Ψ TThhee SSppiinnoorr BBoossee--EEiinnsstteeiinn CCoonnddeennssaattee:: AA DDiippoollaarr MMaaggnneettiicc SSuuppeerrfflluuiidd

Dan Stamper-Kurn UC Berkeley, Physics Lawrence Berkeley National Laboratory, Materials Sciences Ψ Spinor gases

mF = 2 Energy mF = 1

F=2 mF = 0 magnetically Optically m = -1 F trappable trapped F=2 m = -2 spinor gas F

mF = -1

F=1 mF = 0

Optically mF = 1 trapped F=1 spinor gas B What do we want to know?

≈ χ χ χ ’ ≈ ρ ρ ρ ’ + + + + − +1,+1 +1,0 +1,−1 ∆ σ ,σ σ ,π σ ,σ ÷
∆ ÷
ρ = ρ ρ ρ χ = ∆ χ + χ χ − ÷ ∆ 0,+1 0,0 0,−1 ÷ π ,σ π ,π π ,σ ∆ρ ρ ρ ÷ ∆ ÷ « −1,+1 −1,0 −1,−1 ◊ ∆χ − + χ − χ − − ÷ « σ ,σ σ ,π σ ,σ ◊ FC F = cosθ spin F m=1 coh erences y φ “orientation”

x nematicity N m=2 coh erences “alignment” θ θ
≈ iφ −iφ ’ R∆e cos m =1 − e sin m = −1 ÷ « 2 2 ◊ + F′ = 2 1 1 1

12 2 2 y F =1

z x in this case small

signal 2 (# photons/pixel) = » + ( + + ε )ÿ S Nγ 1 A (
1) Fy Fy ⁄ ∝ n2D Direct imaging of Larmor precession

m

0 0 3 l a n g i s t s a r t n y o F c 0.5 - ~ e s a h p

0 k a e time (50 s/frame) P Aliased sampling: 2 x 20 kHz – Observed rate = 38.097(15) kHz Spinor gas magnetometry B N atoms

F F F φ Probe:

t = 0 t = τ

N N 0 0

N N B 0 1

N N 0 0

t = 0 t = τ

≈ > 1 ’ 1 1 Area-resolving: ∆B = ∆ ÷× = S × ∆ µ ÷ A « g B DTcoh n2 D ◊ Ttotal A Ttotal A Ψ Experimental demonstration

Clarify fundamental and technical limits on precision single-particle and collective light scattering, experimental tricks unbiased phase estimation methods

Measure background noise confirm areal and temporal scaling of sensitivity

Measure localized magnetic field quantum-mechanical diffusion of magnetization actually measure

limits to dynamic range magnetic background field with uncontrolled long-range inhomogeneities actually measure fictitious Z eeman shift from focused, off- resonance laser Ψ —Field“ measurements

B

AC Stark shift = “Fictitious” magnetic field Comparing atoms & SQUIDs Existing low freq.

magnetometers

actual duty cycle

D ~ 0.003 (SQL)

unity duty cycle

D = 1

noise in backg round images

—High resolution magnetometry with a spinor BEC,“ arXiv:cond-mat/0612685 Ψ Bose-Einstein condensates with spin

mF = 2 Energy mF = 1

F=2 mF = 0 magnetically trappable, Optically mF = -1 trapped F=2 mixtures decay m = -2 87 spinor gas F Long lived in Rb (Cornell group)

mF = -1

F=1 mF = 0

Optically mF = 1 trapped F=1 spinor gas B

Quantum fluids are described by vector order parameter (somewhat similar to 3He) FFC Ψ(x) Ψ(x) so what? Symmetries of order parameter broken symmetries + excitations, phase transitions, topologies

Symmetries in interactions

F=1 e.g. F=1 spinor, low B F=1

Ftot = 0 Ftot = 1 Ftot = 2

ultracold gas a0 no interactions a2 (s-wave) leads to spin waves single spatial mode a not allowed a (e.g. BEC) 0 2 Energy Scales in a Spinor Condensate Ψ (circa 2006)

• Spin dependent interaction energy

FC 2 = E spin c2 n F 8 Hz, or 400 picokelvin!

∝ ∆a Klausen, Bohn, Greene, PRA 64, 053602 (2001)

• Non-linear (quadratic) Zeeman shift E = q F 2 quad z -1 + +1 0 + 0 q = (72 Hz / G 2 )h × B 2

isotropic anisotropic interaction energy dominates quadratic term dominates

2 0 = q (∝ B ) q 2 c2 n Quenching across a symmetry-breaking transition James Higbie, Sabrina Leslie, Lorraine Sadler, Mukund Vengalattore

FC 2 = − + 2 E c F q Fz

gradual (2nd order) transition = unmagnetized gas Fz 0 Tc (for BEC)

) T (

e unmagnetized, scalar state r u t a r

e ferromagnetic state p

m e T

BEC

q = 2 c quadratic Zeeman shift (q) Quench dynamics; phase-ordering kinetics

3

3 0

m

d l o h d l o h s m 0 5 1 = T s m 0 1 2 = T

r e b m u n t n i o P

0 1 5 0 0 2 5 1 y -

e b o r p

4 . 0 -

y 2 . 0 -

F

∝

x

0 . 0

l a n g i

S

( n i s A C ] ) t 2 . 0

× + ω φ

x y

F n F n e A F i T

( ) 4 . 0

= + ∝

i φ

d l o h d l o h B T s m 0 3 = T s m 0 9 =

z Spontaneously formed =
φ ferromagnetism A n FT • inhomogeneously broken symmetry • ferromagnetic domains, large and small • unmagnetized domain walls marking rapid reorientation

