# From the Kinematics of Precession Motion to Generalized Rabi Cycles

Authors: Danail S. Brezov; Clementina D. Mladenova; Ivaïlo M. Mladenov
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We use both vector-parameter and quaternion techniques to provide a thorough description of several classes of rotations, starting with coaxial angular velocityΩ of varying magnitude. Then, we fix the magnitude and let Ω precess at constant rate about the z-axis, which yields a particular solution to the free Euler dynamical equations in the case of axially symmetric inertial ellipsoid. The latter appears also in the description of spin precessions in NMR and quantum computing. As we show below, this problem has analytic solutions for a much larger class of motions determined by a simple condition relating the polar angle and z-projection of Ω (expressed in cylindrical coordinates), which are both time-dependent in the generic case. Relevant physical examples are also provided.


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## Body

## 1. Introduction: Quaternions and Vector-Parameters

The Clifford algebra of quaternionsH≅Cl0,2(R)≅Cl30(R) is known to be intimately related to the rotation group in R3 and one way to see it is via the standard spin covering map Spin(3)≅SU(2)→Z2SO(3) that is topologically a projection from the unit sphere S3→Z2RP3. The basis of bivectors in Cl3 is spanned by three units satisfying(1)i2=j2=k2=ijk=-1,where 1 denotes the identity element, so one can express each quaternion ζ∈H in the form(2)ζ=ζ0,ζ=ζ01+ζ1i+ζ2j+ζ3k.Hence, as a linear space, H may obviously be identified with R4≅R×R3 by introducing coordinates as shown above. At the same time, quaternions have a specific Clifford multiplication rule that can be expressed in the above notation as(3)ζ0,ζζ~0,ζ~=ζ0ζ~0-ζ·ζ~,ζ0ζ~+ζ~0ζ+ζ×ζ~,where · and × stand for the dot and cross products in R3, respectively. Just like in the case of complex numbers, Clifford conjugation in H is given by sign inversion of the imaginary (vector) part ζ: that is, ζ=(ζ0,ζ)→ζ∗=(ζ0,-ζ), and since there are no zero divisors, the quaternion norm defined as ζ21=ζ∗ζ yields the inverse of every nonzero element ζ-1=ζ-2ζ∗. In particular, restricting to ζ2=1, one obtains the unit sphere in R4, which the composition law (3) endows with the group structure of SU(2). At this point, rotations in R3 appear quite naturally if we identify three-vectors with pure (imaginary) quaternions x∈R3→X~=x1i+x2j+x3k∈su(2) and use the adjoint action of the spin group Adζ:X~→ζX~ζ∗. This effectively transforms the components of x via the linear orthogonal map(4)Rζ=ζ02-ζ2I+2ζζt+2ζ0ζ×,where I denotes the identity in R3, ζζt stands for the usual dyadic (tensor) product, and ζ× is the skew-symmetric linear map associated with ζ via the Hodge duality: that is, ζ×ζ~=ζ×ζ~. The above map clearly preserves orientation and thus represents rotation in R3. Denoting its angle ϕ and the unit vector along its axis n, one may express the associated quaternion as(5)ζ0=cos⁡ϕ2,ζ=sin⁡ϕ2n.Central projection onto the plane ζ0=1 then yields thevector-parameterc=ζ/ζ0=tan⁡(ϕ/2)n that is actually not a vector but rather a point in projective three-space, equipped with an additional group structure. This simple construction appears quite convenient: on the one hand, unlike other known alternatives (such as Euler angles which can be considered as coordinates of points in projective space on the three-torus and lead to singularities, e.g., the so-calledgimbal lock) it gives a topologically adequate description of the orthogonal group SO(3)≅RP3, while, on the other hand, it provides an efficient associative composition law(6)c2,c1=c2+c1+c2×c11-c2·c1inherited from the quaternion multiplication (3) upon central projection. This constitutes a (nonlinear) representation of SO(3) related to the usual matrix realization via R(〈c2,c1〉)=R(c2)R(c1), in which the inverse element is given by -c and the neutral one by the zero vector. Moreover, it yields rational expressions for the matrix entries of R(c) that may be obtained directly from (4) or equivalently by means of the famous Cayley transform(7)Rc=I+c×I-c×=1-c2I+2cct+2c×1+c2·Clearly, the above follows also from Rodrigues’ rotation formula with the aid of Euler’s trigonometric substitution, but the projective approach reveals the major advantages of vector parameterization in a much more direct manner. Recall also that the angular velocity of a rotating rigid body is defined in the body and inertial frames, respectively, as(8)Ω×=R˙Rt,ω×=RtR˙.Straightforward differentiation of the Cayley representation (7) yields the relations (see also [1, 2])(9)Ω=21+c2I+c×c˙,ω=21+c2I-c×c˙while in the inverse direction, one easily derives the Riccati equations for the vector-parameter(10)c˙=12I+cct-c×Ω=12I+cct+c×ω.We refer to [3] for a detailed introduction to quaternions and some of their applications in a wide range or fields, from navigation and attitude control to computer graphics. In quantum mechanics quaternions and Pauli matrices appear naturally in the description of spin systems as we recall below. Moreover, modern physics embraces the idea of quaternionic Hilbert space and quaternionic quantum mechanics (cf. [4, 5]). Many features of complex numbers are preserved except, of course, commutativity, which makes the definition of quaternionic analyticity far nontrivial and yet fruitful (see [6]), for example, in the description of electrodynamics [7] and its relation to the so-calledFueter analyticity. Vector parameterization [8, 9], on the other hand, is much less popular although it has proven quite efficient in the description of (pseudo-)orthogonal groups in dimensions three and four [10], as well as in a wide range of problems related to classical and quantum mechanics [2, 11], special relativity and electrodynamics [12, 13], robotics and navigation [3, 14], and various other fields of science and technology. Here we make use of both descriptions.The text is organized as follows: after this brief introductory part, we consider the kinematical problem of a rotating rigid body with fixed direction of the angular velocity vector using the projective vector-parameter construction discussed above, thus extending an old result due to O’Reilly and Payen [15]. Then, in Section 3, we switch back to quaternions, which lead to linear, rather than Riccati equations, in order to deal with the more complicated settings. Namely, we consider the case of a precessing (about a fixed axis) angular velocity and the most general case, given conveniently in cylindrical coordinates, using techniques from the arsenal of quantum mechanics, which allow us to derive a simple integrability condition. Finally, after a brief note on dynamics, we relate these results to the Rabi oscillator that appears in quantum computation, showing how our solutions yield more freedom in the entire process.


