Raman-Induced Oscillation Between an Atomic and Molecular Gas Dan Heinzen Changhyun Ryu, Emek Yesilada, Xu Du, Shoupu Wan Dept. of Physics, University of Texas at Austin Support: NSF, R.A. Welch Foundation, NASA MRD -US Japan seminar 2006-
BEC II Changhyun Ryu, Wooshik Shim, Emek Yesilada, Shoupu Wan, Xu Du, D. H. Zeeman-slowed 87 Rb beam Dark MOT, molasses Cloverleaf trap RF-induced evaporation BEC with up to 2 10 6 atoms (F = 1, M= 1)
Outline Feshbach Resonance and Raman Photoassociation in Bose Condensates. Raman Photoassociation in a Mott Insulator. - Resolved Contact Energy Shifts Can Determine Fraction of Sites with 1, 2, or 3 atoms. Can Determine Atom-Molecule Scattering Length. - Oscillating Atomic <-> Molecular gas! Bragg spectroscopy of atoms in a 3D optical lattice
Coherent Atom-Molecule Coupling Mechanisms Feshbach Resonance (JILA, MIT, Innsbruck, ENS, Rice, Munich, ) Raman Photoassociation (Texas, Rice, Munich, ) Interatomic Potential R Interatomic Potential R Variation: Radio Frequency Coupling (JILA, MIT, Innsbruck, )
Coherent Atom-Molecule Coupling In a Bose Condensate χ n (coupling) ϕ :atomfield ψ: molecule field H ~ χ ϕ 2 ψ + + h.c. - Analogous to frequency doubling and parametric downconversion - Pairing fields play a role ~ spontaneous down conversion - Can in principle produce macroscopic coherent oscillation - Theory: Drummond, Holland, Timmermanns, Burnett,
Molecule Losses Γ M = Γ L + K inel n a 5 2 S 1/2 + 5 2 P 1/2 Δ 1 e Γ L = Rate of spontaneous laser light scattering (Can be calculated) Interatomic Potential V e (R) ω 2 Ω 2 ε g ω 1 Ω 1 f 5 2 S 1/2 + 5 2 S 1/2 R V g (R) K inel n a = Rate of inelastic collisions with atoms (not calculable, generically expect K inel ~ 10 11 cm 3 /s )
time χ n<< Γ Μ rate equation dynamics density n a n m << n a atom loss rate : χ 2 n/ Γ M time χ n>> Γ Μ Coherently coupled matter waves Collective atom-molecular Rabi oscillation, STIRAP, density n a n m ~ n a
Stimulated Raman Photoassociation 5 2 S 1/2 + 5 2 P 1/2 Δ 1 e ω 1 Interatomic Potential V e (R) ω 2 Ω 2 ε Ω 1 ΔE=k B T/h f 5 2 S 1/2 + 5 2 S 1/2 g R Δ 2 V g (R)
Stimulated Raman resonances in an 87 Rb Bose condensate N(τ) / N 0 N(τ) / N 0 1.0 0.0 1.0 (A) (B) More density Linewidth < 2 khz! Increased linewidth with increased atomic density N(τ) / N 0 N(τ) / N 0 0.0 (C) 1.0 0.0 (D) 1.0 0.0 636.010 636.020 636.030 (ω 2 ω 1 ) / 2π (MHz) Shift in line center with increased atomic density Fit with photoassociation rate theory of Bohn and Julienne
Results (Wynar, et al., Science 287, 1016 (2000)) ε 0 /2π = 636.0094 ± 0.0012 MHz a ma = -180 ± 150 a 0 K inel < 8 x 10-11 cm 3 /s 1,000 more accurate molecular binding energy than previously First measurement of a molecule-condensate interaction Mean field interactions account for shift and most of the broadening (no definite, nonzero K inel )
Calculated Coupling Rate and Spontaneous Photon Loss Rate for Rb 2 [ Assumes K inel << 10-11 cm 3 /s ] Experiments: Tried, limited evidence of collective coherent behavior
Bose-Hubbard Model First Studied: M. Fisher et al., PRB 40, 546 (1989) Zero temperature lattice model in 1, 2 or 3 dimensions Hopping matrix element between adjacent lattice sites J Onsite repulsive contact interaction between bosons U Assume particles remain in lowest band (U << splitting between bands) Optical lattice loaded with dilute gas condensate provides ideal realization of Bose-Hubbard model: D. Jaksch et al., Phys. Rev. Lett. 81, 3108-3111 (1998). Transition first observed: M. Greiner et al., Nature 415, 39 (2002)
Optical Lattice six laser beams produce three orthogonal standing wave fields λ = 830 nm, P up to 200 mw per beam Beam waist 200-300 μm Dipole potential V(x) ~ α(ω) I(x) V = V 0 (sin 2 (kx) + sin 2 (ky) + sin 2 (kz)) condensate V 0 up to 30 E R E R = k B 150 nk = h 3.2 khz Phase transition occurs with V 0c 12-14 E r For V 0 > V 0c, hν L >> U >> J ν L = vibration frequency For V 0 = 20 E R ν L 30 khz U/h 2 khz J/h 8 Hz
Superfluid Phase (perturbed condensate) φ i = <b i > 0 Approx. coherent state Well defined condensate phase Uncertain particle number Very Fast playback
Mott Insulator Phase Aprox. Fock state at each site <b i > 0 as V 0. Very Fast playback
Effect of Trap Potential Energy to add a second atom to a site U V(z) Chemical potential μ Energy to add one atom to a site -z 1 -z 2 0 z 2 z 1 z Calculation of Jaksch et al., Phys Rev. Lett. 81, 3111 (1998)
