Quantum Mechanical Limits on Beam Demagnification & Luminosity - - PowerPoint PPT Presentation

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Quantum Mechanical Limits on Beam Demagnification & Luminosity

• F. Zimmermann
• minimum spot size
• phase-space density
• synchrotron radiation: Oide limit
• beamstrahlung
• ultimate luminosity

Thanks to Ralph Assmann, John Jowett, Francesco Ruggiero!

Frank Zimmermann Quantum Mechanical Limits

CERN

parameter symbol TESLA NLC CLIC bunch population [1010] Nb 2 0.8 0.4 DR energy [GeV] EDR 5 1.98 2.42 emittance from DR [nm] γǫx,y,DR 8000, 20 3000, 30 620, 5 DR bunch length [mm] σz 6 3.6 1.2 DR energy spread [10−3] σE/E 1 0.9 1.1 DR beta [m] βy,DR 60 4 4 FF emittance [nm] γǫx,y,FF 10000, 30 3600, 40 680, 10 IP beam energy [GeV] E∗ 250 250 1500 IP beta [mm] β∗

x,y

15, 0.4 8, 0.11 6, 0.07 IP spot size w/o pinch [nm] σ∗

x,y

554, 5.0 243, 3.0 67, 0.7 free length from IP [m] l∗ 3.0 3.5 4.3 Upsilon Υ 0.05 0.13 8.3 BS photons / e− Nγ 1.56 1.26 2.32

Frank Zimmermann Quantum Mechanical Limits

CERN

Diffraction Limited Spot Size

(inspired by K.-J. Kim, QABP 2000, Capri) classical particle beam σy =

• ǫy
• β∗

y + s2

β∗

y

• light (ZR: Rayleigh length)

σγ

y =

• λ

4π

• ZR + s2

ZR

• quantum particle

σQM

y

=

• ¯

λe 2γ

• β∗

y + s2

β∗

y

• QM important if γǫy → ¯

λe/2 ≈ 0.2 pm; minimum spot size 5–30 pm for CLIC, NLC, TESLA.

Frank Zimmermann Quantum Mechanical Limits

CERN

Shall We Describe Beam by Wigner Function?

W(x, p) = 1 ¯ h

∞

−∞ exp

i

¯ hpy

• φ
• x − y

2

• φ∗
• x + y

2

• dy

where φ(x) wave function; e.g., for uncorrelated Gaussian wave packet: W(x, p, x0, p0) = 1 2πσxσp exp

• −1

2

(x − x0)2

σ2

x

+ (p − p0)2 σ2

p

• where σxσp = ¯

h/2; W corresponds to particle density, but, for general φ, W is not a positive function! temporal evolution of W(x, p, t): Wigner-Moyal equation ≡ Liouville equation (for quadratic potentials)

Frank Zimmermann Quantum Mechanical Limits

CERN

Phase Space Density

each phase space cell of dimension h3 can accommodate 1 polarized electron → density limit: ρps ≡ N γ3ǫxǫyǫz ≤ 1 ¯ λ3

e

parameter symbol TESLA NLC CLIC DR design density [m−3] ρps 2 × 1023 7 × 1023 2 × 1025 density limit [m−3] 1/¯ λ3

e

7 × 1034 7 × 1034 7 × 1034

we are far away from the limit thanks to longitudinal emittance!

Frank Zimmermann Quantum Mechanical Limits

CERN

Synchrotron Radiation in Final Quadrupole(s)

(K. Oide, PRL 61, 1713 (1988))

particles emit synchrotron radiation photons, lose energy, acquire different focal lengths → spot-size blow up

total number of photons emitted N ≈ 5 2 √ 3αγθ ≈ 5 2 √ 3αγKl∗ ǫy/β∗

y

energy loss per unit length dδ ds ≈ 2 3reγ3 K2l∗2ǫy β∗

y

photon energy squared per unit length dδ ds ≈ 55 24re¯ λeγ5 K3l∗3ǫ3/2

y

β∗

y 3/2

Frank Zimmermann Quantum Mechanical Limits

CERN

Synchrotron Radiation in Final Quadr. Cont’d

σ∗

y 2 = β∗ yǫ + 110

3 √ 6πre¯ λeγ5F(KQ, Lq, l∗)