## 2. The Coaxial Angular Velocity (CAV) Motion

Now, let us consider the first equation in (10) and look for solutions with a fixed direction of Ω. Without loss of generality, we may choose for convenience a reference frame, in which Ω is aligned with the z-axis: that is,(11)Ω=0,0,Ωt,Ω=2ωt.This allows for expressing the matrix ODE (10) explicitly in components as(12)c˙1=ωc1c3-c2,c˙2=ωc2c3+c1,c˙3=ω1+c32.From the third equation, one obviously has (we set the initial moment at t=0)(13)c3t=tan⁡ϑ,ϑt=ϑ˚+∫0tωsds,while for the first two we may use the substitution z(t)=c1(t)+ic2(t), which yields(14)z˙t=c3t+iωtzt.The latter clearly possesses a trivial solution z≡0 that is a well-known result: in this case, the rotation takes place about the axis determined by the angular velocity vector Ω with magnitude Ω=2ω. However, for z≠0, one has(15)ddtln⁡z=tan⁡ϑ+iω.Combining the above two cases, one may express the general solution in the form (here the amplitude ρ≥0 and the two phases φ˚, ϑ˚∈[0,2π) are constants of integration)(16)zt=ρsec⁡ϑeiφ,φt=φ˚+∫0tωsdsfrom which it is straightforward to write(17)ct=sec⁡ϑρcos⁡φ,ρsin⁡φ,sin⁡ϑt. Changing the parameter t→τ=tan⁡ϑ(t), it is not hard to show that the integral curves (17) obtained above are actually rays in R3 described by(18)cτ=u+vτ,uτ-v,τt,τ∈RP1,where the constants u and v are most naturally given in polar coordinates as(19)u=ρcos⁡ϑ˚-φ˚,v=ρsin⁡ϑ˚-φ˚.In this way, we end up with a two-parameter family of rays in R3, whose intersection points (x,y,x~,y~)=(u,-v,u+v,u-v) with the planes z=0 and z=1 constitute isosceles right triangles with the origin. In other words, if we denote c(τ)=c0+τv in (18), the constant vectors c0 and v are perpendicular and c02=v02, where v0=v-e^z is the projection of v in the (x,y)-plane. Those are all orbits still periodic in ϑ. In particular, the vertical z-axis and the horizontal projective line at infinity are limiting cases corresponding to ρ→0 and ρ→∞ as discussed below in detail. Note that one may redefine φ˚ via an overall phase shift introduced by a proper choice of initial time, which yields ϑ˚=0, so in physical terms one has only two relevant constants (amplitude and phase) and the general solution of (12) is generated by the time flow of c(t) as orbits of all possible initial points. On the other hand, it is not difficult to see that the whole manifold SO(3)≅RP3 is covered by the above map. Moreover, each point in projective space is reached by the initial data setting t=0 since RP3 is obviously covered by the family of rays (18), in which we may use ϑ˚ as an intrinsic parameter. Now, in order to lift the whole construction up to the spin cover, we may express the two images ζ±=π±-1c∈S3 corresponding to the vector-parameter (17) in the form(20)ζ±=±sin⁡ψeiφicos⁡ψe-iϑicos⁡ψeiϑsin⁡ψe-iφwith a suitable choice of a quaternion basis and using the notation(21)ψ=arctan⁡ρ∈0,π2·Note that the ϑ-orbits on the unit sphere project to circles in the (x,y) and (z,w) planes with equal frequencies but generally different phases and amplitudes. In particular, for ψ=0 and ψ=π/2 the former (resp., the latter) circle shrinks to a point. From the perspective of the Hopf map, one has quaternion ϑ-orbits on S3 whose holonomy shifts the S1 phase ϑ by a whole period for each revolution about the z-axis of the base S2. This becomes more apparent if we write ζ with its coordinates on S3 as(22)ζ±,ζ0±=±sin⁡ψcos⁡φ,sin⁡ψsin⁡φ,cos⁡ψsin⁡ϑ,cos⁡ψcos⁡ϑ.Stereographic projection (from the south pole onto the equatorial plane) yields also the so-calledWiener-Milenkovic conformal rotation vector (see [9] for more details)(23)ς=ζ1+ζ0=tan⁡ϕ4nwhich does not involve infinities at least for ψ≠0, so one may plot the corresponding curves in Euclidean space. Moreover, the rotation angle is not difficult to derive from formula (17): namely,(24)ϕt=2arctan⁡sec⁡ϑρ2+sin2⁡ϑ.As for the corresponding invariant