From the Kinematics of Precession Motion to Generalized Rabi Cycles

Hindawi Advances in Mathematical Physics Volume 2018, Article ID 9256320, 10 pages https://doi.org/10.1155/2018/9256320

Research Article From the Kinematics of Precession Motion to Generalized Rabi Cycles

Danail S. Brezov,1 Clementina D. Mladenova,2 and Iva\lo M. Mladenov 3

1 University of Architecture, Civil Engineering and Geodesy, 1 Hristo Smirnenski Blvd., 1046 Sofa, Bulgaria 2Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 4, 1113 Sofa, Bulgaria 3Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 21, 1113 Sofa, Bulgaria

Correspondence should be addressed to Iva¨ılo M. Mladenov; [email protected]

Received 29 September 2017; Accepted 16 January 2018; Published 28 February 2018

Academic Editor: Manuel Calixto

Copyright © 2018 Danail S. Brezov et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We use both vector-parameter and quaternion techniques to provide a thorough description of several classes of rotations, starting with coaxial angular velocity Ω of varying magnitude. Ten, we fx the magnitude and let Ω precess at constant rate about the �-axis, which yields a particular solution to the free Euler dynamical equations in the case of axially symmetric inertial ellipsoid. Te 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 �-projection of Ω (expressed in cylindrical coordinates), which are both time-dependent in the generic case. Relevant physical examples are also provided.

1. Introduction: Quaternions and thesametime,quaternionshaveaspecifcClifordmultipli- Vector-Parameters cation rule that can be expressed in the above notation as 0 (� , �)(�̃ , ̃�)=(��̃ − � ⋅ ̃�,� ̃� + �̃ � + � × ̃�), Te Cliford algebra of quaternions H ≅�ℓ0,2(R)≅�ℓ3 (R) 0 0 0 0 0 0 (3) 3 is known to be intimately related to the rotation group in R 3 where ⋅ and × stand for the dot and cross products in R , and one way to see it is via the standard spin covering map Z2 respectively.Justlikeinthecaseofcomplexnumbers,Cliford Spin(3) ≅ SU(2) ��→ SO(3) that is topologically a projection conjugation in H isgivenbysigninversionoftheimaginary ∗ 3 Z2 3 � �=(�, �)→� =(�,−�) from the unit sphere S ��→ RP . Te basis of bivectors in (vector) part :thatis, 0 0 ,and since there are no zero divisors, the quaternion norm defned �ℓ3 is spanned by three units satisfying 2 ∗ as ‖�‖ 1 =�� yields the inverse of every nonzero element −1 −2 ∗ 2 2 2 2 � =‖�‖ � . In particular, restricting to ‖�‖ =1,one i = j = k = ijk =−1, (1) 4 obtains the unit sphere in R , which the composition law (3) endows with the group structure of SU(2).Atthispoint, 1 3 where denotes the identity element, so one can express each rotations in R appear quite naturally if we identify three- quaternion �∈H in the form 3 vectors with pure (imaginary) quaternions x ∈ R → �=̃ �1i +�2j +�3k ∈ su(2) and use the adjoint action of the �=(�0, �) =�01 +�1i +�2j +�3k. (2) ̃ ̃ ∗ spin group Ad� : �→�� . Tis efectively transforms the components of x via the linear orthogonal map Hence, as a linear space, H may obviously be identifed with 4 3 2 2 � × R ≅ R × R by introducing coordinates as shown above. At R (�) =(�0 − � ) I +2�� +2�0� , (4) 2 Advances in Mathematical Physics

