Outline:

1. SeminarSpatial Quantum Noise InterferometryinExpanding Ultracold Atomic CloudsbySascha Hoinka14.12.2005

2. Outline: 1. Motivation 2. The second order spatial correlation function 3. HBT-Interferometry: A new powerful tool in absorption imaging 4. Experiments: S. Fölling (Mainz): Particle correlations in a bosonic Mott insulator M. Greiner (JILA): Local and nonlocal fermionic atom pair correlations 5. Applications to more complex systems, E. Altman 6. Summary

3. Greiner et al. Nature.415 (2002) Motivation: Interference in atom optics Absorption imaging of time-of-flight (TOF) expansion is a widely used method for probing ultracold atomic systems Example: BEC in an optical lattice: - Superfluid phase: Macroscopic occupation of a delocalised single particle state leads to a coherent giant matter wave  Long range phase coherence results in a multiple matter wave interference pattern (Bragg-peaks) - Mott phase: Image shows the incoherent sum of localised single particle wave functions  Long range spatial order of atoms shows no pattern  However: Analysis of image noise can reveal higher order interference pattern originating from the spatial ordering in quantum systems

4. uncorrelated atoms bunching antibunching Grondalski et al. Opt. Exp.5 (1999) Two-particle correlation function: Normalised density-density correlation function:  observable in a single absorption image Hanbury Brown & Twiss (HBT) - Interferometry: - The quantum nature of atoms causes intrinsic density fluctuations - Constructive/destructive interference occurs between possible indistinguishable detection paths of bosons/fermions - Density-density correlation signal: : detector separation : atom density : average over a sample of images

5. alat d Grondalski et al. Opt. Exp.5 (1999) HBT-detection in atom optics: Example: Two bosons in an 1D optical lattice: - Atoms in lattice site i and j are released and detected independently at detector D1 and D2 seperated by - The spatial correlation function corresponds to photon (scattered from atoms) coincidence counts between the two detectors - The reciprocal lattice vector l reveals the spatial ordering: pair correlations of: 6 bosonic atoms 2 bosonic atoms

6. Particle correlation in a bosonic Mott insulator (Mainz) Direct measurement of the spatial correlation function of quantum density fluctuations Fölling et al.:Nature 434, 481 (2005) Theory, Altman et al.:PRA 70, (2004)

7. Particle correlation in a bosonic Mott insulator (Mainz) Experimental procedure: - Loading a BEC of 87Rb (~6x105) atoms into an 3D optical trap with simple cubic geometry - Ramping up the lattice in 160ms to ~50Er to create a Mott insulator ( between 1 and 3) - Ballistical expansion after switching off all traps - Recording the average 2D column density profile of the cloud by a resonant laser pulse  Incoherent sum of all localised single particle wave functions  Absence of a long range phase coherence alat= 425nm

8. atom cloud uncorrelated particles bunching of particles Analysing the fluctuations of the cloud: Pixel array of a CCD camera as a HBT-Interferometer: - Each pixel represents a detector - The array records the Optical Density - Coincidence counts of each pixel can be related to the distance d Density-density correlation function: The spatial order of the atoms in the Mott phase cannot be seen in however, it will be revealed by caculating

9. Analysing the fluctuations of the cloud: Image analysis: - Optical density: - Column density : - Number of atoms per bin: (Scattering cross section: ) intensity without atoms intensity of cloud

10. Analysing the fluctuations of the cloud: Image analysis: - Optical density: - Column density : - Number of atoms per bin: (Scattering cross section: ) Calculation of : intensity without atoms intensity of cloud power spectrum autocorrelation function Wiener-Khintchine Theorem: “The power spectrum is equal to the Fourier transform of the autocorrelation function”

11. Imaging the correlation signal C(d): Result:

12. Imaging the correlation signal C(d): Result: Analysing the signal peaks: - The imaging system only has a finte resolution  Convolute peaks by a gaussian (effective bin size)  peak amplitude

13. The role of the pixel size of a CCD camera: The noise of the detected probe-photons per bin is a sum of: Atom shot noise + Photon shot noise + Technical noise Quantum nature of atoms Fluctuations of laser light Dark counts, readout noise