axis, ignoring the common prefactor in (17), we see that it precesses about the vector Ω. Besides, the rate of precession and nutation are synchronized and in the case ω=const., equal to half the magnitude of the angular velocity Ω=2ω. It is convenient also to introduce spherical coordinates for the unit invariant vector n=n(θ,φ) in the form(25)θ=arctan⁡ρcsc⁡ϑ.Using formula (7) and taking into account (21), we express the corresponding rotation matrix explicitly as(26)Rφ,ϑ,ψ=cos⁡2ϑcos2⁡ψ+cos⁡2φsin2⁡ψsin⁡2φsin2⁡ψ-sin⁡2ϑcos2⁡ψsin⁡ϑ+φsin⁡2ψsin⁡2φsin2⁡ψ+sin⁡2ϑcos2⁡ψcos⁡2ϑcos2⁡ψ-cos⁡2φsin2⁡ψ-cos⁡ϑ+φsin⁡2ψsin⁡ϑ-φsin⁡2ψcos⁡ϑ-φsin⁡2ψcos⁡2ψ·Note that the angles φ and ϑ differ only by a phase(27)ϑ=φ+ϑ˚-φ˚,while ψ is a constant of integration, which makes the time dependence of the above matrix entries relatively simple. Moreover, we can manipulate both ψ and ϑ-φ via specific choice of initial data to obtain some particular cases. Firstly, we discuss the three distinct types of solutions pointed out in [15] for the case ω=const. The most obvious one, for which c∥Ω with Ω=ϕ˙ (Type I), corresponds to ψ≡0 (ρ→0) and is usually referred to assteady rotation. It suggests a constant invariant axis oriented along the angular velocity vector and linearly evolving angle ϕ: namely,(28)R∥ϑ=cos⁡2ϑ-sin⁡2ϑ0sin⁡2ϑcos⁡2ϑ0001·On the other hand, setting ψ=π/2, that is, ρ→∞, we obtain the half-turn example considered in [15] (Type II rotations) as R⊥(φ)=2nnt-I, where n=cos⁡φ,sin⁡φ,0t, which may be written explicitly in matrix terms directly from (26) in the form (note that the ϑ-dependence in (26) is completely factored out by demanding ψ to be a right angle)(29)R⊥φ=cos⁡2φsin⁡2φ0sin⁡2φ-cos⁡2φ000-1·This time the rotation takes place about a varying axis that remains perpendicular to Ω and precesses about it, while the angle remains ϕ=π at each particular instant of time.The third type of rotations proposed in [15], in which both the rotation axes and angle are periodic functions of time, is also given by formula (26) if we simply set ω=const. We may consider, for example, the case of coherent phases ϑ=φ, that is, ϑ˚=φ˚, in which(30)c=ρ,ρtan⁡φ,tan⁡φt,and express the respective rotational matrix explicitly: that is,(31)R0φ,ψ=cos⁡2φ-sin⁡2φcos⁡2ψsin⁡2φsin⁡2ψsin⁡2φcos⁡2φcos⁡2ψ-cos⁡2φsin⁡2ψ0sin⁡2φcos⁡2ψ·Specific solutions for different values of the phase shift are constructed similarly. Note also that the above matrices have such a simple form only in the reference frame, in which the angular velocity Ω is aligned with one of the axes. Otherwise, the vector-parameter (17) needs to be rotated correspondingly, relative to the coordinate change.The compact explicit form of the solution (26) allows for expressing the three parameters φ, ϑ, and ψ, which we use as a substitute for the usual Euler angles, in the generic case ψ≠0,π/2 as(32)φ=-12atan2⁡r31,r32+atan2⁡r13,r23,ϑ=12atan2⁡r31,r32-atan2⁡r13,r23,ψ=12arccos⁡r33,where atan2 stands for the proper quadrant inverse tangent (note that ψ∈[0,π/2], so sin⁡2ψ≥0 and cos⁡2ψ is monotonous, thus invertible in this interval) and the rjk’s denote the matrix entries of R(φ,ϑ,ψ) given by formula (26) in the standard basis. Moreover, since(33)1+c2=1+ρ21+tan2⁡ϑas it follows from formula (17), it is straightforward to obtain also the azimuthal angle θ in the spherical representation of the invariant unit vector n and the angle of rotation ϕ, respectively, as(34)θ=arctan⁡csc⁡ϑtan⁡ψ,ϕ=2arccos⁡cos⁡ϑcos⁡ψ.However, in the limiting cases of vanishing and infinite amplitude of precession, corresponding to ψ=0 and ψ=π/2, respectively, (32) yield indeterminacy, so these two cases need to be treated separately. In particular, for ρ→0, one has(35)ψ=0⟹ϑ=12atan2⁡r21,r11,θ=0,ϕ=2ϑ,while the polar angle φ becomes obsolete. Similarly, the limit ρ→∞ yields(36)ψ=π2⟹φ=12atan2⁡r12,r11,θ=π2,ϕ=πand ϑ is no longer relevant as it has been already discussed.


