ENERGY, SCATTERING LENGTH, AND MASS DEPENDENCE

Transcription

1 ENERGY, SCATTERING LENGTH, AND MASS DEPENDENCE OF ULTRACOLD THREE-BODY COLLISIONS J.P. D Incao and B.D. Esry Department of Physics Kansas State University

2 Why three-body collisions?

3 Three-body processes Three-Body Recombination (K 3 ) X + X + X X 2 + X Collision Induced Dissociation (D 3 ) X 2 + X X + X + X Vibrational Relaxation (V rel ) X 2 + X X 2 + X

4 Universality Three-body physics becomes universal for a r 0 and E 1 2µa 2 Rate Non-Universal Features Universal Features K 3 (cm 6 /s) a<0 K 3 (cm 6 /s) a> a (a.u.) nK 10nK 1µK 100nK E 10µK µK B+B+B B 2 +B E 1µK 100nK 10nK 1nK a (a.u.)

5 What was known (m 1 =m 2 =m 3 ) Gross scaling, valid for E 1 2µa 2 Near threshold scaling for all s-wave systems... V rel K 3 (D 3 ) J π E a > 0 a < 0 E a > 0 a < 0 BBB 0 + const a? const (k 4 ) a 4 a 4 1 k 2?? k 6 (k 10 )?? 2 + k 4?? k 4 (k 8 ) a 8? BBB 0 + const?? const (k 4 )?? 1 k 2?? k 2 (k 6 )?? F F F 0 + const a 3.332? k 4 (k 8 )?? 1 k 2?? k 2 (k 6 ) a 6? 2 + k 4?? k 4 (k 8 )?? Bedaque, Braaten, Efimov, Esry, Fedichev, Greene, Hammer, Macek, Nielsen, Petrov, Shylapnikov

6 Scattering length scaling (m 1 =m 2 =m 3 ) Gross scaling, valid for E 1 2µa 2 Near threshold scaling for all s-wave systems... V rel K 3 (D 3 ) J π E a > 0 a < 0 E a > 0 a < 0 BBB 0 + const a const const (k 4 ) a 4 a 4 1 k 2 a const k 6 (k 10 ) a 10 a k 4 a const k 4 (k 8 ) a 8 a BBB 0 + const a const const (k 4 ) a 4 a 4 1 k 2 a const k 2 (k 6 ) a 6 a F F F 0 + const a const k 4 (k 8 ) a 8 a k 2 a const k 2 (k 6 ) a 6 a k 4 a const k 4 (k 8 ) a 8 a D Incao and Esry, PRL (2005)

7 Adiabatic hyperspherical representation He+ 4 He+ 4 He R 2 r r r31 2 Adiabatic hyperspherical treats R as slowly-varying (think Born-Oppenheimer) W ν (R) (10 5 a.u.) He He λ=0 λ=4 λ=6 λ= R (a.u.) In Out [ 1 2µ Hyperradial Schrödinger Equation ] F ν (R) + V νν (R)F ν (R) = EF ν (R) ν ν d 2 dr 2 + W ν(r)

8 Adiabatic hyperspherical representation He+ 4 He+ 4 He R 2 r r r31 2 Adiabatic hyperspherical treats R as slowly-varying (think Born-Oppenheimer) W ν (R) (10 5 a.u.) He He λ=0 λ=4 λ=6 λ= R (a.u.) In Out [ 1 2µ Hyperradial Schrödinger Equation ] F ν (R) + V νν (R)F ν (R) = EF ν (R) ν ν d 2 dr 2 + W ν(r)

9 Schematic potentials R R < r 0 r 0 < R < a a < R (Not Universal) (Universal) (Universal) Type I (attractive) Short Range Details s /4 or s2 ν 1/4 l(l + 1) E vl + λ(λ + 4) + 15/4 or Type II (repulsive) Short Range Details + p2 0 1/4 or p2 ν 1/4 l(l + 1) E vl + λ(λ + 4) + 15/4 or Efimov Physics Free particles Constants depend on symmetry and number of resonant interactions

10 Schematic potentials R R < r 0 r 0 < R < a a < R (Not Universal) (Universal) (Universal) Type I (attractive) Short Range Details s /4 or s2 ν 1/4 l(l + 1) E vl + λ(λ + 4) + 15/4 or Type II (repulsive) Short Range Details + p2 0 1/4 or p2 ν 1/4 l(l + 1) E vl + λ(λ + 4) + 15/4 or Efimov Physics Free particles Constants depend on symmetry and number of resonant interactions

11 Schematic potentials R R < r 0 r 0 < R < a a < R (Not Universal) (Universal) (Universal) Type I (attractive) Short Range Details s /4 or s2 ν 1/4 l(l + 1) E vl + λ(λ + 4) + 15/4 or Type II (repulsive) Short Range Details + p2 0 1/4 or p2 ν 1/4 l(l + 1) E vl + λ(λ + 4) + 15/4 or Efimov Physics Free particles Constants depend on symmetry and number of resonant interactions

12 Schematic potentials: Type I (attractive) Includes: 0 + BBB, 0 + BBB, 0 + BB B /F F F (2 resonant interactions) V αγ V βγ α s 2 ν 1/4 a>0 λ(λ+4)+15/4 V αγ V βγ a<0 λ(λ+4)+15/4 V αβ β s2 0 1/4 V αβ E vl' + l (l+1) V αβ α s2 0 1/4 γ E vl' + l (l+1) β γ E vl' + l (l+1) r 0 a r 0 a R (a.u.) R (a.u.)

