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! CERN Quantum Mechanical Limits Frank Zimmermann
parameter symbol TESLA NLC CLIC bunch population [10 10 ] N b 2 0.8 0.4 DR energy [GeV] E DR 5 1.98 2.42 emittance from DR [nm] γǫ x,y, DR 8000, 20 3000, 30 620, 5 DR bunch length [mm] σ z 6 3.6 1.2 DR energy spread [10 − 3 ] σ E /E 1 0.9 1.1 DR beta [m] β y, DR 60 4 4 FF emittance [nm] γǫ x,y, FF 10000, 30 3600, 40 680, 10 IP beam energy [GeV] E ∗ 250 250 1500 IP beta [mm] β ∗ 15, 0.4 8, 0.11 6, 0.07 x,y IP spot size w/o pinch [nm] σ ∗ 554, 5.0 243, 3.0 67, 0.7 x,y free length from IP [m] l ∗ 3.0 3.5 4.3 Upsilon Υ 0.05 0.13 8.3 BS photons / e − N γ 1.56 1.26 2.32 CERN Quantum Mechanical Limits Frank Zimmermann
Diffraction Limited Spot Size (inspired by K.-J. Kim, QABP 2000, Capri) classical particle beam � � y + s 2 � σ y = ǫ y β ∗ β ∗ y light ( Z R : Rayleigh length) � Z R + s 2 λ � � σ γ y = 4 π Z R quantum particle � � y + s 2 � ¯ λ e σ QM = β ∗ y 2 γ β ∗ y QM important if γǫ y → ¯ λ e / 2 ≈ 0 . 2 pm; minimum spot size 5–30 pm for CLIC, NLC, TESLA. CERN Quantum Mechanical Limits Frank Zimmermann
Shall We Describe Beam by Wigner Function? � i � ∞ W ( x, p ) = 1 x − y x + y � � � � � −∞ exp hpy φ φ ∗ dy h ¯ ¯ 2 2 where φ ( x ) wave function; e.g., for uncorrelated Gaussian wave packet: � � ( x − x 0 ) 2 + ( p − p 0 ) 2 �� 1 − 1 W ( x, p, x 0 , p 0 ) = exp 2 πσ x σ p 2 σ 2 σ 2 x p where σ x σ p = ¯ 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) CERN Quantum Mechanical Limits Frank Zimmermann
Phase Space Density each phase space cell of dimension h 3 can accommodate 1 polarized electron → density limit: N ≤ 1 ρ ps ≡ γ 3 ǫ x ǫ y ǫ z ¯ λ 3 e parameter symbol TESLA NLC CLIC DR design density [m − 3 ] 2 × 10 23 7 × 10 23 2 × 10 25 ρ ps density limit [m − 3 ] λ 3 7 × 10 34 7 × 10 34 7 × 10 34 1 / ¯ e we are far away from the limit thanks to longitudinal emittance! CERN Quantum Mechanical Limits Frank Zimmermann
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 energy loss per unit length 3 r e γ 3 K 2 l ∗ 2 ǫ y ds ≈ 2 dδ β ∗ y photon energy squared per unit length total number of photons emitted λ e γ 5 K 3 l ∗ 3 ǫ 3 / 2 dδ ds ≈ 55 5 5 y 3 αγKl ∗ � 24 r e ¯ N ≈ 3 αγθ ≈ ǫ y /β ∗ √ √ y 3 / 2 β ∗ 2 2 y CERN Quantum Mechanical Limits Frank Zimmermann
Synchrotron Radiation in Final Quadr. Cont’d � ǫ ∗ � 5 / 2 y ǫ + 110 2 = β ∗ y λ e γ 5 F ( K Q , L q , l ∗ ) σ ∗ 6 πr e ¯ √ y β ∗ 3 y CERN Quantum Mechanical Limits Frank Zimmermann
Synchrotron Radiation in Final Quadrupole(s) Cont’d optimum choice of β ∗ y , � 275 � 2 / 7 γ ( γǫ y ) 3 / 7 , β ∗ y = 6 πr e ¯ λ e F ( K Q , L q , l ∗ ) √ 3 increases linearly with γ ! Minimum spot size � 1 / 2 � 275 � 1 / 7 � 7 ( γǫ y ) 5 / 7 σ ∗ y min = 6 πr e ¯ λ e F ( K Q , L q , l ∗ ) √ 5 3 independent of γ ! CERN Quantum Mechanical Limits Frank Zimmermann
Synchrotron Radiation in Final Quadr. Cont’d Minimum emittance γǫ y ≈ ¯ λ e / 2 → minimum Oide spot size σ ∗ y min ≈ 1 . 3 pm! ( F ≈ 5 . 4) CERN Quantum Mechanical Limits Frank Zimmermann
Similar Effect: Synchrotron Radiation in Solenoid Field with Crossing Angle Effect of SR in solenoid body computed by J. Irwin: 3 � 5 ∆ σ ∗ 2 λ e γ 5 � � = c u r e ¯ 1 = 1 c u r e ¯ λ e � B s θ c l ∗ γ � y dsR 36 ( s ) 2 � � . � � σ ∗ 2 σ ∗ 2 σ ∗ ρ ( s ) 20 2 Bρ � � y y y A larger effect can arise from the fringe field of the solenoid. Simulated vertical spot size σ ∗ y at the CLIC collision point vs. θ c (left) and vs. length of fringe field (right), considering solenoid fields of 4 and 6 T. (D. Schulte & F.Z., PAC 2001) CERN Quantum Mechanical Limits Frank Zimmermann
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 CERN Quantum Mechanical Limits Frank Zimmermann