3 � where I denotes the identity in R , �� stands for the while in the inverse direction, one easily derives the Riccati × usualdyadic(tensor)product,and� is the skew-symmetric equations for the vector-parameter linear map associated with � via the Hodge duality: that is, ×̃ ̃ 1 � × 1 � × � � = � × �. Te above map clearly preserves orientation and ċ = (I + cc − c ) Ω = (I + cc + c ) �. (10) 3 2 2 thus represents rotation in R . Denoting its angle � and the n unit vector along its axis , one may express the associated We refer to [3] for a detailed introduction to quaternions quaternion as and some of their applications in a wide range or felds, � from navigation and attitude control to computer graphics. In �0 = cos , quantum mechanics quaternions and Pauli matrices appear 2 naturally in the description of spin systems as we recall below. (5) � Moreover, modern physics embraces the idea of quaternionic � = n. sin 2 Hilbert space and quaternionic quantum mechanics (cf. [4, 5]). Many features of complex numbers are preserved except, Central projection onto the plane �0 =1then yields the of course, commutativity, which makes the defnition of vector-parameter c = �/�0 = tan(�/2)n that is actually not quaternionic analyticity far nontrivial and yet fruitful (see a vector but rather a point in projective three-space, equipped [6]), for example, in the description of electrodynamics [7] with an additional group structure. Tis simple construction and its relation to the so-called Fueter analyticity.Vector appears quite convenient: on the one hand, unlike other parameterization [8, 9], on the other hand, is much less known alternatives (such as Euler angles which can be con- popular although it has proven quite efcient in the descrip- sidered as coordinates of points in projective space on the tion of (pseudo-)orthogonal groups in dimensions three and three-torus and lead to singularities, e.g., the so-called gimbal four[10],aswellasinawiderangeofproblemsrelatedto lock) it gives a topologically adequate description of the classical and quantum mechanics [2, 11], special relativity and 3 orthogonal group SO(3) ≅ RP , while, on the other hand, it electrodynamics [12, 13], robotics and navigation [3, 14], and provides an efcient associative composition law various other felds of science and technology. Here we make use of both descriptions. c2 + c1 + c2 × c1 Te text is organized as follows: afer this brief introduc- ⟨c2, c1⟩ = (6) 1−c2 ⋅ c1 tory part, we consider the kinematical problem of a rotating rigid body with fxed direction of the angular velocity vector inherited from the quaternion multiplication (3) upon central using the projective vector-parameter construction discussed projection. Tis constitutes a (nonlinear) representation of above, thus extending an old result due to O’Reilly and Payen SO(3) related to the usual matrix realization via R(⟨c2, c1⟩) = [15]. Ten, in Section 3, we switch back to quaternions, which R(c2)R(c1), in which the inverse element is given by −c lead to linear, rather than Riccati equations, in order to deal and the neutral one by the zero vector. Moreover, it yields with the more complicated settings. Namely, we consider rational expressions for the matrix entries of R(c) that may the case of a precessing (about a fxed axis) angular velocity be obtained directly from (4) or equivalently by means of the and the most general case, given conveniently in cylindrical famous Cayley transform coordinates, using techniques from the arsenal of quantum mechanics, which allow us to derive a simple integrability 2 � × I + c× (1 − c ) I +2cc +2c condition. Finally, afer a brief note on dynamics, we relate R (c) = = ⋅ (7) I − c× 1+c2 these results to the Rabi oscillator that appears in quantum computation, showing how our solutions yield more freedom Clearly, the above follows also from Rodrigues’ rotation in the entire process. formula with the aid of Euler’strigonometric substitution, but the projective approach reveals the major advantages of vec- 2. The Coaxial Angular Velocity (CAV) Motion tor parameterization in a much more direct manner. Recall also that the angular velocity of a rotating rigid body is Now, let us consider the frst equation in (10) and look defned in the body and inertial frames, respectively, as for solutions with a fxed direction of Ω.Withoutlossof generality, we may choose for convenience a reference frame, Ω× = RṘ �, in which Ω is aligned with the �-axis: that is,

(8) � �× = R�Ṙ . Ω = (0, 0, Ω) ,Ω=2�(�) . (11)

Straightforward diferentiation of the Cayley representation Tis allows for expressing the matrix ODE (10) explicitly in (7) yields the relations (see also [1, 2]) components as

2 �1̇ =�(�1�3 −�2), Ω = (I + c×) c,̇ 1+c2 �̇ =�(�� +�), (9) 2 2 3 1 (12) 2 × � = (I − c ) ċ �̇ =�(1+�2). 1+c2 3 3 Advances in Mathematical Physics 3

From the third equation, one obviously has (we set the initial otherhand,itisnotdifculttoseethatthewholemanifold 3 moment at �=0) SO(3) ≅ RP iscoveredbytheabovemap.Moreover,each � point in projective space is reached by the initial data setting ∘ 3 � (�) = �, � (�) = �+∫ � (�) �, �=0since RP isobviouslycoveredbythefamilyofrays 3 tan d (13) ∘ 0 (18), in which we may use � as an intrinsic parameter. Now, while for the frst two we may use the substitution �(�) = in order to lif the whole construction up to the spin cover, we �± =�−1c ∈ S3 �1(�) + i�2(�),whichyields may express the two images ± corresponding to the vector-parameter (17) in the form �̇ (�) =(�3 (�) + i)�(�) � (�) . (14) � i� � −i� ± sin e icos e �≡0 � =±( ) (20) Te latter clearly possesses a trivial solution that is a � i� � −i� well-known result: in this case, the rotation takes place about icos e sin e the axis determined by the angular velocity vector Ω with magnitude Ω=2�.However,for�=0 ̸ ,onehas with a suitable choice of a quaternion basis and using the notation d ( �) = ( �+ ) �. (15) � � ln tan i �= �∈[0, ]⋅ d arctan 2 (21) Combining the above two cases, one may express the general � solution in the form (here the amplitude �≥0and the two Note that the -orbits on the unit sphere project to circles in ∘ (�, �) (�, �) �∘ �∈[0,2�) the and planes with equal frequencies but gener- phases , are constants of integration) ally diferent phases and amplitudes. In particular, for �=0 � and �=�/2the former (resp., the latter) circle shrinks to a i� ∘ � (�) =� � ,�(�) = �+∫ � (�) � (16) point. From the perspective of the Hopf map, one has quater- sec e d 3 1 0 nion �-orbits on S whose holonomy shifs the S phase � by a whole period for each revolution about the �-axis of the from which it is straightforward to write 2 base S .Tisbecomesmoreapparentifwewrite� with its � S3 c (�) = sec �(�cos �, � sin �, sin �) . (17) coordinates on as (�±,�±)=±( � �, � �, � �, � �) . Changing the parameter �→�=tan �(�),itisnothardto 0 sin cos sin sin cos sin cos cos (22) showthattheintegralcurves(17)obtainedaboveareactually 3 rays in R described by Stereographic projection (from the south pole onto the equatorial plane) yields also the so-called Wiener-Milenkovic � 1 c (�) = (�+V�, �� − V,�) ,�∈RP , (18) conformal rotation vector (see [9] for more details) where the constants � and V are most naturally given in polar � � � = = tan n (23) coordinates as 1+�0 4