ǫ∗

y

β∗

y

5/2

Frank Zimmermann Quantum Mechanical Limits

CERN

Synchrotron Radiation in Final Quadrupole(s) Cont’d

• ptimum choice of β∗

y,

β∗

y =

275

3 √ 6πre¯ λeF(KQ, Lq, l∗)

2/7

γ(γǫy)3/7 , increases linearly with γ! Minimum spot size σ∗

ymin =

7

5

1/2 275

3 √ 6πre¯ λeF(KQ, Lq, l∗)

1/7

(γǫy)5/7 independent of γ!

Frank Zimmermann Quantum Mechanical Limits

CERN

Synchrotron Radiation in Final Quadr. Cont’d Minimum emittance γǫy ≈ ¯ λe/2 → minimum Oide spot size σ∗

ymin ≈ 1.3 pm! (F ≈ 5.4)

Frank Zimmermann Quantum Mechanical Limits

CERN

Similar Effect: Synchrotron Radiation in Solenoid Field with Crossing Angle

Effect of SR in solenoid body computed by J. Irwin: ∆σ∗ 2

y

σ∗

y

= cure¯ λeγ5 σ∗ 2

y

• dsR36(s)2
• 1

ρ(s)

• 3

= 1 20 cure¯ λe σ∗ 2

y

Bsθcl∗γ 2Bρ 5 . A larger effect can arise from the fringe field of the solenoid. Simulated vertical spot size σ∗

y at the CLIC collision point vs. θc

(left) and vs. length of fringe field (right), considering solenoid fields

• f 4 and 6 T. (D. Schulte & F.Z., PAC 2001)

Frank Zimmermann Quantum Mechanical Limits

CERN

Oide Limit - Possible Remedies

• photon statistics → rms spot size overestimates the

luminosity loss (Hirata, Oide, Zotter, Phys. Lett. B 224, 437 (1989))

• adjust strength of quadrupole to compensate for

average energy loss

• install octupoles near final quadr. to compensate

average energy loss at various amplitudes (P. Raimondi)

• suppress the radiation with ρ/γ ≫ β (?)
• suppress the radiation by using ‘ultra-dense’ beam (?)
• strong adiabatic focusing

Frank Zimmermann Quantum Mechanical Limits

CERN

Adiabatic Focuser

(P. Chen, K. Oide, A.M. Sessler, S.S. Yu, PRL 64, 1231 (1999)) adiabatic focusing K(s) = 1 + α2 β(s)2 focusing strength increases with s as 1/β2; amplitude of lower-energy particle never exceeds that of reference particle; holds true for particles which emit radiation; energy loss in 3 regimes: dγ ds = −2 3 α ¯ λe ×        Υ2, Υ ≤ 0.2 (classical) 0.2Υ, 0.2 ≤ Υ ≤ 22 (transition) 0.556Υ2/3, 22 ≤ Υ (quantum) where Υ = γ2¯ λe/ρ; trick is to exploit the quantum regime!

Frank Zimmermann Quantum Mechanical Limits

CERN

Adiabatic Focuser Cont’d

limitation: fractional energy loss must remain small! → maximum emittance for transition regime: γǫtrans ≤ 546¯ λe α α3 (1 + α2

0)2 ;

→ maximum emittance for quantum regime: γǫquant ≤ 153 2322 ¯ λe α α3 (1 + α2

0)2 ;

use α0 = √ 3; find critical emittance to enter quantum regime: γǫc = 33/2153 234222 ¯ λeα3 ≈ 6.2 × 10−6 m ; minimum spot size σ∗

ymin,AF =

2 11¯ λe(γǫy)2 1/3 exp

• −3

33/2 16 ¯ λe α3γǫy 1/2

Frank Zimmermann Quantum Mechanical Limits

CERN

Adiabatic Focuser Cont’d

promise of extremely small spot sizes

for γǫy = 10 nm, adiabatic focuser reaches σ∗

ymin,AF = 10−9 nm! (e− wave nature not cons.)

