Modulated rotating wave

Fig. 6. Tip-path plots for rotating wave (RW) and modulated-rotating-wave (MRW) states in both the laboratory frame and in the rotating frame. The MRW states are quasiperiodic in the laboratory frame and periodic in the rotating frame. The rotating wave is periodic in the laboratory frame and steady in the rotating frame. The parameter values are the same as in Figure 5 with (from left to right) a = 0.72, o = 0.746, o = 0.755, and a = 0.80.

These results leave several basic questions open How to derive a non-Markovian master equation (ME) for arbitrary time-dependent driving and modulation of a thermally relaxing two-level system Would the two-level system (TLS) model hold at all for modulation rates, that are comparable to the TLS transition frequency u)a (between its states e) and g)) which may invalidate the standard rotating-wave approximation (RWA), [to hen-Tannoudji 1992] Would temperature effects, which are known to incur upward g) —> e) transitions, [Lifshitz 1980], further complicate the dynamics and perhaps hinder the suppression of decay How to control decay in an efficient, optimal fashion We address these questions by outlining the derivation of a ME of a TLS that is coupled to an arbitrary bath and is driven by an arbitrary time-dependent field. 

For 7 > 6.1 we are unable to compute stable TWl modes. Rather, we find modulated traveling waves consisting of a single hot spot which alternately speeds up and slows down and its intensity alternately increases and decreases as it rotates around the cylinder, thus describing a modulated traveling wave for a 1-headed spin (MTWl). A space time plot of such a mode is shown in Figure 6 for i = 6.1. 

The state = ( = 0 is a steady state of Equations (11) for all parameter values, and by the choice of the constant terms in expansions for / and g, it is linearly stable for all values of a and a2. For the full system (8) this state corresponds to v = iv = 0, p = po= constant. The trivial steady state coexists with rotating-wave and modulated-wave solutions discussed next. Hence it plays much the same role in the ODE model as the homogeneous steady state plays in excitable media. That is, in all excitable media, there is a homogeneous steady state (e.g. u — v = O the reaction-diffusion model), which is linearly stable for all parameter values, and which coexists with rotating waves and modulated waves when they exist. The trivial steady state in the ODE model has the same character as this homogeneous state in excitable media. 

Overall, the RDE provides an efficient and reproducible mass transport and hence the analytical measurement can be made with high sensitivity and precision. Such well-defined behavior greatly simplifies the interpretation of the measurement. The convective nature of the electrode results also in very short response tunes. The detection limits can be lowered via periodic changes in the rotation speed and isolation of small mass transport-dependent currents from simultaneously flowing surface-controlled background currents. Sinusoidal or square-wave modulations of the rotation speed are particularly attractive for this task. The rotation-speed dependence of the limiting current (equation 4-5) can also be used for calculating the diffusion coefficient or the surface area. Further details on the RDE can be found in Adam s book (17). 

It is weliknown that all static polarizations of a beam of radiation, as well as all static rotations of the axis of that beam, can be represented on a Poincare sphere [25] (Fig. la). A vector can be centered in the middle of the sphere and pointed to the underside of the surface of the sphere at a location on the surface that represents the instantaneous polarization and rotation angle of a beam. Causing that vector to trace a trajectory over time on the surface of the sphere represents a polarization modulated (and rotation modulated) beam (Fig. lb). If, then, the beam is sampled by a device at a rate that is less than the rate of modulation, the sampled output from the device will be a condensation of two components of the wave, which are continuously changing with respect to each other, into one snapshot of the wave, at one location on the surface of the sphere and one instantaneous polarization and axis rotation. Thus, from the viewpoint of a device sampling at a rate less than the modulation rate, a two-to-one mapping (over time) has occurred, which is the signature of an SU(2) field. 