∘ ∘ �=�cos (�−�) , which does not involve infnities at least for �=0 ̸ ,soonemay plot the corresponding curves in Euclidean space. Moreover, ∘ (19) ∘ V =�sin (�−�) . the rotation angle is not difcult to derive from formula (17): namely, Inthisway,weendupwithatwo-parameterfamilyofraysin 3 2 R , whose intersection points (�, �, �,̃ �)̃ = (�, −V,�+V,�−V) � (�) =2arctan (sec �√�2 + sin �) . (24) with the planes �=0and �=1constitute isosceles right c(�) = triangles with the origin. In other words, if we denote As for the corresponding invariant axis, ignoring the com- c +�k c k 0 in (18), the constant vectors 0 and are perpendicular mon prefactor in (17), we see that it precesses about the c2 = k2 k = k − ̂e k and 0 0,where 0 � is the projection of in vector Ω. Besides, the rate of precession and nutation are (�, �) � the -plane. Tose are all orbits still periodic in .In synchronized and in the case �=const.,equaltohalfthe � particular, the vertical -axis and the horizontal projective magnitudeoftheangularvelocityΩ=2�.Itisconvenient �→0 line at infnity are limiting cases corresponding to also to introduce spherical coordinates for the unit invariant and �→∞as discussed below in detail. Note that one may ∘ vector n = n(�, �) in the form redefne � via an overall phase shif introduced by a proper ∘ choice of initial time, which yields �=0,soinphysical �=arctan (� csc �) . (25) termsonehasonlytworelevantconstants(amplitudeand phase) and the general solution of (12) is generated by the Usingformula(7)andtakingintoaccount(21),weexpress time fow of c(�) as orbits of all possible initial points. On the the corresponding rotation matrix explicitly as 4 Advances in Mathematical Physics

2 2 2 2 cos 2� cos �+cos 2� sin � sin 2� sin �−sin 2� cos � sin (� + �) sin 2� 2 2 2 2 R (�, �, �) = ( sin 2� sin �+sin 2� cos � cos 2� cos �−cos 2� sin �−cos (� + �) sin 2�)⋅ (26) sin (� − �) sin 2� cos (� − �) sin 2� cos 2�

Note that the angles � and � difer only by a phase Specifc solutions for diferent values of the phase shif are constructedsimilarly.Notealsothattheabovematriceshave ∘ �=�+�−�,∘ (27) such a simple form only in the reference frame, in which the angular velocity Ω is aligned with one of the axes. Otherwise, while � is a constant of integration, which makes the time the vector-parameter (17) needs to be rotated correspond- dependence of the above matrix entries relatively simple. ingly, relative to the coordinate change. Moreover, we can manipulate both � and �−�via specifc Te compact explicit form of the solution (26) allows for � � � choice of initial data to obtain some particular cases. Firstly, expressing the three parameters , ,and ,whichweuse we discuss the three distinct types of solutions pointed out in as a substitute for the usual Euler angles, in the generic case �=0,�/2 ̸ [15] for the case �=const. Te most obvious one, for which as ̇ c ‖ Ω with Ω=� (Type I), corresponds to �≡0(�→0)and is usually referred to as steady rotation.Itsuggestsaconstant 1 �=− [atan2 (�31,�32)+atan2 (�13,�23)] , invariant axis oriented along the angular velocity vector and 2 linearly evolving angle �:namely, 1 �= [ (� ,� )− (� ,� )] , 2 atan2 31 32 atan2 13 23 (32) cos 2� − sin 2� 0 1 R‖ (�) =(sin 2� cos 2� 0)⋅ (28) �= � , 2 arccos 33 001

