Spinor Bose gases lecture outline

Transcription

1 Spinor Bose gases lecture outline 1. Basic properties 2. Magnetic order of spinor Bose-Einstein condensates 3. Imaging spin textures 4. Spin-mixing dynamics 5. Magnetic excitations We re here

2 Coupling strengths 1, 1, 1, 1, or, or, 1, 0 1, 0 1, 1 1, 1

3 Two more ingredients: fixed magnetization quadratic Zeeman shift non interacting Stenger et al., Nature 396, 345 (1998)

4 Two more ingredients: fixed magnetization quadratic Zeeman shift Stenger et al., Nature 396, 345 (1998)

5 Evidence for antiferromagnetic interactions of F=1 Na Stenger et al., Nature 396, 345 (1998) Miesner et al., PRL 82, 2228 (1999).

6 Many body ground state Is mean-field approximation good? Scalar Bose gas (Bogoliubov, 1947): Quantify the quantum depletion 4 2 Quantum depletion = fraction of atoms outside condensate

7 Many body ground state Is mean-field approximation good? Spinor Bose gas [Law, Pu, and Bigelow, PRL 81, 5727 (1998); others] No barrier to quantum depletion? Is there a Bose-Einstein condensate at all?

8 Many body ground state Spin-1 gas: Law, Pu, and Bigelow, PRL 81, 5727 (1998); others] 2 2 0(ferromagnetic, 87 Rb) 0(antiferromagnetic, 23 Na), 0, 0 N-fold degenerate ground state broken-symmetry mean-field state = coherent superposition of these degenerate states some subtlety if magnetization is exactly conserved unique ground state unbroken rotational symmetry not BEC: fractionated condensate observable by correlations: 0along any axis

9 3. Imaging spin textures a. methods i. dispersive imaging ii. absorptive spin-sensitive in-situ imaging b. equilibration toward ground-state c. topological structures d. magnetization curvature

10 Dispersive birefringent imaging F 2 F 1 phase contrast imaging polarization contrast imaging phase plate polarizer z y x linear

11 Spin echo imaging fine tuning: pulses: π π π/2 π images: M x M y M z time N 2 x 10 6 atoms T 50 nk vector imaging sequence repeat? ~3 m ~200 m ~50 m geometry surfboard

12 Absorptive spin-sensitive in-situ imaging ASSISI

13 Absorptive spin-sensitive in-situ imaging 50 µm

14 Expected ground states zero magnetization 0 longitudinal axis ferromagnetic states transverse plane unmagnetized state Z2 U(1) Majorana representation, Majorana, Nuovo Cimento 9, 43 (1932) 0 SO(2) U (1) 2 n U (1) 0 q

15 F=1 87 Rb gas at thermal equilibrium? prepare fully depolarized thermal gas in uniform magnetic field lower temperature what happens? unpolarized thermal gas T T cn, /3 magnetized something unmagnetized, scalar BEC 0 20 q

16 Development of spin texture q/h = 0 Transverse Longitudinal Time: ms 50 µm

17 Development of spin texture q/h = + 5 Hz Transverse Longitudinal Time: ms 50 µm

18 Development of spin texture q/h = - 5 Hz Transverse Longitudinal Time: ms 50 µm

19 Why is equilibration so slow? Questions and opportunities Partial answer: coarsening dynamics [A. J. Bray, Adv. Phys. 51, 481 (2002)] sometimes: does scaling hold? how to determine this from small, short-lived samples? what are underlying mechanisms?

20 Questions and opportunities Spin correlation functions: Fluctuations and spin susceptilibity Equal time (one image): static spin structure Unequal time (repeated image): dynamical spin structure factor what can we learn from examining within one coarsened region, and from variation between regions? What do such measurements tell us when we re unsure of equilibrium? Precedent and guide from studies of density fluctuations: Hung et al., Extracting density density correlations from in situ images of atomic quantum gases, NJP 13, (2011).

