Magnetic phenomena in a spin-1 quantum gas Dan Stamper-Kurn UC Berkeley, Physics Lawrence Berkeley National Laboratory, Materials Sciences
Spinor gases 37 electrons J = 1/2 Energy F=2 Optically trapped F=2 spinor gas m F = 2 m F = 1 m F = 0 m F = -1 m F = -2 magnetically trappable m F = -1 F=1 m F = 0 37 protons 50 neutrons I = 3/2 F I J Optically trapped F=1 spinor gas B m F = 1
phase-contrast image; dispersive (minimally destructive) B z y F 2 x 1 12 1 2 1 2 F 1 small signal (normalized # photons/pixel) S 1 0 s0n2 D s1n 2D F y
Measuring the vector spin Larmor precession: continuous spin rotation about z-axis resonant RF pulses: a /2 spin rotation about x-axis /2 t=0 spin along x /2 t=0 spin along y /2 t=0 spin along z time time
Interatomic interactions contact interactions magnetic dipolar interactions B R
Interatomic interactions F 1 F total F 1 Low energy only s-wave collisions occur interactions characterized by scattering length Rotational symmetry: interactions depend on total spin, not its orientation 87 Rb: F 0 2 total F total a 0 = 5.39 nm a 2 = 5.31 nm interactions are repulsive slightly less repulsive ferromagnetic
Energy scales in a spinor Bose-Einstein condensate spin-independent contact interactions cn 0 2000 Hz, or 100 nk spin-dependent contact interactions quadratic Zeeman shift c n F 2 2 10 Hz, or 0.5 nk m z 1 2 q F z m z 0 m 1 2 z q F z
Phases and symmetries E c n F q F 2 2 2 z longitudinal axis ferromagnetic states transverse plane unmagnetized state Z2 U(1) SO(2) U(1) U(1) BEC 0 q 2 0 2 c n q Non-equilibrium (quantum) dynamics at a (quantum) phase transition
330 μm B z T hold = 30 ms T hold = 90 ms x Signal F y 0.4 0.2 0.0-0.2 i nft n Fx ify Ae C Asin( t )] probe -y -0.4 0 5 10 15 Point number 20 T hold = 150 ms T hold = 210 ms
Spontaneously formed ferromagnetism A T n F inhomogeneously broken symmetry ferromagnetic domains, large and small unmagnetized domain walls marking rapid reorientation A/ AMAX T hold = 30 60 90 120 150 180 210 ms
Spectrum of stable and unstable modes mz 0 Bogoliubov spectrum Gapless phonon (m=0 phase/density excitation) Spin excitations E ( k q)( k q 2) 2 2 2 S Energies scaled by c 2 n E 2 3 2 1 0-1 q = q -0.5 = 2.5 2.0 1.5 1 0.0 0.5 1.0 k 1.5 2.0 q>2: spin excitations are gapped by qq ( 2) 1>q>2: 0>q>1: q<0: broad, white instability broad, colored instability sharp instability at specific q 0
400 350 300 250 200 150 100 50 0 0 Tuning the amplifier 50 100 150 200 250 400 m Hold time= 170 ms 40 m Quench end point q = -2 Hz 0Hz 2 Hz 5 Hz 10 Hz
G( r) nf r r r rnfnr n r r n r spin-spin correlation function T = transverse measure of area of domain walls L = longitudinal Spontaneous symmetry breaking in a quenched ferromagnetic spinor BEC, Nature 443,312 (2006)
Quantum aspects of spontaneous magnetization mean-field theory atomic gas treated as a classical field (like classical E+B fields) predicts dynamical instability non-zero magnetization will grow does not explain symmetry breaking whence non-zero magnetization? quantum-field theory includes quantum fluctuations fluctuations provide symmetry-breaking seed G(0) = seed x gain quantify seed by extrapolation Lamacraft, PRL 98, 160404 (2007) time after quench
calculated amplification of quantum noise Amplification of spin fluctuations related to atomic scattering properties (Bloch, Chapman) noise is (nearly) quantum noise amplifier is (nearly) quantum-limited (@30dB gain)
Ferromagnetic spin textures generate helical spin pattern (uniform spin current) using inhomogeneous field 0 db z /dz Evolve w/o gradient zero-wavevector helix with gradient non-zero-wavevector helix
Ferromagnetic spin textures energy budget: spin-dependent contact interaction: c2 n F 2 ~ - 0.5 nk, minimized quadratic Zeeman shift: spin current kinetic energy 2 q qfz excess ~ 30 pk; λ = 60 μm 2 λ 50 μm 1 Hz