where atan2 stands for the proper quadrant inverse tangent �=�/2 �→∞ On the other hand, setting ,thatis, ,weobtain (note that �∈[0,�/2],sosin2� ≥ 0 and cos 2� is the half-turn example considered in [15] (Type II rotations) monotonous, thus invertible in this interval) and the ��’s R (�) = 2nn� − I n =( �, �, 0)� as ⊥ ,where cos sin ,which denote the matrix entries of R(�, �, �) given by formula (26) may be written explicitly in matrix terms directly from (26) in the standard basis. Moreover, since in the form (note that the �-dependence in (26) is completely factored out by demanding � to be a right angle) 2 2 2 1+c =(1+�)(1+tan �) (33) cos 2� sin 2� 0 R (�) = ( 2� − 2� 0 )⋅ asitfollowsfromformula(17),itisstraightforwardtoobtain ⊥ sin cos (29) also the azimuthal angle � in the spherical representation 00−1 of the invariant unit vector n and the angle of rotation �, respectively, as Tistimetherotationtakesplaceaboutavaryingaxisthat remains perpendicular to Ω andprecessesaboutit,whilethe �=arctan (csc � tan �) , angle remains �=�at each particular instant of time. (34) Te third type of rotations proposed in [15], in which both �=2arccos (cos � cos �) . the rotation axes and angle are periodic functions of time, is �= also given by formula (26) if we simply set const. We may However, in the limiting cases of vanishing and infnite consider, for example, the case of coherent phases �=�,that �=0 �= ∘ amplitude of precession, corresponding to and ∘ is, �=�,inwhich �/2, respectively, (32) yield indeterminacy, so these two cases need to be treated separately. In particular, for �→0,onehas � c = (�, � tan �, tan �) , (30) 1 �=0�⇒�= atan2 (�21,�11), and express the respective rotational matrix explicitly: that is, 2 �=0, (35) R0 (�, �) �=2�, cos 2� − sin 2� cos 2� sin 2� sin 2� (31) =( 2� 2� 2� − 2� 2�)⋅ sin cos cos cos sin while the polar angle � becomes obsolete. Similarly, the limit 0 sin 2� cos 2� �→∞yields Advances in Mathematical Physics 5

� 1 �= �⇒ � = (� ,� ), where �̂ = (1/2�)Ω stands for the unit vector in the direction 2 2 atan2 12 11 ∘ of Ω and �=�(0)is simply the initial condition. ∘ ∘ � ∘ �= , (36) Denoting also c = c(0) = �/�0,weobtainthetime 2 dependence of the vector-parameter

�=� ∘ ∘ × c + tan �� (I − c ) �̂ c (�) = ⋅ � ∘ (46) and is no longer relevant as it has been already discussed. 1−(c, �̂) tan �� (0) �̂ 3. Quaternion Description of Precessions In particular, if is parallel to ,itremainssuchalongthe fow determined by (46) as one has To derive the kinematic equations in quaternion terms, it � (�) sufces to recall that c = �/�0 andsubstituteitinformula ( ) 2 � (�) (10), which yields upon multiplication by �0 0 ̇ ̇ 2 × × ∘ 2�0� −2�0� =�0 Ω + (�, Ω) � +�0Ω �. (37) cos ��I + sin ��̂ sin ��̂ sin ��̂ =( )( ) � − ��̂� �� ∘ (47) Ten, taking a dot product with and using the identities sin cos cos � � �2 2 2 ̇ ̇ �� =�0 + � =1�⇒�0�0 + (�, �) =0 (38) ∘ sin (�� + �) �̂ =( ), yield the derivative of the scalar (real) part of the quaternion ∘ cos (�� + �) ̇ 2�0 =−(Ω, �) . (39) soweendupwiththetrivialorbitc(�) = tan ��̂,whichgives Combining both expressions, we obtain the vector (imagi- ∘ thesolutionalsointhecaseofasteadyinitialstate(�0 =1). nary) part in the form ∘ ∘ � ⊥ �̂ � =0 c(�) = c∘ + ��̂ × c∘ ̇ Similarly, if and 0 ,onehas tan , 2� = Ω × � +�0Ω, (40) so c(�) remains perpendicular to �̂. More generally, we may ∘ so the kinematic equations have the block-matrix representa- denote p(�) = (c(�), �̂) and p(0) = tan �,aswellasq = c − �̂ = P⊥c tion p �̂ ,andobtain ̇ × � 1 Ω Ω � p (�) = tan � (�) , ( )= ( )( )⋅ (41) �̇ 2 −Ω� 0 � × 0 0 I + tan ��̂ (48) q (�) = ∘ q (0) . Identifying quaternions and four-vectors allows for writing 1−tan � tan �� =̇ A� the above result as and, thus, derive the general Note that, just as before, we may weaken the condition of solution for the propagator in terms of time-ordered matrix constant angular velocity by demanding only �̂ = const.,thus Ω = . exponents. In the particular case const ,however,weend letting �=�(�). Tis introduces just a slight modifcation up with a system of linear homogeneous ODEs with constant of the above solution: namely, in formula (46), we make the coefcients, hence the solution substitution �A � � (�) = e � (0) . (42) �� → ∫ � (�) d�. (49) 2 2 2 0 Note that det(A −�I)=(�+�) ,sowehavetwo A double roots �± =±i�,whichyields,inthecanonical It is useful to point out that the matrix (and therefore, its exponent) may be expressed also in a 2×2block form. In basis, synchronized rotations (with equal frequencies) in two � 4 �̂ = (0, 0, 1) mutually perpendicular planes in R as we have seen in the particular, if ,onehas Ω =0 J�� case 1,2 .Moreover, e 0 0−1 �A =( ), J =( ) 2� � 2� 2�+1 � 2�+1̂ e −J�� (50) A = (−1) � I, A = (−1) � A (43) 0 e 10 and, thus, the propagator may be expressed as which yields the solutions obtained in the previous section. �A More generally, we have e = cos ��I + sin ��Â, (44) �̂ J �̂ I − �̂ J ̂ 3 2 1 where we let I denote the identity in any dimension and Â = A =( ), (51) −1 −�̂ I − �̂ J −�̂ J � A with � = (1/2)|Ω|. Bearing this in mind, for the time 2 1 3 evolution of an arbitrary quaternion �(�),onehas that is, a real 4×4representation of a pure unit quaternion ̂ ̂⋆ ∘ ∘ ∘ and may be mapped onto su(2) as A → A = �̂3i+�̂2j−�̂1k. � (�) = �� + �� (� �̂ + �̂ × �), ⋆ ⋆ cos sin 0 Te action of A =�Â on � is naturally realized via matrix (45) ∘ ∘ multiplication if the latter is represented as a point (�, �) in ̂ 2 �0 (�) = �0 cos �� − (�, �) sin ��, C :namely, 6 Advances in Mathematical Physics