## 3. Quaternion Description of Precessions

To derive the kinematic equations in quaternion terms, it suffices to recall thatc=ζ/ζ0 and substitute it in formula (10), which yields upon multiplication by ζ02(37)2ζ0ζ˙-2ζ˙0ζ=ζ02Ω+ζ,Ωζ+ζ0Ω×ζ.Then, taking a dot product with ζ and using the identities(38)ζ2=ζ02+ζ2=1⟹ζ0ζ˙0+ζ,ζ˙=0yield the derivative of the scalar (real) part of the quaternion(39)2ζ˙0=-Ω,ζ.Combining both expressions, we obtain the vector (imaginary) part in the form(40)2ζ˙=Ω×ζ+ζ0Ω,so the kinematic equations have the block-matrix representation(41)ζ˙ζ˙0=12Ω×Ω-Ωt0ζζ0·Identifying quaternions and four-vectors allows for writing the above result as ζ˙=Aζ and, thus, derive the general solution for the propagator in terms of time-ordered matrix exponents. In the particular case Ω=const., however, we end up with a system of linear homogeneous ODEs with constant coefficients, hence the solution(42)ζt=etAζ0.Note that det(A-λI)=(λ2+ω2)2, so we have two double roots λ±=±iω, which yields, in the canonical basis, synchronized rotations (with equal frequencies) in two mutually perpendicular planes in R4 as we have seen in the case Ω1,2=0. Moreover,(43)A2k=-1kω2kI,A2k+1=-1kω2k+1A^and, thus, the propagator may be expressed as(44)etA=cos⁡ωtI+sin⁡ωtA^,where we let I denote the identity in any dimension and A^=ω-1A with ω=(1/2)Ω. Bearing this in mind, for the time evolution of an arbitrary quaternion ζ(t), one has(45)ζt=cos⁡ωtζ˚+sin⁡ωtζ˚0ω^+ω^×ζ˚,ζ0t=ζ˚0cos⁡ωt-ζ˚,ω^sin⁡ωt,where ω^=(1/2ω)Ω stands for the unit vector in the direction of Ω and ζ˚=ζ(0) is simply the initial condition.Denoting alsoc˚=c(0)=ζ˚/ζ˚0, we obtain the time dependence of the vector-parameter(46)ct=c˚+tan⁡ωtI-c˚×ω^1-c˚,ω^tan⁡ωt·In particular, if ζ(0) is parallel to ω^, it remains such along the flow determined by (46) as one has(47)ζtζ0t=cos⁡ωtI+sin⁡ωtω^×sin⁡ωtω^-sin⁡ωtω^tcos⁡ωtsin⁡ϑ˚ω^cos⁡ϑ˚=sin⁡ωt+ϑ˚ω^cos⁡ωt+ϑ˚,so we end up with the trivial orbit c(t)=tan⁡ϑω^, which gives the solution also in the case of a steady initial state (ζ˚0=1). Similarly, if ζ˚⊥ω^ and ζ˚0=0, one has c(t)=c˚+tan⁡ωtω^×c˚, so c(t) remains perpendicular to ω^. More generally, we may denote p(t)=(c(t),ω^) and p(0)=tan⁡ϑ˚, as well as q=c-pω^=Pω^⊥c, and obtain(48)pt=tan⁡ϑt,qt=I+tan⁡ωtω^×1-tan⁡ϑ˚tan⁡ωtq0.Note that, just as before, we may weaken the condition of constant angular velocity by demanding only ω^=const., thus letting ω=ω(t). This introduces just a slight modification of the above solution: namely, in formula (46), we make the substitution(49)ωt⟶∫0tωsds.It is useful to point out that the matrix A (and therefore, its exponent) may be expressed also in a 2×2 block form. In particular, if ω^=(0,0,1)t, one has(50)etA=eJωt00e-Jωt,J=0-110which yields the solutions obtained in the previous section. More generally, we have(51)A^=ω^3Jω^2I-ω^1J-ω^2I-ω^1J-ω^3J,that is, a real 4×4 representation of a pure unit quaternion and may be mapped onto su(2) as A^→A^⋆=ω^3i+ω^2j-ω^1k. The action of A⋆=ωA^⋆ on ζ is naturally realized via matrix multiplication if the latter is represented as a point (ξ,η) in C2: namely,(52)ξ=ζ1+iζ2,η=ζ3+iζ0.Then, the kinematic equation for ζ may be written as ζ˙=A⋆ζ, or in components(53)ξ˙η˙=iω3ω2-iω1-ω2-iω1-iω3ξη,ωk=ωω^kand for fixed ω^ the corresponding unitary evolution operator takes the form(54)exp⁡∫0tωsdsA^⋆=cos⁡∫0tωsdsI+sin⁡∫0tωsdsA^⋆.