We can say that such a static device is a U( ) unipolar, set rotational axis, sampling device and the fast polarization (and rotation) modulated beam is a multipolar, multirotation axis, SU(2) beam. The reader may ask how many situations are there in which a sampling device, at set unvarying polarization, samples at a slower rate than the modulation rate of a radiated beam The answer is that there is an infinite number, because from the point of the view of the writer, nature is set up to be that way [26], For example, the period of modulation can be faster than the electronic or vibrational or dipole relaxation times of any atom or molecule. In other words, pulses or wavepackets (which, in temporal length, constitute the sampling of a continuous wave, continuously polarization and rotation modulated, but sampled only over a temporal length between arrival and departure time at the instantaneous polarization of the sampler of set polarization and rotation—in this case an electronic or vibrational state or dipole) have an internal modulation at a rate greater than that of the relaxation or absorption time of the electronic or vibrational state. 

The representation of the sampling by a unipolar, single-rotation-axis, U(l) sampler of a SU(2) continuous wave that is polarization/rotation-modulated is shown in Fig. 2, which shows the correspondence between the output space sphere and an Argand plane [28]. The Argand plane, S, is drawn in two dimensions, x and v, with z = 0, and for a set snapshot in time. A point on the Poincare sphere is represented as P(t,x,y,z), and as in this representation t = 1 (or one step in the future), specifically as P(l,x,y,z). The Poincare sphere is also identified as a 3-sphere, S 1, which is defined in Euclidean space as follows ... 

K thus defines a static polarization/rotation—whether linear, circular or elliptical—on the Poincare sphere. The 2, r representation of the vector K gives no indication of the future position of K that is, the representation does not address the indicated hatched trajectory of the vector K around the Poincare sphere. But it is precisely this trajectory which defines the particular polarization modulation for a specific wave. Stated differently a particular position of the vector K on the Poincare sphere gives no indication of its next position at a later time, because the vector can depart (be joined) in any direction from that position when only the static 2, r coordinates are given. 

Within the harmonic approximation for the bending motion the rotational FC factors are proportional to the square of the bending wave-function with argument (7 — je) = jft/mcj, modulated by a sinusoidal factor with wavelength Aj 7r/7e-... 

Schematic diagram showing the integration of a polarization modulated birefringence apparatus within a laser Doppler velocimeter. This shows the side view. L light source (a diode laser was used) PSG rotating half-wave plate design LS lens FC flow cell (flow is into the plane of the figure) CP circular polarizer D detector 2D-T two dimensional translation stage 3D-T three dimensional translation stage LDVP laser Doppler velocimeter probe.

Fig. 9. Polarization curve of an Fe-disc Pt-split-ring electrode with hydrodynamic square wave modulation. In 1 M NaOH with anodic and cathodic scan including capacity of the Fe disc (dashed curve), modulation frequency of rotation co = 0.05 Hz (insert), simultaneous detection of Fe(II) and Fe(III) ions at Pt half rings [12].

Another example for the HMRRD electrode is given in Fig. 9 for Fe in alkaline solutions [12, 27]. The square wave modulation of the rotation frequency co causes the simultaneous oscillation of the analytical ring currents. They are caused by species of the bulk solution. Additional spikes refer to corrosion products dissolved at the Fe disc. This is a consequence of the change of the Nemst diffusion layer due to the changes of co. This pumping effect leads to transient analytical ring currents. Besides qualitative information, also quantitative information on soluble corrosion products may be obtained. The size of the spikes is proportional to the dissolution rate at the disc, as has been shown by a close relation of experimental results and calculations [28-30]. As seen in Fig. 7, soluble Fe(II) species are formed in the po-... 

Blaedel and Engstrom [48] noted that for a quasi-reversible process the current could be simply expressed in terms of the rate constant and mass-transport coefficient. Application of a square wave step in the rotation rate of a RDE (i.e., PRV, see Section 10.4.1.3) resulted in modulation of the diffusion-limited current and hence modulation of the mass-transfer coefficient. By solving the appropriate quadratic equation it was possible to derive a value for the heterogeneous rate constant for the electrochemical cathodic, kf, or anodic, kb, process of interest. Values for the standard heterogeneous rate constant and transfer coefficient were subsequently... 

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