Anharmonic Confinement Induced Resonances: Theory vs Experiment

Anharmonic Confinement Induced Resonances: Theory vs Experiment Peter D. Drummond, Shi-Guo Peng, Hui Hu, Xia-Ji Liu CAOUS Centre, Swinburne University of Technology *Tsinghua University, Beijing IQEC Sydney 2011

A Detective Story in Six Parts 1 Confinement Induced Resonance 2 HCIR: Theory vs experiment 3 ACIR: Theory vs experiment 4 Multiple ACIR resonances 5 Two-dimensional ACIR resonances 6 Summary

Harmonic Confinement Induced Resonance (HCIR) Low-dimensional transverse resonance phenomena Predicted by Olshanii - Phys. Rev. Lett. 81, 938 (1998)! Harmonic resonance: SINGLE internal excited state Like a Feshbach resonance for waveguides Allows quantum engineering by changing trap frequency

HCIR in ANISOTROPIC waveguide Resonances at ε = 0, satisfy an integral equation: F e (ε,0) = 0 dt { exp ( tε2 ) η t [ ] } 1 (1 e ηt )(1 e t ) 1 1 t 3/2 πd =. a 3D d = 2 h/mω y, η = ω x /ω y, a 3D = 3D scattering length Peng, et al, Phys. Rev. A 82, 063633 (2010)

Physics: SINGLE resonance predicted in waveguides 2 1 CIR 0 E/ y -1-2 -3 Bound state of H rel Bound state of H e -4-4 -3-2 -1 0 1 2 3 4 d/a 3D

Innsbruck Cs Experiment: PRL 104, 153203 (2010)

Expected HCIR resonance vs anisotropy 1.1 1.0 (R) a 3D / a y 0.9 0.8 1.00 1.05 1.10 1.15 1.20 1.25 x / y

Innsbruck CIR Measurements don t fit theory! 1.1 1.0 (R) a 3D / a y 0.9 0.8 1.00 1.05 1.10 1.15 1.20 1.25 x / y

Large anisotropy: multiple resonances - also don t fit theory!

Two-dimensional HCIR resonance only for ATTRACTIVE case Predicted by Petrov et al, PRL 84, 2551 (2000). Experimental 2D resonance Observed resonance at Innsbruck for REPULSIVE case!! Similar observation with fermions by Vale group, SUT Resonance observed for ATTRACTIVE case at Cambridge! See: Frohlich et. al, PRL 106, 105301 (2011).

Mystery - what are these new resonances??

ACIR - anharmonic waveguide resonances NEW COLD-ATOM QUANTUM TECHNOLOGIES! Anharmonic confinement induced resonances - ACIR: Even order multiple resonances: Caused by QUARTIC nonlinearity Odd order multiple resonances: Caused by CUBIC nonlinearity

Two types of confinement resonances (a) (b) (a) ACIR: MANY resonances Sala et. al, arxiv:1104.1561; Peng et. al., arxiv:1107.2725 due to COM resonance of atom pairs (b) HCIR: SINGLE resonance Olshanii, PRL 81, 938 (1998) due to internal excitation of relative motion

Anharmonic confinement potential For dipole trapping experiments with optical lattices, the trapping potential near a potential minimum at r=0 is of form: U ext (r) = V x (r)sin 2 ( 2πx λ x 1 2 m [ ω 2 x x2 ( 1+ α xx 2 ) +V y (r)sin 2 ( 2πy d 2 x +ω 2 y y2 ( 1+ α yy 2 d 2 y λ y ) + r V ) x Vx 0 +... )] + + r V y V 0 y

Typical anharmonic parameters Anharmonic quartic parameter: α x,y = 1 3 (2πd x,y/λ) 2 = 8π 2 h/ ( 3λ 2 mω x,y ) Experiment 133 Cs 40 K Trap frequency ω 2π 14.5 khz 2π 80 khz Wavelength λ 1.064 10 6 m 1.064 10 6 m Atomic mass m 2.22 10 25 kg 0.6635 10 25 kg Length d x,y 0.102 10 6 m 0.08 10 6 m Anharmonicity α x,y 0.121 0.075

Theory with tightly bound molecules Anharmonic energy in units of hω y Two non-interacting atoms in a waveguide, η = ω x /ω y : ε a = η(1+3α x /8)+(x y) Bound Feshbach molecule: ε b = h/ ( ma3d 2 ω ) y Feshbach molecule in a waveguide: { ε m = ε b + η N x + 1 2 + 3α x 16 [ N x (N x +1)+ 1 2 ]} +x y ACIR resonance: ε m = ε a

ACIR vs HCIR Resonance 3 + 2 + F.R. >1 E/ y + + The internal excited bound state with ground COM state (0,0) The internal ground bound state with excited COM state (2,0) The internal ground bound state with excited COM state (0,2) 1/a 3D <0 1/a 3D 1/a 3D >0

General theory of ACIR resonance Integral equation for bound state energy ε b : [ d = a ] 3D η exp(εb t/2) dt π 0 t(1 e ηt )(1 e t ) 1 t 3/2. Anharmonic ACIR resonance (N x,n y ) when: { 3(Ny (1+N y )+N x (1+N x )) 9 ε b +2ηN x +N y + α 16 2N y +1+η(2N x +1) 32ε + 3( η 2 +1 ) } 320ε 2 = 0

Predicted ACIR resonance vs anisotropy 1.1 1.0 (R) a 3D / a y 0.9 0.8 1.00 1.05 1.10 1.15 1.20 1.25 x / y

Comparison of ACIR to experiment 1.1 1.0 (R) a 3D / a y 0.9 0.8 1.00 1.05 1.10 1.15 1.20 1.25 x / y

ACIR vs HCIR vs Experiment 1.1 1.0 (R) a 3D / a y 0.9 0.8 1.00 1.05 1.10 1.15 1.20 1.25 x / y

Large anisotropy: theory vs. experiment B(G) 47.8 47.7 47.6 47.5 47.4 47.3 47.2 47.1 47.0 1.4 1.6 1.8 2.0 2.2 (0,1) (0,2) (0,3) (0,4) (1,0) (2,0) x y

Multiple resonance summary Multiple resonances are an ACIR signature! Identified 22 ACIR resonances at large anisotropy. Also 8 unidentified resonances: Internal resonance with anharmonic shifts? Four atom mixing pairs of molecules?

Two-dimensional limit: a 3D vs anharmonicity 0.90 Experimental data from Innsbruck group ACIR prediction 0.85 a 3D / a 0.80 0.75-0.2-0.1 0.0 0.1 0.2

Two-dimensional limit: a 3D vs confinement a 1.4 1.3 Experimental data from Innsbruck group ACIR prediction (R) a 3D (1000a 0 ) 1.2 1.1 1.0 1.3 1.4 1.5 1.6 a (1000a 0 )

New type of low-dimensional nonlinear optics Experiment in excellent agreement with ACIR theory! Anharmonic confinement induced COM resonances - ACIR: Explains 35 newly observed resonances Caused by QUARTIC and CUBIC anharmonic couplings Harmonic confinement induced internal resonances - HCIR: Does not match any Innsbruck data! Molecular loss insensitive to HCIR? Unexplained resonances - Anharmonic shifted internal resonance? Higher order 4-body anharmonic resonance?

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