�=� + � , 1 i 2 derive the exact solution transforming system (53) into a (52) single ODE. Namely, afer diferentiating the frst equation �=�3 + i�0. and substituting ̇ ⋆ � �=A � −2i�0� ̇ Ten, the kinematic equation for may be written as , �� = e (i�+��) (57) or in components ̇ from the second one, the time dependence of the coefcients � i�3 �2 − i�1 � ( )=( )( ), factorsoutandweendupwithalinearhomogeneoussecond- −� − � − � � �̇ 2 i 1 i 3 (53) order ODE with constant coefcients �−2̈ � �+(�̇ 2 +�2 −2�� )�=0. �� =��̂� i 0 0 (58) and for fxed �̂ the corresponding unitary evolution operator With the notation takes the form �=√�2 +�2, � � ⋆ exp (∫ � (�) d� Â )=cos ∫ � (�) d� I 0 0 �=�−�0, (59) (54) � � ̂⋆ + sin ∫ � (�) d� A . �0 = arctan , 0 � we may express the general solution in the compact form 4. Precessing Angular Velocity i� � � (�) = e 0 (� cos �� + � sin ��) (60) Let us now consider the case of varying directional vector �̂ . One relatively simple setting is that of an angular velocity and, respectively, from the above substitution, we have precessing about a fxed axis: for example, −i� � � (�) = 0 ( � (� �� + � ��) � + e tan 0 cos sin Ω =2(�cos 2�0�, � sin 2�0�, �) , �,�,�0 ∈ R . (55) (61) +isec�0 (� cos �� − � sin ��)). Tis time the magnitude Ω=2√�2 +�2 is constant, so one Here, � and � are complex constants determined by the initial has ∘ �=�(0) �=�(0)∘ 2i� � conditions and as −� � 0 ⋆ e A (�) =− ( ) ∘ i −2i� � (56) �e 0 � �=�,

∘ (62) ⋆ ∘ and due to the nonvanishing commutator [A (�), i�=cos �0�−sin �0� �. � ∫ A⋆(�) �] =0̸ 0 d , obtaining the propagator via exponentiating ∘ ∘ � � the above matrix generally involves time-ordering or some TisallowsforwritingthepropagatorU(�) : (�, �) → (�, �) other cumbersome procedure. Nevertheless, one may still in the simple form

i�0� i�0� e (cos �� + isin�0 sin ��) −ie cos �0 sin �� U (�) =( ), (63) −i�0� −i�0� −ie cos �0 sin �� e (cos �� − isin�0 sin ��)

whereforthefrstrowwesubstitutetheexpressionsfor�, � ∼ Tis allows for expressing also the evolution of the vector- ∘ ∘ �, � in the equation for �(�), while for the second one, we parameter in the form (see Figure 1) use the fact that U(�) takesvaluesinSU(2). Considering, for 1 ∘ ∘ c (�) = example, a steady initial state (�, �) = (0, i) from the above, 1−sin �0 tan �� tan �0� one easily obtains � �� cos 0 tan (65) �0 (�) = cos �� cos �0�−sin �0 sin �� sin �0�, ×( cos �0 tan �� tan �0� )⋅ � � = � �� , 1 ( ) cos 0 sin cos 0 sin �0 tan �� + tan �0� (64) �2 (�) = cos �0 sin �� sin �0�, 2 2 2 Taking into account the fact that 1+c = sec ��sec ��, �3 (�) = sin �0 sin �� cos �0�+cos �� sin �0�. one may express the entries of the corresponding rotation Advances in Mathematical Physics 7