## 4. Precessing Angular Velocity

Let us now consider the case of varying directional vectorω^. One relatively simple setting is that of an angular velocity precessing about a fixed axis: for example,(55)Ω=2αcos⁡2ω0t,αsin⁡2ω0t,ωt,α,ω,ω0∈R+.This time the magnitude Ω=2α2+ω2 is constant, so one has(56)A⋆t=-i-ωαe2iω0tαe-2iω0tωand due to the nonvanishing commutator A⋆(t),∫0tA⋆(s)ds≠0, obtaining the propagator via exponentiating the above matrix generally involves time-ordering or some other cumbersome procedure. Nevertheless, one may still derive the exact solution transforming system (53) into a single ODE. Namely, after differentiating the first equation and substituting(57)αη=e-2iω0tiξ˙+ωξfrom the second one, the time dependence of the coefficients factors out and we end up with a linear homogeneous second-order ODE with constant coefficients(58)ξ¨-2iω0ξ˙+α2+ω2-2ωω0ξ=0.With the notation(59)σ=α2+χ2,χ=ω-ω0,ψ0=arctan⁡χα,we may express the general solution in the compact form(60)ξt=eiω0tAcos⁡σt+Bsin⁡σtand, respectively, from the above substitution, we have(61)ηt=e-iω0ttan⁡ψ0Acos⁡σt+Bsin⁡σt+isec⁡ψ0Bcos⁡σt-Asin⁡σt.Here, A and B are complex constants determined by the initial conditions ξ˚=ξ(0) and η˚=η(0) as(62)A=ξ˚,iB=cos⁡ψ0η˚-sin⁡ψ0ωξ˚.This allows for writing the propagator Ut:(ξ˚,η˚)t→(ξ,η)t in the simple form(63)Ut=eiω0tcos⁡σt+isin⁡ψ0sin⁡σt-ieiω0tcos⁡ψ0sin⁡σt-ie-iω0tcos⁡ψ0sin⁡σte-iω0tcos⁡σt-isin⁡ψ0sin⁡σt,where for the first row we substitute the expressions for A,B~ξ˚,η˚ in the equation for ξ(t), while for the second one, we use the fact that U(t) takes values in SU(2). Considering, for example, a steady initial state (ξ˚,η˚)=(0,i) from the above, one easily obtains(64)ζ0t=cos⁡σtcos⁡ω0t-sin⁡ψ0sin⁡σtsin⁡ω0t,ζ1t=cos⁡ψ0sin⁡σtcos⁡ω0t,ζ2t=cos⁡ψ0sin⁡σtsin⁡ω0t,ζ3t=sin⁡ψ0sin⁡σtcos⁡ω0t+cos⁡σtsin⁡ω0t.This allows for expressing also the evolution of the vector-parameter in the form (see Figure 1)(65)ct=11-sin⁡ψ0tan⁡σttan⁡ω0t×cos⁡ψ0tan⁡σtcos⁡ψ0tan⁡σttan⁡ω0tsin⁡ψ0tan⁡σt+tan⁡ω0t·Taking into account the fact that 1+c2=sec2⁡σtsec2⁡ωt, one may express the entries of the corresponding rotation matrix using formula (7). In particular, in the resonant limit ω0→ω, one has ψ0=0 and σ=α; hence, c(t)=tan⁡αt,tan⁡αttan⁡ω0t,tan⁡ω0tt, which finally yields(66)Rtχ=0=cos⁡2ω0t-cos⁡2αtsin⁡2ω0tsin⁡2αtsin⁡2ω0tsin⁡2ω0tcos⁡2αtcos⁡2ω0t-sin⁡2αtcos⁡2ω0t0sin⁡2αtcos⁡2αt.As we show in the next section, a similar expression holds also for variable α and ω0 (under certain restrictions) if we substitute αt→∫0tα(s)ds and ω0t→ω0(t). One may also derive the rotation matrix using directly formula (4), written in components as(67)Rζ=1-2ζ22+ζ322ζ1ζ2-ζ0ζ32ζ1ζ3+ζ0ζ22ζ1ζ2+ζ0ζ31-2ζ12+ζ322ζ2ζ3-ζ0ζ12ζ1ζ3-ζ0ζ22ζ2ζ3+ζ0ζ11-2ζ12+ζ22.Figure 1