14. The role of the pixel size of a CCD camera: The noise of the detected probe-photons per bin is a sum of: Atom shot noise > Photon shot noise + Technical noise ! Quantum nature of atoms Fluctuations of laser light Dark counts, readout noise Each of the atoms in a bin must scatter a large number of photons. Adjustable by varying the intensity and pulse duration of the probe laser. Noise of the CCD camera can be neglected if the number of detected photons per pixel is high. Raises the signal-to-noise ratio (SNR). “Adjusting“ the relative atom shot noise level: Spatial low and high pass filter adjust the effective bin size atom shot noise photon shot noise

15. Conclusion so far: - In the Mott insulator phase strong periodic quantum correlations occur in the density fluctuations and can be revealed - The density-density correlation function is a powerful tool for a direct measurement and identification of quantum phases - Correlation amplitudes can be observed if the atom number distribution is detected at the atom shot noise level

16. Measurement of local and nonlocal atom shot noise correlations in time of flight (TOF) absorption images Probing pair-correlated fermionic atoms (JILA) Greiner et al.:PRL 94, (2005) Theory, Altman et al.:PRA 70, (2004)

17. 1. LOCAL atom-atom correlations 2. NONLOCAL Probing pair-correlated fermionic atoms (JILA) Creation and detection of pairs of fermionic 40K atoms: - Trapping and cooling a dilute gas of 40K atoms to ultracold temperatures - Using an optical dipole trap (FORT) with radial ~ 300Hz and axial ~ 4Hz as the confining potential - Initially preparing an incoherent mixture of atoms in the and spin states with a magnetic Feshbach resonance (FR) at 202.10G - Creation of weakly bounded molecules by sweeping the magnetic field across the FR - Analysing the shot noise after dissociating the molecules

18. Probing pair-correlated fermionic atoms (JILA) 1. Experiment: Detecting local pair correlations of two atoms: - Switching off the trap, the molecule cloud (~3x105) expands ballistical - Dissociating molecules by increasing the magnetic field across the FR: resulting atoms form an entangled singlet state of: , - Taking absorption images “quasi-instantaneously”: TOF FR B(t) pairs molecules Second image after spin flip (rf -puls)

19. r  Analysing the absorption images: Utilizing spherical symmetry of the cloud: - Extracting the raw noise signal: - 1D correlation function for relative rotation about the cloud center: fluctuation of the atom number in spin state  per bin

20. Revealing local pair correlations: Shot noise images from atom pairs in the two different energy eigenstates: evidence for correlations: high low concentration of atoms

21. Perfect spatial pair correlation Revealing local pair correlations: Shot noise images from atom pairs in the two different energy eigenstates: evidence for correlations: high low concentration of atoms Normalised correlation function: effective bin size

22. FORT FORT Second experiment: atom pairs in the BEC regime Detecting nonlocal pair correlations of two atoms: - For detecting a reliable correlation signal it is important: i) Maximize the ratio between the relative and the center-of-mass momenta  Dissociating molecules by detuned rf-photodissociation ( ) : ii) Minimize the collision rate during initial expansion  Rapidly (50s) switch magnetic field to low interactions - Seperate imaging of expanding atoms in spin states and :  TOF B@ 202.07G B@ 198G rf-pulse

23. extracting the noise extracting the noise Nonvanishing correlation signal occurs at due to equal but opposite momenta between pair correlated atoms Again image processing: TOF images show a sperical shell of pair correlated (~105) atoms: - Images taken after only 1.4ms and 1.7ms after dissociation - Residual atoms and molecules in the center

24. -1 0 1 d(l) Applications to more complex systems Altman et al. PRA 70, (2004) Fermionic superfluidity (BCS regime): - Cooper pairs are defined by a macroscopic occupation of zero momentum pairs - The density profile of the expanding cloud contains the BCS momentum distribution (k, -k)  Correlation peak at diametrically opposite points occurs by comparing the two noise images Antiferromagnetic order in an optical lattice: - Density fluctuations of an expanding atomic cloud contains information of the spin order  Apperance of additional second order peaks at half the reciprocal lattice vector l

25. Summary: • - Analysing the density fluctuations in absorption images can reveal • hidden spatial order in atomic systems • Pair correlations of bosons in optical lattice and fermions in BEC regime • - HBT-Measurement could be one possibility to analyse novel quantum • phases in bosonic and fermionic atomic systems • - Density-density correlation function have proved to be a powerful • tool for probing ultracold atomic gases Grondalski et al. Opt. Exp.5 (1999) Fölling et al. Nature 434, 481 (2005) Greiner et al. PRL 94, (2005) Altman et al. PRA 70, (2004)