Raman photoassociation of two atoms in an optical lattice site Proposal: D. Jaksch et al., Phys. Rev. Lett. 89, 040402 (2002) V(R) ½ μω 2 R 2 V m (R) R Continuum discrete levels of atoms in lattice site ω/2π 30 khz = lattice vibration frequency Enhanced freebound coupling Eliminates inelastic collisions
Photoassociation in a Mott Insulator Lattice Height 80 ms V 0 20E R time Photoassociation laser pulse, duration t Measure atom number N(t) BEC with N(0) atoms
Single Color Photoassociation Optical Lattice On V 0 = 22 E R N(t)/N(0) = Aexp(-t/τ 1 ) + Bexp(-t/τ 2 ) τ 1 = 1.56 ms τ 2 = 82.6 ms 40% of atoms in multiply occupied sites, remainder in singly occupied sites. Optical Lattice Off N(t)/N(0) = 1/(1+t/τ) τ= 4.39 ms PA = probe of short range correlations in a gas
Raman Frequency Scan N(τ)/N(0) 1.0 0.8 0.6 1.0 0.8 N(0) 0.5 million N=2 per site 30% N=3 per site 23% N(0) 0.25 million N=2 per site 33% 0.6 636.123 636.126 636.129 636.132 636.135 (w 2 -w 1 )/2π (MHz)
N = 2 atoms per site N = 3 atoms per site U aa 3U aa U am Atom-molecule collisional loss
N = 2 atoms per site N = 3 atoms per site U aa 3U aa 0 U am Atom-molecule collisional loss Measured shift Measured U aa a am = 5 ± 20 a 0 Greater width of N=3 peak: inelastic collision loss, greater power broadening. Estimate that K inel ~ few 10 11 cm 3 /s
Number of atoms vs. PA time, on Raman resonance with N = 2 peak 1.0 0.8 N(0)~0.28 million atoms N(τ)/N(0) 0.6 0.4 0.2 0.0 0 1 2 3 4 5 6 Raman PA time (ms) Central core of gas oscillates between an atomic and a molecular quantum gas! Ultimate control of atomic pairs all degrees of freedom exactly controlled
Bragg spectroscopy ω 1, k 1 ω 2, k 2 Two far-detuned laser beams imposed on the gas sample θ Stimulated absorption of one photon from one laser beam and stimulated emission into the other laser beam BEC q, ω Momentum transfer hq = hk 2 hk 1 Energy transfer hω = hω 2 hω 1 Frequency difference determined by two acousto-optical modulators ω(q) response of quantum gas to perturbation Stenger et al., PRL 82, 4569
Dispersion relation ω(q) 7000 For a weakly interacting quantum gas system ω/2π (Hz) 6000 5000 4000 3000 ξ 1 = 8πna Free particle 2 2 h q h ω( q) = (2μ + 2m 2 h q 2m µ is chemical potential; m is atomic mass 2 ) 2000 For small q, collective excitations 1000 0 Phonon Experimental setting 0 2e+6 4e+6 6e+6 8e+6 1e+7 q (m -1 ) c h ω( q) = hcq μ =, speed of sound m ξ is healing length Steinhauer et al., PRL 88, 120407 For large q, single-particle excitations 2 2 h q hω ( q) = + μ 2 m
Bragg spectroscopy of Superfluid in 3-D Lattice Temperature increase due to the excitations Measure the gas temperature with TOF imaging 1.1 1.0 V Optical lattice Bragg beam BEC fraction 0.9 0.8 0.7 t Two photon excitation rate ~500 Hz 2 Γ I Ω R = 4 Δ I Г is natural linewidth; Δ is frequency detuning (430 GHz); I is laser intensity (~100 mw/cm 2 ); I sat is saturation intensity. sat 0.6 0.5 0.4 V 0 = 5.5 Er 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Frequency (khz) Gaussian fitting: resonant frequency is taken as the center value of the fitting Pulse duration 3-20 ms
Theory Bogoliubov theory (Biao Wu, IOP, Beijing) Breaks down for V 0 > 3 E r As optical lattice depth increases, μ* increases due to tighter confinement, m* increases due to the decreased band width
Excitation spectra at different optical lattice depths 1.0 1.0 1.0 0.9 0.9 0.9 BEC fraction 0.8 0.7 0.6 BEC fraction 0.8 0.7 0.6 BEC fraction 0.8 0.7 0.6 0.5 No OL 0.5 V 0 = 3.3 Er 0.5 V 0 = 5.5 Er 0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Frequency (khz) 0.4 0.0 0.5 1.0 1.5 2.0 Frequency (khz) 0.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Frequency (khz) 1.0 0.9 1.0 0.9 0.30 0.28 BEC fraction 0.8 0.7 0.6 BEC fraction 0.8 0.7 0.6 Peak width (khz) 0.26 0.24 0.5 V 0 = 7.7 Er 0.5 V 0 = 9.9 Er 0.22 0.4 0.0 0.5 1.0 1.5 Frequency (khz) 0.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Frequency (khz) 0.20 0 2 4 6 8 10 Optical lattice depth (in Er)
Conclusion Photoassociation in Mott insulator provides measure of singly, doubly, and triply occupied lattice sites confirm large fraction of multiply occupied sites. Resolved spectrum determines atom-molecule interactions Oscillating atomic <-> molecular gas! Bragg spectra in lattice show breakdown of Bogoliubov theory at surpisingly low lattice depths.