Proceedings of the 1999 Particle Accelerator Conference, New York, 1999 QUANTUM ASPECTS OF ACCELERATOR OPTICS Sameen A. Khan∗, Dipartimento di Fisica Galileo Galilei, Universit`a di Padova, Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Padova Via Marzolo 8, Padova 35131, ITALY Abstract symmetric and quadrupole lenses respectively. This for- malism further outlined the recipe to obtain a quantum the- Present understanding of accelerator optics is based mainly ory of aberrations. Details of these and some of the related on classical mechanics and electrodynamics. In recent developments in the quantum theory of charged-particle years quantum theory of charged-particle beam optics has beam optics can be found in the references [8]-[17]. I shall been under development. In this paper the newly developed briefly state the central theme of the quantum formalism. formalism is outlined. The starting point to obtain a quantum prescription is Charged-particle beam optics, or the theory of transport to build a theory based on the basic equations of quantum of charged-particle beams through electromagnetic sys- mechanics appropriate to the situation under study. For tems, is traditionally dealt with using classical mechanics. situations when either there is no spin or spinor effects This is the case in ion optics, electron microscopy, acceler- are believed to be small and ignorable we start with the ator physics etc [1]-[4]. The classical treatment of charged- scalar Klein-Gordon and Schr¨odinger equations for rela- particle beam optics has been extremely successful, in the tivistic and nonrelativistic cases respectively. For electrons, designing and working of numerous optical devices from 1 protons and other spin- 2 particles it is natural to start with electron microscopes to very large particle accelerators, in- 1 the Dirac equation, the equation for spin- particles. In cluding polarized beam accelerators. It is natural to look 2 practice we do not have to care about the other (higher spin) for a prescription based on the quantum theory, since any equations. physical system is quantum at the fundamental level! Such In many situations the electromagnetic fields are static a prescription is sure to explain the grand success of the or can reasonably assumed to be static. In many such de- classical theories and may also help towards a deeper un- vices one can further ignore the times of flights which are derstanding and designing of certain charged-particle beam negligible or of not direct interest as the emphasis is more devices. To date the curiosity to justify the success of on the profiles of the trajectories. The idea is to analyze the the classical theories as a limit of a quantum theory has evolution of the beam parameters of the various individ- been the main motivation to look for a quantum prescrip- ual charged-particle beam optical elements (quadrupoles, tion. But, with ever increasing demand for higher lumi- bending magnets, ···) along the optic axis of the system. nosities and the need for polarized beam accelerators in ba- This in the language of the quantum formalism would re- sic physics, we strongly believe that the quantum theories, quire to know the evolution of the wavefunction of the which up till now were an isolated academic curiosity will beam particles as a function of ‘s’, the coordinate along the have a significant role to play in designing and working of optic axis. Irrespective of the starting basic time-dependent such devices. equation (Schr¨odinger, Klein-Gordon, Dirac, ···)thefirst It is historically very curious that the, quantum ap- step is to obtain an equation of the form proaches to the charged-particle beam optics have been very modest and have a very brief history as pointed out ∂ i¯h ψ (x,y; s)=Hˆ (x,y; s) ψ (x,y; s) , in the third volume of the three-volume encyclopaedic text ∂s (1) book of Hawkes and Kasper [5]. In the context of accelera- tor physics the grand success of the classical theories orig- where (x, y; s) constitute a curvilinear coordinate system, inates from the fact that the de Broglie wavelength of the adapted to the geometry of the system. For systems with (high energy) beam particle is very small compared to the straight optic axis, as it is customary we shall choose the typical apertures of the cavities in accelerators. This and optic axis to lie along the Z-axis and consequently we have related details have been pointed out in the recent article s = z and (x, y; z) constitutes a rectilinear coordinate sys- of Chen [6]. A detailed account of the quantum aspects tem. Eq. (1) is the basic equation in the quantum formalism of beam physics is to be found in the Proceedings of the and we call it as