is yet another common mode of magnetization process. This simply means that all the spins rotate together. 

As illustrated earlier in the text (Figure 10.5), molecules released from the centrifuge generate an oscillatory Raman signal, characteristic of the coherent rotation with well-defined relative phase relation between the quantum states inside a rotational wave packet. Time-resolved coherent Raman response from a wave packet centered at A = 69 in oxygen is plotted at the bottom of Figure 10.9a. Knowing the wave packet composition from the state-resolved detection discussed above. 

The development of a new form of spectroscopy based on the exploitation of the time evolution of the coherence associated with the rotational motion of an excited molecule. Conventional spectroscopies depend on the measurement of differences between the energy levels of a molecule, which become more and more difficult to measure and to interpret as the size of the molecule increases. In contrast, the intervals between recurrences in the coherent rotational motions of large molecules are directly related to the moments of inertia of the molecules and can be used to determine their structures. 

Figure 10. Nucleation modes in homogeneous magnets (a) coherent rotation in a sphere, (b) curling in a sphere, and (c) curling in a cylinder. The arrows show the local magnetization M = Mz ez + m, where ez is parallel to the axis of revolution of the ellipsoid (cylinder).

Comparison of Eqs. (10) and (11) yields the radius Rcoh for the transition from coherent rotation to curling. For R < i coh, the exchange energy dominates, and the nucleation is realized by coherent rotation, whereas for R > i coh the nucleation behavior is dominated by flux closure and realized by curling. For spheres and wires (cylinders), one obtains Rcoh = 5.099 /ex and... 

For the above mentioned FePt particles, the particle diameter is clearly smaller than the critical particle size given by Eq. (8) for coherent rotation. Furthermore the strength of the magnetostatic interaction field acting on nearest neighbor particles is only about 2% of the anisotropy field for a particle distance of 2 nm. Thus the Stoner-Wohlfarth theory can be applied. 

To maintain thermal stability, hence a condition EB/kBT= In (for) needs to be fulfilled. For z = 10 years storage, 109-10u Hz [28] and ignoring dispersions, i.e. assuming monodisperse particles, this becomes Es/kBT= 40-45. Reversal for isolated, well-decoupled grains to first order can be described by coherent rotation over EB. This simple model, as first discussed by Stoner and Wohlfarth in 1948 [29], considers only intrinsic anisotropy and external field (Zeeman) energy terms. For perpendicular geometry one obtains the following expression ... 

The resistance to magnetization reversal indicates that an energy barrier separates the initial and the final magnetic states. This energy barrier is a consequence of magnetic anisotropy. This can be illustrated within the so-called Stoner-Wohlfarth (SW) model, in which reversal is assumed to occur by in-phase rotation of all moments (coherent rotation) [7], For HoPP, antiparallel to M, the energy may be expressed as ... 

Other reversal processes than coherent rotation may be envisaged and, thus, the validity of the above approach be questioned. However, in the case... 

Figure 1. Angular variation of the normalized coercive field Hc(0)/Ha ( ) Stoner-Wohlfarth model and (o) 1/cos 6 dependence, approximately observed in usual hard magnets. The value of Hc(0) is arbitrary, //C(0)///A = 0.2 has been assumed. At large 6 values, when //c(0) > //sw(0), coherent rotation is favoured again. Inset definitions of the various angles involved in Eqs. 1-3.

Additionally to the reduction of the coercive field with respect to the anisotropy field, the occurrence of nucleation or pinning-depinning may be characterized by the fact that the angular dependence of the coercive field is very different from the one corresponding to coherent rotation. Rather it may be expressed as [ 0] ... 

In this section, we concentrate on the fundamental impact of particle size reduction on magnetization processes in individual particles. Although not directly related to coercivity, the classical effect of single domain particle formation is described. At small particle size, reversal by coherent rotation tends to be favoured with respect to nucleation/pinning-depinning finally thermal activation effects and macroscopic quantum tunnelling are discussed. 

Rc is of the order of 30 nm for Co metal, but of 5 nm only for a high anisotropy system such as Nd2Fei4B. Whilst the SW model of coherent rotation has been used for many years as a model paradigm to illustrate the phenomenon of coercivity, the first unambiguous experimental evidence for... 

Even below TB, the magnetization of nanoparticles may be strongly affected by thermal activation. Very small Co particles showing coherent rotation allowed thermal activation effects to be analyzed quantitatively [110]. From 40 mK up to 12 K, the coercive field was found to be a function of the expression theoretically expected for thermal activation, Tln(T/T0]213. 

Free-layer switching during the write process can be basically modeled in the Stoner-Wohlfarth coherent-rotation model (Ch. 4), which yields an astroid switching curve [71]. Figure 16(a) shows an ellipsoidal cell, with magnetization M and applied field H. In the Stoner-Wohlfarth approximation, the cell energy per unit volume is... 

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