� Ω (�) =2(�(�) cos 2�0 (�) ,�(�) sin 2�0 (�) ,�(�)) . (68)

Weuseastandardtechniqueinphysicstofactoroutthe ⋆ ⋆ dependence on the diagonal part A0 = A�=0 of (56) known as the interaction picture (cf. [16]). It involves expressing the ⋆ propagator as (the matrix A0 is not necessarily constant; we onlydemandthatitcommuteswithitsintegral;see[16])

Figure 1: Te evolution of the vector-parameter (65) and the trace � (3, 14, 25) ⋆ of the corresponding rotation acting on the vector . U (�) = U0 (�) U� (�) , U0 (�) = exp ∫ A0 (�) d�, (69) 0

where U�(�) solves the reduced equation (this is easy to see matrix using formula (7). In particular, in the resonant limit via straightforward diferentiation) �0 →�,onehas�0 =0and �=�;hence,c(�) = (tan ��, �� , � �)� tan tan 0 tan 0 ,whichfnallyyields ̇ ⋆ ⋆ −1 ⋆ ⋆ −1 �� (�) = A� (�) �� (�) , A� = U0 A U0 − A0 ,�� = U0 �. (70)

R (�)�=0 With the aid of formulas (53) and (68), we express cos 2�0�−cos 2�� sin 2�0� sin 2�� sin 2�0� (66) � 2� � 2�� 2� �− 2�� 2� � i ∫ �(�)d� =(sin 0 cos cos 0 sin cos 0 ). e 0 0 U (�) =( ), 0 � 0 sin 2�� cos 2�� −i ∫ �(�)d� 0 e 0 (71) As we show in the next section, a similar expression holds 0 −2i� ⋆ e also for variable � and �0 (under certain restrictions) if we A (�) =−�( ) � � i �� → ∫ �(�) � � �→�(�) 2i� 0 substitute 0 d and 0 0 .Onemayalso e derive the rotation matrix using directly formula (4), written � in components as �(�) = ∫ �(�) �−�(�) with the notation 0 d 0 .Notethatinthe �= . A⋆(�) R (�) special case const the matrix � commutes with its integral and may thus be exponentiated trivially. Ten, the 2 2 evolution operator U(�) = U0(�)U�(�) has the simple form 1−2(�2 +�3 )2(�1�2 −�0�3)2(�1�3 +�0�2) 2 2 (67) =(2(�1�2 +�0�3)1−2(�1 +�3 )2(�2�3 −�0�1)). U (�) 2 2 2(�1�3 −�0�2)2(�2�3 +�0�1)1−2(�1 +�2 ) � � i(� +�) i(� −�) e 0 cos ∫ � (�) d�−ie 0 sin ∫ � (�) d� (72) 0 0 =( � � ) −i(� −�) −i(� +�) 5. Generic Angular Velocity −ie 0 sin ∫ � (�) d� e 0 cos ∫ � (�) d� 0 0 Our construction may be generalized as follows: let us consider a time-varying angular velocity vector in the form andinparticular,for�=0, the corresponding rotation (note once more that here Ω(�) = 2√�2 +�2) matrix is reduced to

� � cos 2�0 (�) − sin 2�0 (�) cos 2 ∫ � (�) d� sin 2�0 (�) sin 2 ∫ � (�) d� 0 0 � � ( ) R (�)�=0 = sin 2�0 (�) cos 2�0 (�) cos 2 ∫ � (�) d�−cos 2�0 (�) sin 2 ∫ � (�) d� ⋅ (73) 0 0 � � 0 sin 2 ∫ � (�) d� cos 2 ∫ � (�) d� ( 0 0 )

� � =� � 2� � , We conclude this section with a comment on the case of 1 ( ) ( ) sin ( ) �(�) (74) nonconstant , which yields some iterative procedure �2 (�) =−�(�) cos 2� (�) , (such as time-ordered exponentiation) for obtaining the propagator in the interaction picture. Here we choose to work which allows us to calculate the corrections explicitly in some with a version of Magnus expansion, usually referred to as the special cases. Consider, for example, small precessions about continuous BCH formula (see [16]). Note that in the standard the �-axis with �=2�(�̇0 −�)= −2��,wherė �≪1is ⋆ quaternion basis, we have the representation A� =�1j +�2k regarded as a perturbation parameter. A straightforward with computation yields 8 Advances in Mathematical Physics