Dissolving spin textures A/ AMAX 30 60 90 120 150 180 30 60 90 120 150 180 ms initial texture = uniform initial texture = wound up
Long range vs short range order Initial texture Final texture
Long range vs short range order long range short range
Possible role of dynamical instabilities Lamacraft, Demler et al.: (arxiv:0710.1848, arxiv:0710.2499) spiral state is dynamically unstable But, where did the energy come from? 30 ms 180 ms
Dipolar interactions: magnetism in a quantum fluid M self-field: 0 0 F B @ 3 10 14 cm -3 B M g n 17 μg 2 2 energy per particle: d 0 FB U g n h x 12 Hz Comparison to other energy scales: total interaction energy: μ ~ h x 2000 Hz Pfau, Santos, Lewenstein, others: 52 Cr (6 μ B ), polar molecules (>137 μ B ) spin-dependent interaction energy: μ ~ h x 12 Hz Yi and Pu, PRL 97, 020401 (2006); Kawaguchi, Saito, Ueda PRL 97, 130404 (2006) 87 Rb is an essentially dipolar spinor quantum fluid
tempering dipolar interactions B R 3 1 2 1 U J cos 1, 2, 1, 2, 1, 2, 2 2 F zf z F xf x F yf y 2 Magic angle spinning (hard for us) Stochastic spin-flip narrowing: repeated RF (/2) pulses with random phase (easy for us)
Evolution with/without dipolar interactions with dipole without dipole with dipole without dipole
F=1 87 Rb gas at thermal equilibrium prepare fully depolarized thermal gas in uniform magnetic field lower temperature what happens? T unpolarized thermal gas T cn, /3 magnetized something 2 c n h20 2 Hz unmagnetized, scalar BEC 0 q0? q
preparing unmagnetized gases initial gas (T > T c ) /2 pulse decohere (diffusion + field gradient) repeat several times m z = +1 m z = 0 m z = -1 1:2:1 mixture 1:1:1 mixture
1:1:1 mixture thermal equilibrium phases F/ FMAX temperature T (BEC)
1:2:1 mixture thermal equilibrium phases F/ FMAX temperature T (BEC)
100 µm Low temperature phase of F=1 87 Rb: spatial correlations G tot spin-spin correlation ( r) r nf r r rnf r n r r n r Crystalline order!? F/ FMAX theory ideas from Joel Moore, Dung-Hai Lee, Ashvin Vishwanath, Jason Ho
Ising-like lattice Ising axis varies across cloud z y x
E1: Spinor BEC Jennie Guzman Sabrina Leslie Christopher Smallwood Mukund Vengalattore (James Higbie) (Lorraine Sadler) E2,E3: Cavity QED Thierry Botter Daniel Brooks Joseph Lowney Zhao-Yuan Ma Kater Murch Tom Purdy (Kevin Moore) (Subhadeep Gupta) E4: Ring-trap interferometry Joanne Daniels Ed Marti Ryan Olf Tony Oettl Enrico Vogt Tiger Wu
Open questions future directions Complete phase diagram of 87 Rb spin-1 gas q, T, magnetization (F z ), dimensionality Spin lattice better characterization of structure, domain boundaries, vortices? superfluid? Phase transitions, quantum and thermal quantum atom optics critical phenomena Spinor gas magnetometry surpass atomic shot noise; spatially resolved spin squeezing Strongly correlated systems: quantum magnetism in optical lattices collaboration w/moore, Vishwanath + others
GT ( x, z) phase separation occurs/symmetry broken spontaneously in disconnected radial bands
magnetic dipole energy Dipolar interactions in the spin helix 1 k min( r, r ) helix x y 2 / k helix U d 2 2 2 d 0 F B U g n U d
1.0 Condensate fraction of m=0 atoms Thermal population vs. T/Tc (Tc defined for the m=0 fraction) 4 0.8 0.6 3 0.4 0.2 x10 6 2 0.0 1 0.00 0.25 0.50 0.75 1.00 1.25 3.5 3.0 0.4 0.6 0.8 T/Tc 1.0 1.2 1.4 2.5 x10 6 2.0 1.5 1.0 Thermal populations vs Vodt 0.5 0.2 0.3 0.4 0.5 Vodt [V] 0.6 0.7 0.8
Ferromagnetic order Thermal population vs. T/Tc (Tc defined for the m=0 fraction) 4 200 3 x10 3 150 x10 6 2 100 50 Spin lattice order 1 0 0.4 0.6 0.8 1.0 0.4 0.6 0.8 T/Tc 1.0 1.2 1.4 Order parameters for the (1,2,1) mixtures and corresponding thermal fractions in the three components (RGB = +1, 0, -1)