The evolution of the vector-parameter (65) and the trace of the corresponding rotation acting on the vector (3,14,25).


## 5. Generic Angular Velocity

Our construction may be generalized as follows: let us consider a time-varying angular velocity vector in the form (note once more that hereΩ(t)=2ω2+α2) (68)Ωt=2αtcos⁡2ω0t,αtsin⁡2ω0t,ωtt.We use a standard technique in physics to factor out the dependence on the diagonal part A0⋆=Aα=0⋆ of (56) known as theinteraction picture (cf. [16]). It involves expressing the propagator as (the matrix A0⋆ is not necessarily constant; we only demand that it commutes with its integral; see [16])(69)Ut=U0tUIt,U0t=exp⁡∫0tA0⋆sds,where UI(t) solves the reduced equation (this is easy to see via straightforward differentiation)(70)ζ˙It=AI⋆tζIt,AI⋆=U0-1A⋆U0-A0⋆,ζI=U0-1ζ.With the aid of formulas (53) and (68), we express(71)U0t=ei∫0tωsds00e-i∫0tωsds,AI⋆t=-iα0e-2iχe2iχ0with the notation χ(t)=∫0tω(s)ds-ω0(t). Note that in the special case χ=const. the matrix AI⋆(t) commutes with its integral and may thus be exponentiated trivially. Then, the evolution operator U(t)=U0(t)UI(t) has the simple form(72)Ut=eiω0+χcos⁡∫0tαsds-ieiω0-χsin⁡∫0tαsds-ie-iω0-χsin⁡∫0tαsdse-iω0+χcos⁡∫0tαsdsand in particular, for χ=0, the corresponding rotation matrix is reduced to(73)Rtχ=0=cos⁡2ω0t-sin⁡2ω0tcos⁡2∫0tαsdssin⁡2ω0tsin⁡2∫0tαsdssin⁡2ω0tcos⁡2ω0tcos⁡2∫0tαsds-cos⁡2ω0tsin⁡2∫0tαsds0sin⁡2∫0tαsdscos⁡2∫0tαsds·We conclude this section with a comment on the case of nonconstant χ(t), which yields some iterative procedure (such as time-ordered exponentiation) for obtaining the propagator in the interaction picture. Here we choose to work with a version of Magnus expansion, usually referred to as thecontinuous BCH formula (see [16]). Note that in the standard quaternion basis, we have the representation AI⋆=u1j+u2k with(74)u1t=αtsin⁡2χt,u2t=-αtcos⁡2χt,which allows us to calculate the corrections explicitly in some special cases. Consider, for example, small precessions about the z-axis with α=2ε(ω˙0-ω)=-2εχ˙, where ε≪1 is regarded as a perturbation parameter. A straightforward computation yields(75)UIt≈1-ε2sin2⁡ΔχI+ε2sin⁡2Δχ-2Δχi+εcos⁡2χ0tj+εsin⁡2χ0tk,where we ignore terms of order ε3 and use the notation Δχ(t)=χ(t)-χ(0). At the same time, U0(t) remains as before and we still have the expression U(t)=U0(t)UI(t).


