LSLLS­271

Presented at the 15th ICFA Advanced Beam Dynamics Workshop

on Quantum Aspects of Beam Physics, January 4­9, 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. in this section.

3.1 The Hawking-Unruh Picture

8

Following up on the celebrated observation byHawking that a black hole is

9

also a black body radiator, Unruh found that an accelerated particle nds

itself b eing in contact with a heat bath of temp erature T given by

ha

k T = ; 8

B

2c

where k is the Boltzman constant, h is the Planck constant, a is the acceler-

B

ation, and c is the sp eed of light.

For an electron moving on a circular tra jectory of radius  in a magnetic

2 2

eld, the acceleration is a = c =. Therefore, it sees a black body dis-

0

tribution of photons with characteristic frequency ! the prime represents

c

quantities in the instantaneous rest frame given by

2

h c

0

: 9 ' k T = h!

B

c

2

These photons, which app ear to b e real to the accelerated electron, are clearly

not real in the lab oratory frame. However, they can b e scattered by the elec-

tron to b ecome real photons in the lab oratory frame with the well-known 6

Figure 5: The Hawking-Unruh picture of synchrotron radiation. An accelerating charge nds

itself surrounded by thermal photons, which can b e scattered by the electron to app ear as

the real photons of synchrotron radiation.

3 0

' h c=, with synchrotron radiation characteristic frequency of h! ' h !

c

c

1

acharacteristic angular op ening . This is illustrated in Fig. 5.

To develop the picture further, let us consider the radiated power. In

a black body distribution, there is on average ab out one photon per mo de

03 0

o ccupying a mo de volume  , where  is the wavelength. Thus, the scattered

power is

0

h!

c

P '  c; 10

s

03



where  is the cross section. On the other hand, the synchrotron radiation

power is given by Larmor's formula,

2

2 2 a

2 02

P = = e h! : 11

s

c

3

3 c 3

Equations 10 and 11 b ecome equal if

02

 =  : 12

This result may surprise some readers. For atoms with b ound electrons,

02

it is well known that the scattering cross section at resonance is  . The case

0

of Hawking radiation by a black hole is also similar, where  is give by r ,

g

the horizon radius. Equation 12 can b e interpreted as the statement that a

free electron can only \hold" photons p er mo de, rather than one photon p er

mo de in the atomic case.

We emphasize that the Hawking-Unruh picture develop ed in the ab oveisa

particular picture for understanding the synchrotron radiation pro cess from a

di erent p ersp ective. There are other p erhaps more familiar pictures. Thus the

synchrotron radiation pro cess can b e viewed as the \shedding" of Weiszacker-

Williams photons as the electron b ends in the magnetic eld. It can also b e

viewed as the scattering of virtual photons contained in the b ending magnetic 7

eld. In this case, however, the scattering cross section should b e taken as a

Thomson cross section.

The Hawking-Unruh picture is well suited for a simple understanding of

equilibrium electron b eam parameters in storage rings, as we will discuss now.

3.2 Simple Understanding of Equilibrium Electron Distribution in Storage

Rings

Consider rst the equilibrium energy distribution. Since the accelerating elec-

trons are in equilibrium with a heat bath with temp erature given by Eq. 8,

the equilibrium kinetic energy in the b eam frame must b e

 

02

p 3

0 2

h
E i = = k T ' h c=: 13

B

2m 2

0

On the other hand, the energy spread in the lab oratory frame
E is related

0

to the energy spread
E in the b eam rest frame via

   

2

0

E
E

: 14 =2

2

E mc

Inserting Eq. 14 into Eq. 13, one obtains

 

2

2

E

'  ; 15

c

E 

7

repro ducing the well-known result up to a factor of order unity. Equation

15 was derived by M. Zolotorev during the workshop.

Next, consider the uctuation of the b etatron oscillation. In the y -direction

vertical to the orbit plane the electron angular divergence is given by

   

2

2

p

2m 2m k T p

y B y

2

= = : 16 h i =

2 2

p p 2m p 2

z

z z

Therefore, the equilibrium normalized emittance b ecomes



2 n

=  h i' " ; 17

c

y



where  is the b etatron wavelength divided by2 . This is another well-known

result. 8

Equation 17 is valid b ecause the disp ersion vanishes in the vertical di-

rection. In the x-direction the horizontal direction, however, the disp ersion

e ect dominates. In that case we use Eq. 15,

 

2

2

E

2 2 2

=  ; 18 hx i =

c

E 

2

where is the horizontal disp ersion = =. Therefore

3

1 

n 2

" = hx i' : 19

c

x

 

Again this agrees qualitatively with the well-known result.

3.3 Suppression of Quantum Fluctuation

For most electron storage rings the quantity = is much larger than unity.

