Spin-injection Spectroscopy of a Spin-orbit coupled Fermi Gas
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1 Spin-injection Spectroscopy of a Spin-orbit coupled Fermi Gas Tarik Yefsah Lawrence Cheuk, Ariel Sommer, Zoran Hadzibabic, Waseem Bakr and Martin Zwierlein July 20, 2012 ENS
2 Why spin-orbit coupling? A little bit of History Last century : classification of quantum states in terms of spontaneous symmetry breaking [Anderson 1997] In 1980 : Quantum Hall state. First topological state characterized by a topological invariant [von Klitzing et al. 1980] : a new topological class predicted and discovered, where timereversal symmetry is preserved. Spin-Orbit coupling plays a crucial role [M.Z. Hasan and C.L. Kane, RMP, 2010] Also : Modified interactions, unconventional pairing, Majorana fermions Cold atoms : (very often) constitute optimal system thanks to purity and controle
3 Spin-orbit Hamiltonian Electron moving in an electric field creates a momentum-dependent magnetic fields in the moving frame In 2D semiconductor electric field can arises from structure Rashba Dresselhaus Provides a good description of 2D SOC in solids
4 Engineering SO coupling Laser 1 Laser 2,, How does the Hamiltonian look like? Reminder 2-level system + electric field e g RWA approx. : By adiabatic elimination of the excited state Raman Beams the Raman process can be described as the interaction of a 2-level system with Flip spin + imparts momentum a field
5 Engineering SO coupling Local pseudo-spin rotation of angle around the z-axis Global rotation Momentum dependent Zeeman field equal Rashba and Dresselhaus contributions Y. J. Lin et al. Nature 471, (2011)
6 Engineering SO coupling,, Define quasi-momentum q quasi-momentum real space
7 Engineering SO coupling 2 (q -Q / 2) 2,, Define quasi-momentum q 2m + d 2 2 (q +Q / 2) 2 2m - d 2 0 quasi-momentum space
8 Experimental Setup Fermionic 6 Li atoms sympathetically cooled by 23 Na Relevant states are 2 nd and 3 rd lowest states at 11G Interactions are negligible ( )
9 Coupling spin and momentum via Raman Fermion Laser 1 Laser 2 2 v/ v Vary detuning Short pulse 1 v/ State-selective imaging after TOF provides spin and momentum information
10 Image analysis Start with a mixture of and, and apply a Raman pulse for a given Check for the linear dependence of the transfer with momentum q (Doppler shift )
11 Pulsing on Raman Beams Start with state
12 Pulsing on Raman Beams Start with state Atomic system is coherent over many cycles Momentum-dependent Rabi oscillations Probability of transfer : with
13 Start with state Set large initial detuning ( Adiabatic Sweep ) and then sweep
14 Start with state Set large initial detuning ( Adiabatic Sweep ) and then sweep
15 Spin-injection spectroscopy How to characterize Hamiltonian? Can topology be measured? Condensed matter: transport, (spin-)arpes, STM Cold atom analog: momentum resolved RF (Jin, Koehl) (=photoemission spectroscopy) Photoemission Spectroscopy probes dispersion E(k)
16 What has been done so far Y. J. Lin et al. Nature 471, (2011) Ian Spielman s group P. Wang et al arxiv: (Jing Zhang s group)
17 Spin-injection spectroscopy Can Topology be measured? Spin-injection spectroscopy: Measures spin, energy, momentum 1. Inject atoms from reservoir 2. Project into free space 3. Spin-selective imaging Reconstruct E(k) along with color of band
18 Experimental Setup 1 st and 4 th states used as reservoir states R R
19 Spin-injection spectroscopy
20 Spin-injection spectroscopy
21 Spin-injection spectroscopy
22 Spin-injection spectroscopy
23 Spin-injection spectroscopy
24 Spin-injection spectroscopy
25 Spin-injection spectroscopy
26 Spin-injection spectroscopy
27 Spin-injection spectroscopy
28 Spin-injection spectroscopy
29 Spin-injection spectroscopy
30 Spin-injection spectroscopy
31 Spin-injection spectroscopy Increasing Raman Intensity
32 Spin-injection spectroscopy Increasing Raman Intensity
33 Creating a Spinful Lattice Add RF coupling -> lattice system with full bandgaps and spinful bands K. Jimenez-Garcia et al PRL 108, (2012)
34 Creating a Spinful Lattice Add RF coupling -> lattice system with full bandgaps and spinful bands The Spin-Orbit band structure is periodically repeated K. Jimenez-Garcia et al PRL 108, (2012)
35 Bandstructure of Raman + RF lattice In repeated scheme
36 Bandstructure of Raman + RF lattice Degenerate point inside spin orbit gap
37 Bandstructure of Raman + RF lattice Bandgap opens between 2 nd and 3 rd band
38 Bandstructure of Raman + RF lattice Larger RF, gap between lowest bands
39 Increasing RF Intensity Spin-injection Spectra Increasing Raman Intensity
40 Increasing RF Intensity Spin-injection Spectra Increasing Raman Intensity
41 Increasing RF Intensity Spin-injection Spectra Increasing Raman Intensity
