LSLLS-271 Presented at the 15th ICFA Advanced Beam Dynamics Workshop on Quantum Aspects of Beam Physics, January 4-9, 1998. QUANTUM FLUCTUATIONS IN BEAM DYNAMICS KWANG-JE KIM Accelerator Systems Division, Advanced Photon Source Argonne National Laboratory 9700 South Cass Avenue, Argonne, IL 60439 E-mail: [email protected] Quantum e ects could b ecome imp ortant for particle and photon b eams used in high-luminosity and high brightness applications in the current and next generation accelerators and radiation sources. This pap er is a review of some of these e ects. 1 Intro duction The main aim of mo dern particle accelerators for high energy physics is to pro duce high energy particle b eams and collide them with a high luminosity: 2 N L = f 1 c 4 x y where f is the collision frequency, N is the numb er of particles in each bunch, c and is the rms b eam size at the collision p oint in the x y direction. x y Similarly, the main aim of mo dern synchrotron radiation facilities is to pro duce x-ray photons with high brightness: f N b B = 2 2 2 " " x y where is the ne structure constant, f is the bunch rep etition frequency, and b " " is the rms emittance in the x y direction. If the radiation source is at x y the symmetry p oint of a straight section, as is usually the case, the emittance is given by 0 0 " = ; " = : 3 x x x y y y 0 0 where and are, resp ectively, the rms b eam size and angular x y x y divergence in the x y direction. In Eq. 2, we have neglected the radi- ation phase space area compared to the particle phase space area|a good approximation for x-ray photons. The requirement of either high luminosity for particle collision or high brightness for radiation pro duction leads to the requirement of small emit- tance. This is evident for a high-brightness radiation source in view of Eq. 2. It is also true for high luminosity b ecause of the practical limit on angular divergence to achieve a small sp ot size . x 1 Figure 1: Schematic illustration of various quantum e ects in particle and photon b eams and their interaction. Quantum mechanical e ects b ecome imp ortant in the dynamics of ex- tremely high-brightness b eams b ecause the b eam b ecomes sensitive to quantum recoils and also b ecause of p ossible wavefunction degeneracy. Various quantum e ects in the interaction of high-brightness particle and radiation b eams are schematically summarized in Fig. 1. This pap er contains discussions of some of these topics. Section 2 gives a review of basic phase-space constraints due to quantum mechanics. Section 3 is a review of a well-understo o d topic, the quantum uctuation in storage rings, from the view that the synchrotron radiation is a manifestation of the Hawking-Unruh radiation. The approach provides a fresh lo ok at the equilibrium emittance section 3.2 and p olarization in electron storage rings section 3.3. Section 4 deals with the quantum e ects in free- electron lasers FELs. A simple discussion of the gain reduction due to the quantum recoil is given in section 4.1. Quantum e ects are also imp ortant for photon statistics and start-up noise in self-ampli ed sp ontaneous radiation, which is the sub ject of section 4.2. Max Zolotorev played an imp ortant role in preparation of this pap er, es- p ecially for the material in section 3. He derived Eq. 15, which is the basis of the rest of section 3.2, and was available for extensive discussions during the development of section 3.4 and other parts of the pap er. This pap er can b e regarded as an incomplete summary of the discussions 2 that to ok place in Working Group A. Some of the interesting and imp ortant topics discussed are not included here, including the spin p olarization in elec- tron and proton b eams by D. Barb er, quantum chaos prob ed byrfcavities by F. Zimmerman, and bunch di usion in electron storage rings by J. Byrd. 2 Wave Functions and Phase-Space Distribution of Quantum Par- ticles 1 In quantum mechanics, a particle in free space b ehaves as a wave packet . The wavenumb er for 1 is given by q = ; c where is the Compton wavelength = h=mc, which, for an electron, is 3:8 c 13 10 m. The Heisenb erg uncertainty principle can b e written as n n n " =2; " =2; " =2: 4 c c c x y z n Here " are the normalized emittances: i n n n " = " ; " = " ; " = ; 5 x y z x y z where the transverse emittances are given by Eq. 3, and in longitudinal n emittance " , is the rms packet length, and is the rms value of , z 0 z where is the nominal energy. 