This has two consequences: rst, the horizontal emittance given by Eq. 19

is much larger than the vertical emittance given by Eq. 17, and second, the

vertical emittance is much larger than  .

c

However, if one considers an ideal limit  ! 1, with and  xed,

then the equilibrium emittance would vanish according to Eqs. 17 and 19.

In this straight focusing channel the emittance do es not vanish strictly but

is limited by the ultimate emittance given by Eqs. 6 or 7 according to

whether the particles are Fermions or Bosons. This case has b een analyzed

10

using quantum mechanics . However, the straight fo cusing channel app ears

to b e dicult to implement b ecause the required length of the channel is to o

long to b e practical.

On the other hand, it is not necessary to have a straightchannel. All that

is necessary to achieve the ultimate emittance is

=

 1: 20 =



11

Huang et al. have carried out a quantum mechanical analysis of the so-

called bent focusing system and found that the quantum uctuation b ecomes

exp onentially suppressed as b ecomes larger than 1, leading to the ultimate

emittance. It remains to b e seen whether a practical storage ring design will

emerge from this idea. 9

3.4 Understanding the Equilibrium Polarization in Electron Storage Rings

An elementary particle has the magnetic dip ole moment

eh

 = g S; 21

2mc

where S is the spin vector and g is a dimensionless constant that is very close

to 2 for electrons. Thus the two spin states of an electron, S = 1=2, will split

z

0

in energy under the external magnetic eld B = B e . The level spacing is

o z

o

given by

eh

0

B ' g h! : 22
E = g

o

c

2mc

Electrons in the upp er state would make transitions to the lower state. The

electrons from this argument are exp ected to b ecome 100 p olarized after a

sucient time.

This is of course not completely true. It is known that the maximum

12

degree of p olarization in an electron storage ring is given by

8

p

P = =0:924: 23

5 3

A lucid discussion of the success as well as the failure of the simple mo del based

on spin splitting and a discussion of a more accurate semiclassical calculation

13

can b e found in Jackson .

The fact that the degree of p olarization is not 100 can be understo o d

14

easily by the Hawking-Unruh picture . The limiting p olarization in this case

would b e

g
E=k T

B

1 e 1 e

' P = : 24

g

E=k T

B

1+e

1+e

Although Eq. 24 predicts numerically a higher degree of p olarization than

Eq. 23, it nevertheless provides qualitative understanding on why the p olar-

ization is not pure.

It is p erhaps instructive to see in more detail how the p olarization evolves.

Figure 6 shows the two electron energy levels under an external DC magnetic

eld B . The spin ip is caused by the black b o dy photons that are p olarized

0

so that the oscillating magnetic eld is p erp endicular to B . If the photon

0

energy is resonant with the level spacing,h!  =
E , spin ip o ccurs due to the

same phenomena as in nuclear magnetic resonance. In the presence of average

photon number hni = exph! =k T , the up-transition rate is prop ortional to

B

hni, while the down-transition rate is prop ortional to hni +1. The equilibrium 10

Figure 6: Illustration of electron energy levels in the external magnetic eld and the spin

ip due to the p erturbation by the oscillating magnetic eld B p erp endicular to B  from

o

the apparent thermal photons.

p olarization is therefore

hni +1hni 1 1 hni

P = = ' : 25

1+2hni 1+2hni 1+hni

This reduces to Eq. 24.

4 Quantum E ects on Free-Electron Lasers

It is well known that the quantum mechanical correction on the classical gain

formula b ecomes imp ortant when the photon energy is comparable to, or larger

than, the gain bandwidth. A less explored topic is the role of quantum mechan-

ics in the statistics of FEL photons, esp ecially the self-ampli ed sp ontaneous

emission.

4.1 Quantum Recoil and FEL Gain

The pro cesses of emission and absoption of a photon in the presence of n

photons are schematically illustrated in Fig. 7. The sp ontaneous emission

probability as a function of frequency ! is p eaked at the resonance frequency

! and can b e written as W ! ! . The total emission probability W is

R s R e

15

given by

W =n +1W ! ! : 26

e s e

Here the term prop ortional to n is due to the induced emission, and ! is the

e

resonant emission frequency taking into account the recoil e ect:

h!

0

! = ! 1 ; 27

e 0

E

e 11

Figure 7: Photon emission a and absorption b in the presense of n-photons.

where ! is the resonance frequency neglecting the recoil e ect, and E is the

0 e

electron energy. Similarly, the total absorption probabilityis

W = nW ! ! ; 28

a s a

where

h!

0

! = ! 1+ 29

a 0

E

e

is the resonant absorption frequency.

Thus the net emission probabilityis

W = W W = W ! ! +n[W ! !  W ! ! ]: 30

e a s e s e s a

Neglecting the rst term in the RHS of Eq. 30 the sp ontaneous emission

term for the case n  1, the gain is given by

W

= W ! !  W ! ! : 31 G =

s e s a

n

Consider now the case of the stimulated emission pro cess in an undulator,

as in the case of the usual free-electron laser. The bandwidth of the sp onta-

neous undulator emission is

1
!