## 6. A Note on Dynamics

Consider the Euler equations describing free rigid body rotation (cf. [17])(76)Ω˙1=μ1Ω2Ω3,Ω˙2=μ2Ω1Ω3,Ω˙3=μ3Ω1Ω2,where the constants μk are expressed in terms of the eigenvalues Ik of the inertial tensor as(77)μ1=I2-I3I1,μ2=I3-I1I2,μ3=I1-I2I3·It is interesting to view our previous considerations on the kinematics determined by a particular choice of angular velocity from a dynamical perspective. We shall further assume that the axes chosen in the parameterization of Ω(t) in formula (68) coincide with the principal axes of the corresponding inertial ellipsoid. To begin with, it is almost straightforward to show that if the direction of the angular velocity given by the unit vector ω^ is preserved, then its magnitude Ω is constant as well. For a free rotation described by formula (63), on the other hand, the Euler dynamical equations (76) yield a symmetric inertial ellipsoid with I1=I2=I and I3=1+ω0/ωI. In particular, in the limit σ→0, the solution (63) obviously converges to the one for the case Ω=const. and ω^=(0,0,1)t as it should be expected. However, considering the general setting (68), we may express (76) in the form(78)α˙=2αtan⁡2ω0μ1ω+ω˙0=2αcot⁡2ω0μ2ω-ω˙0,ω˙=μ3α2sin⁡4ω0.Further on, we easily derive(79)ω˙0t=ωtμ2cos2⁡2ω0t-μ1sin2⁡2ω0t,so in the particular case of rotationally symmetric inertial ellipsoid with μ2=-μ1=μ and μ3=0, we have ω˙0=μω, but then the third equality yields ω=const., so ω0=μωt up to an additive phase. Finally, substituting into the equation for α(t), we find that α=const. as well, so the solution is a linear precession with frequency 2ω0 about the z-direction, for which the angular velocity magnitude Ω=2α2+ω2 is also constant. On the other hand, we may assume only α=const., which immediately leads to μ2=-μ1 and due to the triangle inequality for the principal moments of inertia, this inevitably yields μ3=0, but then ω˙=0 and we obtain the previously considered case once more.Let us now investigate the settingμ1=μ2=μ, which immediately yields the relations(80)I3=I12+I22I1+I2,μ=I2-I1I2+I1,μ3=I12-I22I12+I22·This symmetry reduces system (78) to(81)α˙=2μαωsin⁡4ω0,ω˙=μ3α2sin⁡4ω0,ω˙0=μωcos⁡4ω0and we readily find a conservation law in the form(82)2μω2-μ3α2=C.Choosing further α˚=α(0) allows us to obtain also(83)αt=α˚sec⁡4ω0,ωt=±12μC+μ3α˚2sec⁡4ω01/2,∫dt=±2μ∫sec⁡4ω0dω0C+μ3α˚2sec⁡4ω01/2.However, the above solution does not provide explicit expressions for the propagator as it yields ω˙0(t)≠ω(t). A natural way to reach beyond the trivial setting of linear precession and still be able to use formula (72) is to introduce external forces (whose resultant moment is denoted with f) to the Euler dynamical system (76). Let us consider the simple case of rotational inertial ellipsoid I1=I2=I, that is, μ2=-μ1=μ⇒μ3=0, as it allows for arranging χ=const. in a straightforward manner and, thus, reconstruct the rotation matrix from the angular velocity vector using expressions (72) and (4). Namely, we have ω˙0(t)=ω(t), that is, χ=const., as long as we set μ=1 and f2=f1tan⁡2ω0. Then, the solution takes the simple form(84)αt=α0+12I∫0tf~sds,ωt=ω0+12I3∫0tf3sds,ω0t=∫0tωsds-χ,where f~=f1sec⁡2ω0 and in particular, setting f2=f1=0, we end up with a precessional motion with α=const., while ω(t) and ω0(t) can still be determined from (84).