2 2 2 U� (�) ≈(1−� sin Δ�) I +� (sin 2Δ� − 2Δ�) i a linear precession with frequency 2�0 about the �-direction, (75) Ω=2√�2 +�2 �� for which the angular velocity magnitude is +� 2�� j +� 2�� k, cos �0 sin �0 also constant. On the other hand, we may assume only �= . � =−� 3 const ,whichimmediatelyleadsto 2 1 and due to the where we ignore terms of order � and use the notation triangle inequality for the principal moments of inertia, this Δ�(�) = �(�)−�(0) U (�) . At the same time, 0 remains as before inevitably yields �3 =0,butthen�=0̇ andweobtainthepre- andwestillhavetheexpressionU(�) = U0(�)U�(�). viously considered case once more. Let us now investigate the setting �1 =�2 =�,whichim- 6. A Note on Dynamics mediately yields the relations

Consider the Euler equations describing free rigid body rota- 2 2 �1 +�2 tion (cf. [17]) �3 = , �1 +�2 Ω̇ =�Ω Ω , 1 1 2 3 � −� �= 2 1 , ̇ � +� (80) Ω2 =�2Ω1Ω3, (76) 2 1 2 2 Ω̇ =�Ω Ω , � −� 3 3 1 2 � = 1 2 ⋅ 3 2 2 �1 +�2 where the constants �� are expressed in terms of the eigenval- ues �� of the inertial tensor as Tis symmetry reduces system (78) to � −� � = 2 3 , 1 � 1 �̇ = 2�� sin 4�0, � −� 3 1 �=�̇ �2 4� , �2 = , (77) 3 sin 0 (81) �2 �̇0 =��cos 4�0 �1 −�2 �3 = ⋅ �3 andwereadilyfndaconservationlawintheform It is interesting to view our previous considerations on the kinematics determined by a particular choice of angular 2 2 2�� −�3� =�. (82) velocity from a dynamical perspective. We shall further assume that the axes chosen in the parameterization of Ω(�) ∘ in formula (68) coincide with the principal axes of the Choosing further �=�(0)allows us to obtain also 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 �̂ is preserved, then its � (�) = � �sec 4�0�, Ω magnitude is constant as well. For a free rotation described 1/2 1 ∘ 2 � � by formula (63), on the other hand, the Euler dynamical � (�) =± (� + �3� �sec 4�0�) , √2� � � equations (76) yield a symmetric inertial ellipsoid with �1 = (83) �2 =�and �3 =(1+�0/�)�. In particular, in the limit �→0, 2 4� � the solution (63) obviously converges to the one for the case ∫ �=±√ ∫ sec 0d 0 . � d 1/2 Ω= . �̂ = (0, 0, 1) � ∘ 2 � � const and as it should be expected. How- (� + � � � 4� �) ever, considering the general setting (68), we may express (76) 3 �sec 0� in the form However, the above solution does not provide explicit expres- �=2�̇ tan 2�0 (�1�+�̇0)=2�cot 2�0 (�2�−�̇0), sions for the propagator as it yields �̇0(�) ≠ �(�).Anatural 2 (78) �=�̇ 3� sin 4�0. way to reach beyond the trivial setting of linear precession and still be able to use formula (72) is to introduce external Further on, we easily derive forces (whose resultant moment is denoted with f)tothe Euler dynamical system (76). Let us consider the simple case 2 2 �̇0 (�) =�(�) (�2 cos 2�0 (�) −�1 sin 2�0 (�)), (79) of rotational inertial ellipsoid �1 =�2 =�,thatis,�2 =−�1 = �⇒�3 =0,asitallowsforarranging�=const. in a straight- so in the particular case of rotationally symmetric inertial forward manner and, thus, reconstruct the rotation matrix ellipsoid with �2 =−�1 =�and �3 =0,wehave�̇0 =��, from the angular velocity vector using expressions (72) and but then the third equality yields �=const.,so�0 =��up (4). Namely, we have �̇0(�) = �(�),thatis,�=const.,aslong to an additive phase. Finally, substituting into the equation as we set �=1and f2 = f1 tan 2�0.Ten,thesolutiontakes for �(�),wefndthat�=const. as well, so the solution is the simple form Advances in Mathematical Physics 9

� 1 ̃ � (�) =�(0) + ∫ f (�) d�, may consider solution (72) corresponding to a varying mag- 2� 0 neticfeldsatisfyingonlythecondition�=const.,whichmay 1 � be shown (at least perturbatively) to provide the Rabi reso- � (�) =�(0) + ∫ (�) �, nance in this more general case. On the other hand, a decom- 2� f3 d (84) 3 0 position of the above type is possible only for �=0,which � yields �0 (�) = ∫ � (�) d�−�, 0 U (�)