 ; 32

! N

0 u

where N is the numb er of undulator p erio ds. Therefore, when

u

h!

0

 1; 33

E =N

e u 12

the gain can b e expressed as a frequency-derivative of the sp ontaneous emission

16

sp ectrum. This form of the classical gain is known as Madey's theorem . On

the other hand, when

h!

0

 1; 34

E =N

e u

then the quantum recoil is signi cant, and the gain is reduced from the classical

17;18

case .

From the inequalityofEq. 34, one sees that the quantum recoil is im-

p ortant for short wavelength FELs based on low-energy electron b eam, suchas

the x-ray FEL using extreme high-brightness e-b eam based on eld emission

19

from microtips .

4.2 Self-Ampli ed-Spontaneous Emission

The self-ampli ed sp ontaneous emission SASE is receiving much attention

recently as a promising next generation light source for intense, quasi-coherent

x-rays. Classical analysis of SASE has b een extensively develop ed in the areas

20 21;22

of exp onential growth and start-up from the electron shot noise . Under

a certain set of assumptions, including Eq. 33, it was shown that the evolu-

tion of the quantum Heisenb erg op erator closely parallels that of the classical

23

case . Thus, the eld amplitude op erator is given by

a =f  a0 + f  0 + f  P 0; 35

1 2 3

where  is the normalized time, f s are functions that grow exp onentially in  ,

and

X X

i i

i i

e = p e ; P = : 36

i

i i

In the ab ove, the sum is over all the electrons, and and p are, resp ectively,

i i

th

the phase and momentum op erators for the i electron [ ;p ]=i.

i i

The rst terms gives rise to the coherent state with Glaub er statistics.

The second and third terms are due to the electrons' shot noise.

When

3

n n n

" " "  N  =2 =2; 37

c

x y z

then  and P can b e regarded as classical sto chastic variables. In this case,

the evolution and statistics of the SASE eld are those given by the classical

analysis. On the other hand, if

3

n n n

 N  =2 =2; 38 " " "

c

z y x 13

then the degeneracy of the quantum wave function needs to be taken into

account. Thus the wavefunction of the N -Fermion system must b e written as

P

X

1

p

j i = j z ij z i::: j z i; 39

1 2 n

N !

P

where  z  is the one-particle wavefunction, and the sum is over all p ermu-

P

tation of fz ;:::;z g for Bosons, 1 ! 1.

1 n

A rigorous analysis of SASE, taking into account the wave function degen-

eracy app ears quite complicated. However, an understanding of the e ect can

be seen as follows: The e ective start-up noise of SASE intensity is prop or-

tional to

N

X

2

ik z

j

ik z z 

1 2

 je j i: 40 j i = N + N N +1h h j e

j =1

The rst term in the ab ove corresp onds to the classical noise. In the sec-

ond term, the bar denotes averaging over particles. When the inequality Eq.

37 is valid, the second term can b e calculated with classical averaging and is

negligible when the bunch length is longer than the radiation wavelength. On

the other hand, when the inequality Eq. 38 is valid, a prop er quantum cal-

culation with suitably anti-symmetrized wavefunction, Eq. 39, is necessary.

In the simple case N = 2, this can b e carried out explicitly, with the result

2 2 n

 k " =

c

z z

e  e

ik z z 

1 2

: 41 h je j i'

n

= "

c

z

1  e

Here the upp er lower sign corresp onds to the Fermion Boson case. This

n n

reduces to the classical result when    . For    , the noise is the

c c

z z

reduced increased in the case of the Ferimionic Bosonic system.

Note addedinproof.|After the manuscript was completed, it came to the

author's attention by Z. Huang that results equivalent to Eqs. 15, 17, and

19 were derived in 1997 by K. McDonald \The Hawking-Unruh Temp era-

ture and Quantum Fluctuations in Particle Accelerators," pro ceedings of 1997

Particle Accelerator Conference. Also, in 1986 J.S. Bell and J. M. Leinaas

\The Unruh E ect and Quantum Fluctuations of Electrons in Storage Rings,"

CERN TH 4468-86 derived a result equivalent to Eq. 17 for a weakly fo cused

storage ring.

Acknowledgments

I am grateful to the memb ers of Working Group A of the Workshop on Quan-

tum Asp ects of Beam Physics held in Monterey, CA, January 4-9, 1998; to 14

Max Zolotorev for many discussions and, in particular, for his contribution to

the materials in section 3; to Pisin Chen, without whose insistence this pap er

would not have b een written; and to Z. Huang for p ointing out b oth the spin

factor in the RHS of Eq. 6 and MacDonald's work.

This work is supp orted by the U.S. Department of Energy, Oce of Basic

Energy Sciences, under Contract No. W-31-109-ENG-38.

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