0 Equation 4 can be interpreted as the fact that the phase-space area o ccupied by a quantum particle cannot be smaller than =2. The equality c holds for a Gaussian wave packet. There is some ambiguity in assigning the relative magnitudes of classical b eam emittance and the single-particle, quantum emittance. The ambiguity can b e removed byintro ducing the emittance based on the entropy concept, 2 as was done for the classical wave optics . The concept of phase space for a quantum mechanical particle can b e based 3 on the Wigner distribution . Although the Wigner distribution cannot b e as- signed to a real probability distribution b ecause it could b e negative lo cally,it is nevertheless useful b ecause of its similarity to the real probability distribu- tion of classical particles in phase space. Thus the transformation prop erties of the quantum phase-space distribution as the particle travels through free space and quadrup ole fo cusing elements, etc., are identical to those in classi- cal particle distribution as illustrated in Figure 2. Through a free space, the quantum phase-space distribution is tilted along the x-axis exactly as in the 3 Figure 2: Phase-space representation of a quantum particle and its b ehavior while passing through some accelerator elements. 0 case of a classical distribution. Similarly, it is tilted down the x -axis through a fo cusing element, again, exactly as in the classical distribution. A collection of quantum particles are describ ed by a collection of wave packets centered along the classical particle tra jectories as shown in Figure 3. The phase-space distribution is then a convolution of the classical b eam distribution and the quantum mechanical distribution of a single particle, as shown in Figure 4. The total phase-space area is minimal when the phase-space ellipses of the classical and the quantum distributions are similar and have the same orientation, i.e., when they are matched. Once matched, they remain matched b ecause the transformation prop erties of the classical and quantum phase space are the same, as mentioned b efore. The total phase-space volume of an N -particle b eam is limited by the statistics. For Fermions wehave n n n 3 " " " N =2 =2S +1; 6 c x y z where S is the spin. For Bosons, on the other hand, the limit is given by n n n 3 " " " =2 : 7 c x y z Equations 6 and 7 give the degeneracy limits. Close to these limits, quan- tum mechanical e ects will b ecome imp ortant. The collection of quantum mechanical wave packets discussed here is sim- ilar to the optical wave packets generated byacharged particle b eam, suchas the synchrotron radiation b eam. The synchrotron radiation b eam optics based 4 on the Wigner distribution was discussed earlier . The identity of the transformation prop erties b etween the quantum me- chanical or optical phase space and the classical phase space via the Wigner 4 Figure 3: Quantum b eam as a collection of wave packets wavy lines each of which is centered along a classical tra jectory dotted lines. Figure 4: Total phase space of a quantum b eam solid ellipse as a convolution of the single- particle quantum phase space represented by small concentric ellipses and the classical b eam phase space dotted ellipse. 5 distribution is valid for linear transp ort elements such as free space and ideal quadrup oles. In the presence of ab errations, the equivalence needs to b e mo d- 5 i ed . 3 Understanding the Quantum Fluctuations in Electron Storage Rings In electron storage rings the equilibrium b eam emittance is determined by the balance of the damping e ect due to the classical synchrotron radiation and the quantum uctuation due to the discreteness of the photon emission pro- 6;7 cess. This is a well-understo o d topic . During the workshops, the nature of Hawking-Unruh radiation was the sub ject of extensive debate. Here we adopt the p oint of view that Hawking-Unruh radiation is simply another description of the well-known synchrotron radiation. With this p oint of view, it is p ossible to develop a \simple" understanding of the equilibrium b eam parameters in storage rings. The main goal of this section is to provide a new physical understanding of well-understo o d phenomenon. Therefore we are interested in the parametric relationships and order of magnitude, and will not b e careful ab out numerical factors of 2 , etc.