̃ = 2� = =0 � � where f f1 sec 0 andinparticular,settingf2 f1 , i�0 ∫ � � �− ∫ � � � e 0 cos ( ) d isin ( ) d (88) we end up with a precessional motion with �=const., while =( )( 0 0 ) −i� � � �(�) � (�) 0 0 and 0 can still be determined from (84). e −isin∫ � (�) d� cos ∫ � (�) d� 0 0 7. Larmor Precession and Rabi Cycles andinparticular,thetransitionprobabilityismeasuredas 2 2 � �↑ =|U12| = sin ∫ �(�)d�.For�=const. the time average Te quaternion representation of the kinematical problem 0 is clearly given by ⟨�↑⟩ = 1/2. Te choice of �(�) and described above has many advantages, as we could see. One �(�) of them, which we have not discussed so far, is that it allows allowsforaveryprecisecontroloverthespinofthe for a straightforward implementation of our solutions to corresponding two-level system (qubits) and they will proba- other physical models governed by analogous equations. One bly fnd good use in quantum computation (see, for example., particularly famous example appears in the context of Pauli’s [19]). description of spin 1/2 particles in an external magnetic feld �, for which the kinematics of the spin state Ψ is determined 8. Concluding Remarks by the equation Let us note that our representation Ω in the standard � ⋆ ̇ quaternion basis yields Ω → A =�3i +�2j −�1k ∈ su(2), Ψ= �Ψ, (85) 2 sincewechosetorelate�3 to the Cartan subalgebra. On the � other hand, one may use the ordering �→(�0,�1,�2,�3) or, where � is a physical constant (the gyromagnetic ratio) and equivalently, choose (�, �) = (�0 + i�1,�2 + i�3), and then the � ⋆ the magnetic induction may be written as a purely imag- quaternion associated with Ω may be written simply as A = inary quaternion, so the evolution of the above system de- � i +�j +�k Ω =�B 1 2 3 . Tis choice of coordinates has its obvious scribes a rotor with angular velocity .Ten,wemay advantages: for example, the propagator U(�) is a solution to use our solutions (with the proper rescaling) in a completely the frst-order scalar (in quaternion terms) initial value prob- diferent physical context: for the description of spin pre- lem cession in a coaxial (54) or a precessing magnetic feld. Te ̇ ⋆ former is a famous quantum mechanical efect known as �=A �, � (0) = I, (89) Larmor precession, while the latter was considered previously by Rabi in [18]. His solution (that won him a Nobel prize back where the quaternion � is expressed in the above basis as in 1944) has a crucial impact on magnetic resonance imaging, �=�01+�1i+�2j+�3k. In our representation, this clear picture quantum computing, and optics. A major part of the idea is somehow distorted, but it agrees well with the classical is based on the so-called Rabi resonance, which yields the description of the symmetric top or the usual treatment of highest probability for a transition from a ↓ to a ↑ spin state. the spin in quantum mechanics, which has its own merits. We Te resonance corresponds to our solution with �=0,which also point out that both the perturbation technique and the may be interpreted also as a decomposability condition for dynamical considerations may be applied also to the vector- U(�) into a pair of orthogonal pure quaternions parameter description of the problem, but the quaternion realization here reduces the Riccati equations (10) to linear i�� i�� e cos �� −ie sin �� ones, which provides a serious simplifcation. U (�) =( ) − −i�� −i�� One major advantage of the above approach is simplicity ie sin e cos (compared, for example, with [15]), which usually allows for a (86) i�� straightforward generalization of the results. In this particu- e 0 cos �� −isin�� =( )( )⋅ lar case, one may easily extend the feld of scalars from R to C, −i�� 0 e −isin�� cos �� preserving much of the obtained so far, but of course, dealing with new issues, such as isotropic directions. Tis exercise is ∘ ∘ With the initial condition (�, �) = (0, �), this yields also the not utterly pointless since the proper Lorentz group in special relativityisknowntobeisomorphictothecomplexorthog- decomposition + onal group in dimension three: that is, SO(3, C)≅SO (3, 1). R (c) = R� (2��) R� (2��) (87) Tus, electromagnetic and relativistic efects such as Tomas precession, for instance, can be modeled in the complexifed into successive precessions about the �� and the �� axis, rigid body framework (see [13]). Another nontrivial step to- with angular velocities 2� and 2�, respectively. Finally, one wards generalization would be considering the dual analogue 10 Advances in Mathematical Physics

of the above results in the context of nonhomogeneous [18] I. I. Rabi, “Space quantization in a gyrating magnetic feld,” Euclidean isometries and screw kinematics (cf. [20]). Finally, Physical Review A: Atomic, Molecular and Optical Physics,vol. one may also focus on the applications in optics, nuclear- 51, no. 8, pp. 652–654, 1937. magnetic resonance, and quantum computation. However, a [19] E. Kyoseva and N. V. Vitanov, “Arbitrarily accurate passband detailed treatment of each of these subjects alone would be a composite pulses for dynamical suppression of amplitude volume-demanding task; therefore, we leave such considera- noise,” Physical Review A: Atomic, Molecular and Optical tions for future research. Physics,vol.88,no.6,ArticleID063410,2013. [20] J. Wittenburg, Kinematics: Teory and Applications,Springer, Berlin, Germany, 2016. Conflicts of Interest Te authors declare that there are no conficts of interest regarding the publication of this paper.

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