42 Increasing RF Intensity Spin-injection Spectra Increasing Raman Intensity
43 Increasing RF Intensity Spin-injection Spectra Increasing Raman Intensity
44 Increasing RF Intensity Spin-injection Spectra Increasing Raman Intensity
45 Increasing RF Intensity Spin-injection Spectra Increasing Raman Intensity
46 Increasing RF Intensity Spin-injection Spectra Increasing Raman Intensity
47 Spin-injection Spectra
48 Spin-injection Spectra
49 Spin-injection Spectra Raman gap RF gap Raman + RF gap
50 Reconstructing the Bandstructure In addition to dispersion, can reconstruct eigenstates TOF gives eigenstate in the basis of free space spin/momentum states
51 Reconstructing the Bandstructure In addition to dispersion, can reconstruct eigenstates TOF gives eigenstate in the basis of free space spin/momentum states
52 Reconstructing the Bandstructure In addition to dispersion, can reconstruct eigenstates TOF gives eigenstate in the basis of free space spin/momentum states
53 Reconstructing the Bandstructure In addition to dispersion, can reconstruct eigenstates TOF gives eigenstate in the basis of free space spin/momentum states
54 Reconstructing the Bandstructure In addition to dispersion, can reconstruct eigenstates TOF gives eigenstate in the basis of free space spin/momentum states
55 Summary and Outlook Summary: SO-coupled Fermi gas Spinful lattice Spin-injection spectroscopy Band and eigenstate reconstruction Future: Interactions : p-wave Pairing in 1D tubes : Majorana edge mode? For details see: L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Y, W. Bakr, M. W. Zwierlein, arxiv:
56 Collaborators Lawrence Cheuk Ariel Sommer We thank these organizations for their support: DARPA, NSF, ONR, AFOSR, Sloan Foundation Zoran Hadzibabic Waseem Bakr Martin Zwierlein
57
58 Experimental Setup Fermionic 6 Li atoms sympathetically cooled by 23 Na Relevant states are 2 nd and 3 rd lowest states at 11G
59 Spin-injection spectroscopy Experiment Theory
60 SO-coupling in a Fermi gas Direct demonstration of SO-coupling through Rabi oscillations Controlled adiabatic loading of SO-coupled bands. Reversibility of loading shows adiabaticity.
61 The spin-orbit Hamiltonian The SO Hamiltonian Raman Coupling Hamiltonian maps to 1D spin-orbit Hamiltonian with
62 Spin-injection spectroscopy Spin-injection spectroscopy on a spinful lattice
63 The spin-orbit Hamiltonian Raman coupled atomic system maps to SO Hamiltonian. Rotating-Frame approximation: Write in terms of COM momentum q (spindependent transformation):
64 The spin-orbit Hamiltonian Raman coupled atomic system maps to SO Hamiltonian. Rotating-Frame approximation:
65 The spin-orbit Hamiltonian Raman coupled atomic system maps to SO Hamiltonian. Rotating-Frame approximation: Write in terms of COM momentum q (spindependent transformation): Amplitude of Raman beams give splitting Detuning imbalances the two wells
66 The spin-orbit Hamiltonian Raman coupled atomic system maps to SO Hamiltonian. Rotating-Frame approximation: Write in terms of COM momentum q (spindependent transformation): Amplitude of Raman beams give splitting Detuning imbalances the two wells
67 SO-coupling in a Fermi gas When SO coupling is ramped slowly: Spin composition follows effective magnetic field Process is reversible By changing detuning, either upper band or lower band
68 Spin-injection spectroscopy How to characterize Hamiltonian? Can we measure topology? Condensed matter: transport, ARPES, STM Cold atom analog: photoemission spectroscopy (PES) has been PES probes E(k) Transfer to hyperfine states outside system with RF Measure momentum in TOF Use RF frequency, free particle dispersion and momentum to reconstruct E(k)
69 Creating a Spinful Lattice Add RF coupling -> lattice system with full bandgaps and spinful bands K. Jimenez-Garcia et al PRL 108, (2012)
70 Detecting Spin Texture Image Sequence: TOF + state-selective imaging
71 Detecting Spin Texture Image Sequence: TOF + state-selective imaging
72 Detecting Spin Texture Image Sequence: TOF + state-selective imaging Some parameter
73 Engineering SO coupling Raman Beams couple two hyperfine states SO coupling along one direction Recoil momentum: Q Recoil energy : E R = 2 Q 2 2m
74 SO-coupling in Ultracold Atoms Realized in bosons: Modified dispersion Synthetic higher-order partial waves Synthetic magnetic field Y. J. Lin et al. Nature 471, (2011) R. A. Williams et al. Science 335, (2011) Y. J. Lin et al Nature (2009). Recently realized in fermions P. Wang et al arxiv: L. W. Cheuk at el arxiv:
75 Bandstructure of Raman + RF lattice
76 Bandstructure of Raman + RF lattice
77 Bandstructure of Raman + RF lattice
78 A Spin Diode Spin diode when the Fermi level is inside the spin gap
79 Experiment vs Simulation
80 Why spin-orbit coupling?
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