6. Interference of BECs

Transcription

1 6. Interference of BECs

2 Josephson effects Weak link: tunnel junction between two traps. Josephson oscillation An initial imbalance between the population of the double well potential leads to periodic oscillation of the particle number in the wells. The difference has a mean value of zero. Experiment in an optical dipole trap with ~1000 atoms ( 87 Rb condensate). Ψ i 1 = ( E1Ψ1 + KΨ2) Ψ i 2 = ( E2Ψ2 + KΨ1) j = sin γ j c Th. Anker et al., Phys. Rev. Lett. 94, (2005)

3 Josephson effects Nonlinear interaction between atoms leads to macroscopic self-trapping : If the difference of the initial imbalance is larger than a critical value, the population imbalance oscillates around a non-zero value. Experiment: Th. Anker et al., Phys. Rev. Lett. 94, (2005) Theory: Smerzi (PRl, PRA)

4 Interference of two condensates Separation between the condensates: d Relative momentum of the condensates after an expansion time τ: d Δp m τ The corresponding waveleght is the periode of the fringe pattern: Δx hτ md M.R. Andrews et al., Science 275, (1997).

5 Interference of two condensates M.R. Andrews et al., Science 275, (1997).

6 Atomlaser Output coupling of a coherent beam of atoms by radio-frequency waves Natural lenght scale Stationary solution (Airy function) I. Bloch et al., Nature 403, 166 (2000)

7 Atomlaser Excellent spatial coherence I. Bloch et al., Nature 403, 166 (2000) O. Vainio et al., PRA 73, (2006)

8 Scheme of Bragg diffraction as Raman process Two photon transition between two momentum states of the ground state: k p 1 initial k 2 Δ p = p + (k k ) final initial 1 2 With k 1 k 2 ( k) and M. Weidemüller, C. Zimmermann (Eds.) Interaction in ultracold gases, Whiley-VCH 2003, Weinheim

9 Bragg pulses Oscillatory behavior with the two photon Rabi-frequency: with the resonant single photon Rabi-frequencies and Γ=1/τ

10 Bragg pulse interferometer π π π 2 2 pulses First pulse split the condensate into equal populations with p=0 and p=2ħk. Second pulse inverts the states. Third pulse recombines the wavepackets, Output ports: 1. Population in p=0 2. Population in p=2ħk If there is no phase shift, the total population is in p=0. This is a single particle interferometer scheme. Using a BEC makes just the detection easier.

11 Bragg pulse interferometer measuring the phase of a Bose-Einstein condensate wave function J. E. Simsarian et al., PRL 85, (2000)

12 Bragg pulse interferometer (open interferometer) π π 2 2 pulses Suitable for measuring the phase distribution of the condensate using the interference of partially overlapping condensates as output. The interference pattern is interchanged for the two momentum components, as expected for complementary ports of the interferometer.

13 Atom Michelson interferometer based on Bragg pulses Wang et al., PRL 94, (2005)

14 Diffraction from a magnetic lattice 1. BEC prepared 30µm below the lattice 2. BEC forced to oscillate towards the lattice 3. Observation of diffraction & interference after 20ms TOF

15 Periodic potential of the meander: Phase imprinting y z x Wave function directly after phase imprinting Phase modulation index Expansion in momentum eigenfunctions of the axial motion (Bessel functions of first kind): Number of atoms in the n th diffraction order proportional to:

16 Bessel functions Number of atoms in the n th diffraction order proportional to:

17 Diffraction and interference 20ms time of flight without interference includes interference cond-mat

18 Measuring forces τ 1 ϕ = () Utdt τ = 20 ms 0 2π phase shift corresponds to grad(b) ~ 7.2 mg/cm Δϕ=π/8 phase spread corresponds to Δa ~ g

19 Spatial interferometer: magnetic double well Y. Shin et al. PRA 72, (2005) Coherence is lost after breaking the tunnel coupling due to excitation of the condensate at the splitting point.

20 Extreme sensitivity to spatial potential variations U(z) test potential z time (s) 0.6 aspect ratio r/z after 15 ms time of flight cigar pancake Gross-Pitaevskii simulation predicts chaos H. Ott, J. Fortágh, S. Kraft, A. Günther, D. Komma, C. Zimmermann PRL 91, (2003)

21 Spatial interferometer: rf-double well T. Schumm et al., Nature Physics 1, 57 (2005) Coherence is preserved after the splitting for ~4 oscillation periods (2ms). Splitting scheme works, but still a fast dephasing.

22 Long phase coherence time due to number squeezing Expected coherence time for condensates in a coherent state: τ c > 200 ms Jo et al., cond-mat/

23 Number squeezing Phase diffusion rate in a condensate: Derivative of the chemical potential Standard deviation of the relative atom number Coherent state: Squeezed state: s: squeezing parameter Jo et al., cond-mat/

24 Vision: on-chip single atom interferometer