the beam-optical equation; H and ψ as the recently held 15th Advanced ICFA Beam Dynamics Work- beam-optical Hamiltonian and the beam wavefunction re- shop [7]. spectively. The second step requires to obtain a relationship A beginning of a quantum formalism starting ab initio for any relevant observable {hOi (s)} at the transverse- s {hOi (s )} with the Dirac equation was made only recently[8]-[9]. The plane at to the observable in at the transverse formalism of Jagannathanet al was the first one to use the plane at sin,wheresin is some input reference point. This Dirac equation to derive the focusing theory of electron is achieved by the integration of the beam-optical equation lenses, in particular for magnetic and electrostatic axially in (1) ∗ ˆ Email: [email protected] ψ (x, y; s)=U (s, sin) ψ (x, y; sin) , (2) 0-7803-5573-3/99/$10.00@1999 IEEE. 2817 Proceedings of the 1999 Particle Accelerator Conference, New York, 1999 which gives the required transfer maps ers of the de Broglie wavelength (λ0 =2π¯h/p0). Lastly, and importantly the question of the classical limit of the hOi (s ) →hOi (s) in quantum formalism; it reproduces the well known Lie alge- = hψ (x, y; s) |O| ψ (x, y; s)i , D E braic formalism of charged-particle beam optics pioneered ˆ † ˆ by Dragt et al [18]. = ψ (x, y; sin) U OU ψ (x, y; sin) (3). We start with the Dirac equation in the presence of static (φ(r), A(r)) The two-step algorithm stated above may give an over- electromagnetic field with potentials simplified picture of the quantum formalism than, it actu- ˆ HD |ψDi = E |ψDi , (4) ally is. There are several crucial points to be noted. The first-step in the algorithm of obtaining the beam-optical where |ψDi is the time-independent 4-component Dirac equation is not to be treated as a mere transformation which spinor, E is the energy of the beam particle and the Hamil- t s ˆ eliminates in preference to a variable along the optic tonian HD, including the Pauli term in the usual notation axis. A clever set of transforms are required which not only is eliminate the variable t in preference to s but also gives us ˆ 2 the s-dependent equation which has a close physical and HD = βm0c + cα · pˆ − µaβΣ · B , (5) mathematical analogy with the original t-dependent equa- tion of standard time-dependent quantum mechanics. The where πˆ = pˆ − qA = −i¯h∇− qA. After a series of trans- imposition of this stringent requirement on the construction formations (see [14] for details) we obtain the accelerator of the beam-optical equation ensures the execution of the optical Hamiltonian to the leading order approximation second-step of the algorithm. The beam-optical equation is E E ∂ such, that all the required rich machinery of quantum me- i¯h ψ(A) = Hˆ (A) ψ(A) , ∂z chanics becomes applicable to compute the transfer maps 1 characterizing the optical system. This describes the essen- ˆ (A) 2 H ≈ −p0 − qAz + πˆ⊥ tial scheme of obtaining the quantum formalism. Rest is 2p0 mostly a mathematical detail which is built in the powerful γm0 + Ωs · S , algebraic machinery of the algorithm, accompanied with p0 some reasonable assumptions and approximations dictated 1 with Ωs = − qB + Bk + γB⊥ (6). by the physical considerations. For instance, a straight op- γm0 tic axis is a reasonable assumption and paraxial approx- πˆ 2 =ˆπ2 +ˆπ2 =2m µ /¯h γ = E/m c2 imation constitute a justifiable approximation to describe where ⊥ x y, 0 a , 0 ,and 1 ˆ (A) the ideal behaviour. S = 2 ¯hσ . We can recognize H as the quantum me- Before explicitly looking at the execution of the algo- chanical, accelerator optical, version of the well known rithm leading to the quantum formalism in the spinor case, semiclassical Derbenev-Kondratenko Hamiltonian [19] in we further make note of certain other features. Step-one of the leading order approximation. We can obtain corrections the algorithm is achieved by a set of clever transformations to this by going an order beyond the first order calculation. and an exact expression for the beam-optical Hamiltonian It is straightforwrd to compute the transfer maps for is obtained in the case of Schr¨odinger, Klein-Gordon and a specific geometry and the detailed discussion with the Dirac equations respectively, without resorting to any ap- quantum corrections can be found in [14]. In the classical proximations! We expect this to be true even in the case of limit we recover the Lie algebraic formalism [18]. higher-spin equations. The approximations are made only One practical application of the quantum formalism at step-two of the algorithm, while integrating the beam- would be to get a deeper understanding of the polarized optical equation and computing the transfer maps for aver- beams.