13 Schematic potentials: Type II (repulsive) Includes: 0 + F F F, higher partial waves V αγ V βγ β α p 2 ν 1/4 p 2 0 1/4 a>0 λ(λ+4)+15/4 V αγ V βγ α p 2 0 1/4 a<0 λ(λ+4)+15/4 V αβ E vl' + l (l+1) V αβ V αβ γ E vl' + l (l+1) β γ E vl' + l (l+1) r 0 a r 0 a R (a.u.) R (a.u.)

14 Getting rates from the potentials We want: K 3 T fi 2 k and V 4 rel T fi 2 k Inelastic transitions proceed via tunneling to position of peak coupling U ν (R) X+X+X X 2 +X P 2 /(2µ U) Use WKB: R [ T fi 2 exp 2 ( 2µ W ν (R) + 1/4 ) ] E dr

15 Getting rates from the potentials We want: K 3 T fi 2 k and V 4 rel T fi 2 k Inelastic transitions proceed via tunneling to position of peak coupling U ν (R) X+X+X X 2 +X P 2 /(2µ U) Use WKB: R [ T fi 2 exp 2 ( 2µ W ν (R) + 1/4 ) ] E dr

16 Approximate rates: pathways Type I: Recombination Pathways (K 3 ) BBB, BBB α a>0 I II r 0 a (α) X+X+X (β) X 2 +X (α) X+X+X (β) X 2 +X IV (α) X+X+X (γ) X 2 +X V αγ V βγ β X+X+X X 2 +X III (α) X+X+X (β) X 2 +X V (α) X+X+X (γ) X 2 +X V αβ V αβ Path Probability γ I+II k 2λ a 2λ+4 sin 2 [s 0 ln(a/r 0 )+Φ] X 2 +X III k 2λ a 2λ+4 ( r 0a ) 2sν r 0 R (a.u.) a IV k 2λ a 2λ+4 V k 2λ a 2λ+4 ( r 0a ) 2sν

17 Type I rates Includes: 0 + BBB, 0 + BBB, 0 + BB B /F F F (2 resonant interactions) Recombination [ a > 0: K 3 k 2λ a 2λ+4 A η sin 2 ( [s 0 ln(a/r 0 ) + Φ]+ B r0a ) ] 2sν+ η Cη [ ] a < 0: K 3 k 2λ a 2λ+4 sinh(2η) sin 2 [s 0 ln( a /r 0 )+Φ]+sinh 2 (η) Vibrational Relaxation [ ] a > 0: V rel k 2l a 2l+1 sinh(2η) sin 2 [s 0 ln( a /r 0 )+Φ]+sinh 2 (η) a < 0: V rel k 2l r 2l+1 0

18 Type II rates Includes: 0 + F F F, higher partial waves Recombination [ ( a > 0: K 3 k 2λ a 2λ+4 A η +B r0a ) 2p0+Cη ( r0a ) 2pν+Dη ( r0a ) ] 2p0 +2p ν η a < 0: K 3 k 2λ a 2λ+4 ( r0 a ) 2p0 Vibrational Relaxation a > 0: V rel k 2l a 2l+1 ( r 0a ) 2p0 a < 0: V rel k 2l r 2l+1 0

19 Unequal masses (m 1 =m 2 m 3 ) Only three things change for BBX and F F X systems: (1) Allowed values of l and λ (symmetry) (2) s 0, p 0, s ν, and p ν (3) Efimov results now for r 0 R a δ a δ = a [ δ(δ + 2) δ + 1 ] 1 2 δ = m X m B,F

20 Unequal masses: rates V rel K 3 (D 3 ) J π E a > 0 E a > 0 a < 0 BBX 0 + const a const(k 4 ) a 4 a 4 1 k 2 a 3 2p 0 k 2 (k 6 ) a 6 a 6 2p k 4 a 5,a 5 2p 0 k 4 (k 8 ) a 8,a 8 a 8, a 8 2p 0 F F X 0 + const a 1 2p 0 k 4 (k 8 ) a 8 a 8 2p 0 1 k 2 a 3,a 3 2p 0 k 2 (k 6 ) a 6,a 6 a 6, a 6 2p k 4 a 5 2p 0 k 4 (k 8 ) a 8 a 8 2p 0 Only feature spacing affected (s 0 )