φ

A / AMAX

Thold = 30 60 90 120 150 180 210 ms Spontaneously formed =
φ ferromagnetism A n FT • inhomogeneously broken symmetry • ferromagnetic domains, large and small Continuous spin texture • unmagnetized domain walls marking rapid ~50 micron pitch reorientation

φ

Alternating spin domains

and domain walls

A / AMAX

30 ms 60 90 120 150 180 210 Thold = 30 60 90 120 150 180 210 ms Ψ Ferromagnetism via spinodal decomposition m = 0 + m = ±1 m =1 + m = −1

Polar (Na) Separate Mix MIT

Ferro (Rb) Mix Separate G-Tech

= +

mx 1

= mz 0 + = = − mx 1

Exponential timescale

τ = > / ∆E =13.7(3)ms S = − = + mx 1 mx 1

Unmagnetized domain boundary of size πξ = 8.3µm ≅ ξ = µ Ferromagnetic domain of size S b S 2.6 m Timmermans, PRL 81, 5718 (1998); Saito and Ueda, PRA 72, 023610 (2005) Ψ Spectrum of stable and unstable modes

Bogoliubov spectrum = Gapless phonon (m=0 phase/density excitation) mz 0 Spin excitations Energies E 2 = (k 2 + q)(k 2 + q − 2) S scaled by c2n

3 qq = =q - =102. 501 2 E2 1 0 -1

0.0 0.5 1.0 1.5 2.0 k

q>2: spin excitations are gapped by q(q − 2) 1>q>2: broad, —white“ instability 0>q>1: broad, —colored“ instability q<0: sharp instability at specific q≠0 »
*
ÿ F T ,L (r +δ r)n(r +δ r)F T ,L (r)n(r) spin-spin …ƒ Ÿ G (δ r) = Re r correlation T ,L … +δ Ÿ … ƒ n(r r)n(r) Ÿ function r ⁄

rise time = 15(4) ms – compare to 13.7(3) ms

measure of area of domain walls GT (x, z)

phase separation occurs/symmetry broken spontaneously in disconnected radial bands Ψ Topological defect formation across a symmetry- breaking phase transition

Kibble (1976), Zurek (1985)

5π φ = “causal horizon” 3 φ = 0

4π φ = 3 φ = 0 2π φ = 3

“Cosmology in the laboratory”

Liquid crystals [Chuang et al, Science 251, 1336 (1991)] Superfluid helium [Hendry et al., Nature 368, 315 (1994)] Spontaneously formed spin vortices
F T candidates:

Mermin-Ho vortex (meron)

mz=0 core

F L “Polar core” spin vortex

Thold = 150 ms Nature 443, 312 (2006) Spontaneously formed spin vortices

candidates:

≈ × ’ FFC a(r) 1 Ψ = ∆ × −iφ ÷ ∆b(r) e ÷ ∆ −2iφ ÷ «c(r) × e ◊

Mermin-Ho vortex (meron)

mz=0 core

≈a(r) × eiφ ’ FFC ∆ ÷ Ψ = ∆b(r) × 1 ÷ ∆ −iφ ÷ «c(r) × e ◊ “Polar core” spin vortex Broken chiral symmetry; Saito, Kawaguchi, Ueda, PRL 96, 065302 (2006) Ferromagnetic spin textures generate helical spin pattern (uniform spin current) using inhomogeneous field

Evolve dB /dz 0 z

t n e i d a

r g

o / w

t n e i d a r g

h t i w Ψ Ferromagnetic spin textures clean starting point for studying dynamics

energy budget: spin-dependent contact interaction:

FC 2 − c2 n F ~ - 0.5 nK, minimized

quadratic Zeeman shift:

2 q qF = excess ~ 30 pK; N = 60 Om z 2

spin current kinetic energy

N ≥ 50 Om ν ≤ 1 Hz φ Ψ Dissolving spin textures

A / AMAX

30 60 90 120 150 180 30 60 90 1 20 1 50 180 ms initial texture = uniform initial texture = wound up Dipolar interactions: magnetism in a quantum fluid

@ 4 1014 cm-3 M ≈ µ = (µ µ ) self-field: B 0M 0 gF B n 23 G

= µ 2 µ 2 energy per particle: E ( 0 gF B )n kB x 0.8 nK

Comparison to other energy scales:

total interaction energy: O ~ kB x 200 nK Ω 52 Pfau, Santos, Lewenstein, others: Cr (6 OB), polar molecules (>137 OB)

spin-dependent interaction energy: O ~ kB x 0.8 nK Ω Yi and Pu, PRL 97, 020401 (2006); Kawaguchi, Saito, Ueda PRL 97, 130404 (2006)

ferromagnetic spin texture: K.E. < kB x 30 pK Ψ Present/future directions/speculation

Equilibrium phase diagram of spinor BEC vs. q, T, … effects of long-range, anisotropic interactions effects of dimensionality

Phase transitions, quantum and thermal ascertain role of quantum vs thermal fluctuations: 2 temperatures quantum atom optics: is it still a BEC? critical exponents

Phenomenology of spinor BEC dynamics domain walls, various spin vortices, interactions b/w them

Spinor gas magnetometry surpass atomic shot noise; spatially resolved spin squeezing applications to …? Ψ

Daniel Brooks, Jennie Guzman, Sabrina Leslie, Kevin Moore, Kater Murch, Tom Purdy, Lorraine Sadler, Ed Marti, Ryan Olf, Subhadeep Gupta, Mukund Vengalattore

Looking for great students and postdocs http://physics.berkeley.edu/research/ultracold

AFOSR