Frank Zimmermann Quantum Mechanical Limits

CERN

Adiabatic Focuser Cont’d

a problem with this approach

final gradient K and corresponding plasma density npl ≈ γK/(2πre); ‘plasma’ may become denser than a solid!

Frank Zimmermann Quantum Mechanical Limits

CERN

IP Physics: Beamstrahlung & L Spectrum

Upsilon parameter Υavg = 2¯ hωc 3E

• ≈ 5

6 γre2N ασz(σx + σy)

• Number of beamstrahlung photons/electron

Nγ ≈ 5ασz 2γ¯ λe Υ (1 + Υ2/3)1/2 ≈ 2 αreN σx + σy ; last approximation applies if Υ is small (Υ ≤ 1). Fraction of the luminosity at the design center-of-mass energy: ∆L/L ≈ 1/N 2

γ(1 − e−Nγ)2

Average energy loss δB ≈ 1 2NγΥ (1 + Υ2/3)1/2 (1 + (1.5Υ)2/3)2

Frank Zimmermann Quantum Mechanical Limits

CERN

Ultimate Luminosity

L = frepnbN 2

b

4πσ∗

xσ∗ y

= 1 4π Pwallη Eb Nb σ∗

xσ∗ y

where η = Pbeam/Pwall = frepNbnb/Pwall. (1) ignore beamstrahlung & Oide effect; assume minimum emittances: γǫx,y,z ≈ ¯ λe/2, β∗

x,y ≈ σz, and σz ≈ (γǫz)/γ/(∆p/p)rms:

L1 = 1 4π PwallηNb Eb 4γ2 ¯ λ2

e

∆p p

• rms

where, e.g., (∆p/p)rms may be final-focus bandwidth (≈ 0.0028) (2) more realistic: maintain constant Nγ: Nb/σx ≈ const., γǫy,z ≈ ¯ λe/2, β∗

y ≈ σz, σz ≈ (γǫz)/γ/(∆p/p)rms, and small Υ:

L2 ≈ 1 8πα PwallηNγ Eb 2γ ¯ λe

• (∆p/p)rms

Frank Zimmermann Quantum Mechanical Limits

CERN

potential luminosity increase over design for the two cases: HL,1 = L1 Ldesign ≈ 4γ2 ¯ λ2

e

σ∗,design

x

σ∗

y,design(∆p/p)rms

γ γ0 2 HL,2 = L2 Ldesign ≈ 2γ ¯ λe σ∗

y,design

• (∆p/p)rms

γ γ0

• luminosity should increase as γ2 from present design → energy reach

parameter symbol TESLA NLC CLIC case (1) for γ = γ0 HL,1 1.3 × 1018 3.3 × 1017 7.7 × 1017 energy reach ∞ ∞ ∞ case (2) for γ = γ0 HL,2 1.1 × 108 6.4 × 107 (8.9 × 107) energy reach [GeV] 2.7 × 1019 1.6 × 1019 (1.3 × 1020) we can reach the Planck scale (1.2 × 1019 GeV)! (J. Irwin, private communication, 1996).

Frank Zimmermann Quantum Mechanical Limits

CERN

Conclusions

• quantum nature of electrons allows focusing

the spot sizes down to 1 pm or less (Picobeam workshop?)

• from fundamental principles, even the Planck

scale can be reached with L ∝ γ2!

• but synchrotron radiation in final quad.’s

(Oide effect) & beamstrahlung at IP constrain already present designs

• new ideas and approaches are welcome!

Frank Zimmermann Quantum Mechanical Limits