21 Unequal masses: rates V rel K 3 (D 3 ) J π E a > 0 E a > 0 a < 0 BBX 0 + const a const(k 4 ) a 4 a 4 1 k 2 a 3 2p 0 k 2 (k 6 ) a 6 a 6 2p k 4 a 5,a 5 2p 0 k 4 (k 8 ) a 8,a 8 a 8, a 8 2p 0 F F X 0 + const a 1 2p 0 k 4 (k 8 ) a 8 a 8 2p 0 1 k 2 a 3,a 3 2p 0 k 2 (k 6 ) a 6,a 6 a 6, a 6 2p k 4 a 5 2p 0 k 4 (k 8 ) a 8 a 8 2p 0 Only relaxation constant at threshold

22 Unequal masses: rates V rel K 3 (D 3 ) J π E a > 0 E a > 0 a < 0 BBX 0 + const a const(k 4 ) a 4 a 4 1 k 2 a 3 2p 0 k 2 (k 6 ) a 6 a 6 2p k 4 a 5,a 5 2p 0 k 4 (k 8 ) a 8,a 8 a 8, a 8 2p 0 F F X 0 + const a 1 2p 0 k 4 (k 8 ) a 8 a 8 2p 0 1 k 2 a 3,a 3 2p 0 k 2 (k 6 ) a 6,a 6 a 6, a 6 2p k 4 a 5 2p 0 k 4 (k 8 ) a 8 a 8 2p 0 6 (a) BBX 6 (b) FFX 4 J π = 1 - J π = J π = 0 + p 0 J π = p 0 p 0 2 p 0 p s 0 δ c -s 0 J π = s 0 δ c J π = δ = m X /m B δ = m X /m F

23 Long-lived BF molecules B-F V BF +F rel 133 Cs- 6 Li a Rb- 6 Li a Na- 6 Li a 3.05 But Cs- 40 K a Rb- 40 K a 3.12 V BF +B rel a sin 2 [s 0 ln( a /r 0 ) + Φ] 7 Li- 6 Li a Na- 40 K a Li- 40 K a 4.83 So, free bosonic atoms quench BF molecules rapidly! H- 6 Li a 4.89 H- 40 K a 6.85 X- Y a 7.00

24 Long-lived BF molecules B-F V BF +F rel 133 Cs- 6 Li a Rb- 6 Li a Na- 6 Li a 3.05 But Cs- 40 K a Rb- 40 K a 3.12 V BF +B rel a sin 2 [s 0 ln( a /r 0 ) + Φ] 7 Li- 6 Li a Na- 40 K a Li- 40 K a 4.83 So, free bosonic atoms quench BF molecules rapidly! H- 6 Li a 4.89 H- 40 K a 6.85 X- Y a 7.00

25 Observing Efimov features Numerically exact results for Boson recombination: B + B + B B 2 + B a VII a VI a V a IV a III a II a I a I a II a III a IV a V a VI a VII K 3 (cm 6 /s) nk 100 nk 0.1 nk 1 nk 1 µk 10 µk 100 µk 1 mk a 4 Analytical Formula u VI u V u IV u III u II u I a 4 scaling Analytical Formula 0.1 nk 1 nk 10 nk 1 mk 10 µk 100 µk 100 nk 1 µk u VI u V u IV u III u II u I Three-Body Shape Resonances Multichannel Interference Minima a (a.u.) a (a.u.) Efimov features (minima or maxima) separated by factor e π s 0 D Incao and Esry, PRL (2004)

26 Observing Efimov features Answer: Change systems! Use boson-fermion mixture with m B m F (s 0 1) K 3 k 2 a 2 a 4 e π/s 0 B+B+F B+F+F K 3 B+B+F a 4 e π/s 0 F+F+B k 2 a a a

27 Observing Efimov features Smaller spacing is better... B F e π/s Cs 6 Li Rb 6 Li 6.86 MIT 23 Na 6 Li Cs 40 K Li 6 Li > 100 JILA 87 Rb 40 K > Na 40 K > Li 40 K > identical bosons 22.7

28 Observing Efimov features Smaller spacing is better... B F e π/s 0 a c (a.u.) E c (nk) 133 Cs 6 Li Rb 6 Li MIT 23 Na 6 Li Cs 40 K Li 6 Li > JILA 87 Rb 40 K > Na 40 K > Li 40 K > identical bosons /m

29 Numerical results Model calculation 133 Cs Cs + 6 Li K 3 /a 4 (10-39 cm 2 /s) s-wave p-wave e π/s a (a.u.)

30 Summary Can predict energy, scattering length, and mass dependence of all rates with s-wave scattering lengths Predict possibility of long-lived BF molecules Identify boson-fermion mixtures as most favorable for observing Efimov physics

31 Summary Can predict energy, scattering length, and mass dependence of all rates with s-wave scattering lengths Predict possibility of long-lived BF molecules Identify boson-fermion mixtures as most favorable for observing Efimov physics

32 Summary Can predict energy, scattering length, and mass dependence of all rates with s-wave scattering lengths Predict possibility of long-lived BF molecules Identify boson-fermion mixtures as most favorable for observing Efimov physics

33 Summary Can predict energy, scattering length, and mass dependence of all rates with s-wave scattering lengths Predict possibility of long-lived BF molecules Identify boson-fermion mixtures as most favorable for observing Efimov physics