21 q/h = - 5 Hz (easy axis feromagnet) e.g. examining 50 µm above ~ Gaussian fluctuations Δ lim ms below from data: lim 0 has diverged

22 Topological structures Symmetry breaking: Hamiltonian of system has symmetry that state of system does not there are symmetry operations that do change the state, but do not change its energy continuous manifold of degenerate (equilibrium, ground, metastable) states Examples of continuous symmetries: Transformation Group Symmetry breaking system multiply wavefunction by complex phase 1 scalar superfluid rotations in 3D 3 ferromagnetic superfluid

23 Topological structures mapping between order parameter space and a d-dimensional contour that cannot be smoothly undone e.g. 1d contour, U(1) vortices U(1) n windings possible (homotopy group is )

24 Vortices in a spin-1 ferromagnetic superfluid 2 phase 0 0 translate toward gas spherical quadrupole magnetic field

25 Vortices in a spin-1 ferromagnetic superfluid 2 phase phase , -1 vortices are topologically equivalent. Also +2 vortex is equivalent to no structure at all! (homotopy group is )

26 Vortices in a spin-1 ferromagnetic superfluid 2 phase , -1 vortices are topologically equivalent. Also +2 vortex is equivalent to no structure at all! (homotopy group is ) polar-core spin vortex

27 Spontaneously formed spin vortices F T candidates: Mermin-Ho vortex (meron) m z =0 core T hold = 150 ms F L Polar core spin vortex

28 Making a spin texture A. Leanhardt, et. al. PRL (2003)

29 Making a spin texture A. Leanhardt, et. al. PRL (2003)

30 Making a spin texture A. Leanhardt, et. al. PRL (2003)

31 Making a spin texture Direct image of magnetization texture: longitudinal transverse Mag 0 Between: magnetized radially (sideways) Core: magnetized up Periphery: magnetized down A. Leanhardt, et. al. 31 PRL (2003)

32 Choi, J.Y., W.J. Kwon, and Y.I. Shin, Observation of Topologically Stable 2D Skyrmions in an Antiferromagnetic Spinor Bose Einstein Condensate. PRL 108, (2012)

33 Ferromagnet or polar spinor condensate: skyrmion, or not skyrmion? Ferromagnetic spinor condensate: Order parameter = Order parameter = ^lz = 1 m ^ ^ l ^n = skyrmion (topological) skyrmion (not topological)

34 Other localized spin structures

35 Spinor superfluid hydrodynamics Lamacraft PRA 77, (2008) Barnett, Podolsky and Refael, PRB 80, (2009) Modification of irrotational flow condition: 1 2 = direction of magnetization magnetization curvature (topological density) creates Lorentz-like force [similar to topological Hall effect in solid-state; see PRL 102, (2009)]

36 In-situ measurement of magnetization curvature skyrmion like spin texture 3.99

37 In-situ measurement of magnetization curvature spin helix spin texture 0.24

38 4. Spin mixing dynamics a. microscopic spin mixing oscillations b. SMA, mean-field dynamics c. spin-mixing instability

39 Microscopic spin mixing Consider just two particles in a tight trap. 4 say initially both atoms are in 0 Ψ 0 0, 2, 0 superposition of states with two different total spin: F 2, , 2, ,0,1 F 0, , 2, ,0,1 evolves at evolves at Ψ , 2, ,0,1

40 spin mixing of many atom pairs Widera et al., PRL 95, (2005)

41 Mean-field macroscopic spin mixing Zhang et al., PRA 72, (2005) derive spinor Gross-Pitaevskii equation (lots of papers, looks complicated) Identify energy landscape and dynamical variables: 1 2 exp 3 unimportant unimportant only one density to keep track of: only one phase to keep track of: 2 2 2

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43 M. S. Chang et al, Nature Physics 1, 111 (2005)

44 Liu, Y., S. Jung, S.E. Maxwell, L.D. Turner, E. Tiesinga, and P.D. Lett, Quantum Phase Transitions and Continuous Observation of Spinor Dynamics in an Antiferromagnetic Condensate. PRL 102, (2009.

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