Adiabatic Focuser (P. Chen, K. Oide, A.M. Sessler, S.S. Yu, PRL 64, 1231 (1999)) adiabatic focusing K ( s ) = 1 + α 2 0 β ( s ) 2 focusing strength increases with s as 1 /β 2 ; amplitude of lower-energy particle never exceeds that of reference particle; holds true for particles which emit radiation; energy loss in 3 regimes: Υ 2 , Υ ≤ 0 . 2 (classical) dγ ds = − 2 α × 0 . 2Υ , 0 . 2 ≤ Υ ≤ 22 (transition) 3 ¯ λ e 0 . 556Υ 2 / 3 , 22 ≤ Υ (quantum) where Υ = γ 2 ¯ λ e /ρ ; trick is to exploit the quantum regime! CERN Quantum Mechanical Limits Frank Zimmermann
Adiabatic Focuser Cont’d limitation: fractional energy loss must remain small! → maximum emittance for transition regime: γǫ trans ≤ 5 4 6¯ α 3 λ e 0 0 ) 2 ; (1 + α 2 α → maximum emittance for quantum regime: γǫ quant ≤ 15 3 α 3 ¯ λ e 0 0 ) 2 ; (1 + α 2 2 3 22 α √ use α 0 = 3; find critical emittance to enter quantum regime: γǫ c = 3 3 / 2 15 3 λ e α 3 ≈ 6 . 2 × 10 − 6 m ; 2 3 4 2 22 ¯ minimum spot size � 2 � 1 / 2 � � � 1 / 3 � 3 3 / 2 λ e ¯ λ e ( γǫ y ) 2 σ ∗ y min , AF = 11¯ exp − 3 α 3 γǫ y 16 CERN Quantum Mechanical Limits Frank Zimmermann
Adiabatic Focuser Cont’d promise of extremely small spot sizes for γǫ y = 10 nm, adiabatic focuser reaches y min , AF = 10 − 9 nm! (e − wave nature not cons.) σ ∗ CERN Quantum Mechanical Limits Frank Zimmermann
Adiabatic Focuser Cont’d a problem with this approach final gradient K and corresponding plasma density n pl ≈ γK/ (2 πr e ); ‘plasma’ may become denser than a solid! CERN Quantum Mechanical Limits Frank Zimmermann
� IP Physics: Beamstrahlung & L Spectrum Upsilon parameter � � γr e 2 N � 2¯ hω c � ≈ 5 Υ avg = � � 3 E 6 ασ z ( σ x + σ y ) � � � � � � � � � � � � � � � � � � � � � � � � � � � � Number of beamstrahlung photons/electron N γ ≈ 5 ασ z Υ αr e N (1 + Υ 2 / 3 ) 1 / 2 ≈ 2 ; 2 γ ¯ λ e σ x + σ y last approximation applies if Υ is small (Υ ≤ 1). Fraction of the luminosity at the design center-of-mass energy: γ (1 − e − N γ ) 2 ∆ L / L ≈ 1 /N 2 Average energy loss 2 N γ Υ (1 + Υ 2 / 3 ) 1 / 2 δ B ≈ 1 (1 + (1 . 5Υ) 2 / 3 ) 2 CERN Quantum Mechanical Limits Frank Zimmermann
Ultimate Luminosity L = f rep n b N 2 = 1 P wall η N b b 4 πσ ∗ x σ ∗ 4 π E b σ ∗ x σ ∗ y y where η = P beam /P wall = f rep N b n b /P wall . (1) ignore beamstrahlung & Oide effect; assume minimum emittances: γǫ x,y,z ≈ ¯ λ e / 2, β ∗ x,y ≈ σ z , and σ z ≈ ( γǫ z ) /γ/ (∆ p/p ) rms : 4 γ 2 L 1 = 1 P wall ηN b � ∆ p � 4 π E b λ 2 ¯ p e rms where, e.g., (∆ p/p ) rms may be final-focus bandwidth ( ≈ 0 . 0028) (2) more realistic: maintain constant N γ : N b /σ x ≈ const . , γǫ y,z ≈ ¯ λ e / 2, β ∗ y ≈ σ z , σ z ≈ ( γǫ z ) /γ/ (∆ p/p ) rms , and small Υ: 1 P wall ηN γ 2 γ � L 2 ≈ (∆ p/p ) rms 8 πα E b ¯ λ e CERN Quantum Mechanical Limits Frank Zimmermann
potential luminosity increase over design for the two cases: � γ � 2 ≈ 4 γ 2 L 1 σ ∗ , design H L, 1 = σ ∗ y, design (∆ p/p ) rms x λ 2 L design ¯ γ 0 e � γ � L 2 ≈ 2 γ � H L, 2 = σ ∗ (∆ p/p ) rms y, design L design ¯ λ e γ 0 luminosity should increase as γ 2 from present design → energy reach parameter symbol TESLA NLC CLIC 1 . 3 × 10 18 3 . 3 × 10 17 7 . 7 × 10 17 case (1) for γ = γ 0 H L, 1 energy reach ∞ ∞ ∞ 1 . 1 × 10 8 6 . 4 × 10 7 (8 . 9 × 10 7 ) case (2) for γ = γ 0 H L, 2 2 . 7 × 10 19 1 . 6 × 10 19 (1 . 3 × 10 20 ) energy reach [GeV] we can reach the Planck scale (1 . 2 × 10 19 GeV)! (J. Irwin, private communication, 1996). CERN Quantum Mechanical Limits Frank Zimmermann
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 ∝ γ 2 ! • but synchrotron radiation in final quad.’s (Oide effect) & beamstrahlung at IP constrain already present designs • new ideas and approaches are welcome! CERN Quantum Mechanical Limits Frank Zimmermann
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