## 7. Larmor Precession and Rabi Cycles

The quaternion representation of the kinematical problem described above has many advantages, as we could see. One of them, which we have not discussed so far, is that it allows for a straightforward implementation of our solutions to other physical models governed by analogous equations. One particularly famous example appears in the context of Pauli’s description of spin1/2 particles in an external magnetic field B, for which the kinematics of the spin state Ψ is determined by the equation(85)Ψ˙=γ2BΨ,where γ is a physical constant (the gyromagnetic ratio) and the magnetic induction B may be written as a purely imaginary quaternion, so the evolution of the above system describes a rotor with angular velocity Ω=γB. Then, we may use our solutions (with the proper rescaling) in a completely different physical context: for the description of spin precession in a coaxial (54) or a precessing magnetic field. The former is a famous quantum mechanical effect known as Larmor precession, while the latter was considered previously by Rabi in [18]. His solution (that won him a Nobel prize back in 1944) has a crucial impact on magnetic resonance imaging, quantum computing, and optics. A major part of the idea is based on the so-called Rabi resonance, which yields the highest probability for a transition from a ↓ to a ↑ spin state. The resonance corresponds to our solution with χ=0, which may be interpreted also as a decomposability condition for U(t) into a pair of orthogonal pure quaternions(86)Ut=eiωtcos⁡αt-ieiωtsin⁡αt-ie-iωtsin⁡αte-iωtcos⁡αt=eiωt00e-iωtcos⁡αt-isin⁡αt-isin⁡αtcos⁡αt·With the initial condition (ξ˚,η˚)=(0,i), this yields also the decomposition(87)Rc=Rz2ωtRx2αtinto successive precessions about the OX and the OZ axis, with angular velocities 2α and 2ω, respectively. Finally, one may consider solution (72) corresponding to a varying magnetic field satisfying only the condition χ=const., which may be shown (at least perturbatively) to provide the Rabi resonance in this more general case. On the other hand, a decomposition of the above type is possible only for χ=0, which yields(88)Ut=eiω000e-iω0cos∫0tαsds-isin⁡∫0tαsds-isin⁡∫0tαsdscos⁡∫0tαsdsand in particular, the transition probability is measured as P↑=U122=sin2⁡∫0tα(s)ds. For α=const. the time average is clearly given by P↑=1/2. The choice of α(t) and ω(t) allows for a very precise control over the spin of the corresponding two-level system (qubits) and they will probably find good use in quantum computation (see, for example., [19]).


## 8. Concluding Remarks

Let us note that our representationΩ in the standard quaternion basis yields Ω→A⋆=ω3i+ω2j-ω1k∈su(2), since we chose to relate ω3 to the Cartan subalgebra. On the other hand, one may use the ordering ζ→(ζ0,ζ1,ζ2,ζ3)t or, equivalently, choose (z,w)=(ζ0+iζ1,ζ2+iζ3), and then the quaternion associated with Ω may be written simply as A⋆=ω1i+ω2j+ω3k. This choice of coordinates has its obvious advantages: for example, the propagator U(t) is a solution to the first-order scalar (in quaternion terms) initial value problem(89)ζ˙=A⋆ζ,ζ0=I,where the quaternion ζ is expressed in the above basis as ζ=ζ01+ζ1i+ζ2j+ζ3k. In our representation, this clear picture is somehow distorted, but it agrees well with the classical description of the symmetric top or the usual treatment of the spin in quantum mechanics, which has its own merits. We also point out that both the perturbation technique and the dynamical considerations may be applied also to the vector-parameter description of the problem, but the quaternion realization here reduces the Riccati equations (10) to linear ones, which provides a serious simplification.One major advantage of the above approach is simplicity (compared, for example, with [15]), which usually allows for a straightforward generalization of the results. In this particular case, one may easily extend the field of scalars from R to C, preserving much of the obtained so far, but of course, dealing with new issues, such as isotropic directions. This exercise is not utterly pointless since the proper Lorentz group in special relativity is known to be isomorphic to the complex orthogonal group in dimension three: that is, SO(3,C)≅SO+(3,1). Thus, electromagnetic and relativistic effects such as Thomas precession, for instance, can be modeled in the complexified rigid body framework (see [13]). Another nontrivial step towards generalization would be considering the dual analogue of the above results in the context of nonhomogeneous Euclidean isometries and screw kinematics (cf. [20]). Finally, one may also focus on the applications in optics, nuclear-magnetic resonance, and quantum computation. However, a detailed treatment of each of these subjects alone would be a volume-demanding task; therefore, we leave such considerations for future research.



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*Source: 9256320-2018-02-28.xml*