austen lamacraft- low energy dynamics of spinor condensates
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faculty.virginia.edu/austen/
Low energy dynamicsof spinor condensates
AustenAusten LamacraftLamacraft
Cornell, October 2009Cornell, October 2009
Phys. Rev. A 77, 063622 (2008)
& arXiv:0909.5620
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Outline
Magnetism in ultracold atomic physics
– Example of spin-1 Bose gas
Dynamics of Bose ferromagnets – Berkeley experiment and role of dipolar forces
– Nonequilibrium description seems necessary!
Dynamics of novel phases at higher spin
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Magnetism in atomic gases: what’s new?
In the solid state we (mostly) care about the quantummechanics of electrons. These are fermions
By contrast, atoms (considered as particles) may be
bosons or fermions Possibility of Bose-Einstein condensation - bosons
accumulate in lowest energy state
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Exotic magnetism
87Rb has nuclear spin I=3/2, electron spin S=1/2
– Possible total spin F=1 or 2
What are magnetic properties of F=1 or 2 Bose gas?
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Magnetism in Bose gases
BEC: (nearly) all atoms sit in same quantum state
This state is called the condensate wavefunction
But what if lowest energy state is degenerate?
Condensate wavefunction is a spin vector (spinor) and must pick a direction in spin space
Bose condensates with spin are always magnets
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Why higher spin is fun
Spin 1/2 (e.g. of electron) points in some direction
– To make electron magnetism more interesting caninvoke non-trivial arrangements on lattice (e.g. Néel)
Spin 1 doesn’t necessarily “point” anywhere
– spin-1 matrices
– Yet evidently there is still a director or nematic axis!
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Which spin state wins?
Must consider interatomic interactions
Atoms can collide with total spin 0 or 2
– Total spin 1? Antisymmetric and blocked by Bose statistics
– Spin 2
– Spin 0
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Spin dependent interactions
Energy of state includes a piece
– For c 2 <0 (e.g. 87Rb): maximize Ferromagnet
– For c 2 >0 (e.g. 23Na): minimize Polar state
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Mean field ground states: spin 1
Work in cartesian components where
Ferromagnet
– maximal for
Polar state
– minimal for
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Order parameter manifolds
Ferromagnet
– Order parameter manifold is SO(3) : any orthogonal triad
specifies a rotationS 3 with opposite points identified (RP 3 )
Polar
– Since any state may be written
– Notice that
“twisted sphere bundle over the circle”
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The Bose ferromagnet: 87Rb
– Stamper-Kurn group, Berkeley
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(( ))
The Mermin-Ho relation
Geometrical interpretation: constant on vorticity lines
The velocity
is not irrotational as in single-component case
On the spin coherent states
Vorticity lines fill fluid, not confined to vortices!
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Equations of motion - motivation IEquations of motion - motivation I
Practical question: how to study dynamics?
TDGPE usually sufficient in dilute systems
Avoid working with 2s+1 component spinor?
Would prefer a description just of spinwaves andsuperfluid flow , even at high spin
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Equations of motion - motivation IIEquations of motion - motivation II
Normal fluids
approximately incompressible at low Mach number
Scalar superfluids
Leaves only possibility of isolated vortex lines
In the spinor case this limit is non-trivial!
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Equations of motion of Bose Ferromagnet
AL, PRA 77 63622 (2008)
SpinwavesSpinwaves have quadratic dispersion around uniform statehave quadratic dispersion around uniform state
(( ))
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Relevance of dipolar forces?
M.M. VengalttoreVengalttore et al.et al. arXiv:0901.3800
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Easily include dipolar forces
Larmor frequency dwarfs other scales
– Average dipole-dipole energy over rapid precession
– q =0 part is easy axis anisotropy
(exercise in demagnetizing factors)
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Effect on spinwaves
Boundary between stability and instabilityBoundary between stability and instability
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Higher spin phases - analogy to non-
collinear Néel states
Two sublattice Néel state has two parameter OP
Three sublattice Néel state has three!
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Spin Lagrangian: <S >=0 phases
Express spin configuration in terms of rotation fromreference state
For polar phase
O(3) sigma model (Zhou, 2001)
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Describing higher spin magnets:the Majorana representation
A geometric way to visualize arbitrary spin states
Generalization of the Bloch sphere
Spin s 2s points on the unit sphere
“Constellation”
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Spin states as symmetric spinors
Form a spin s from symmetric states of 2s spins 1/2
General state a totally symmetric spinor
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The Majorana polynomial
This spinor may be written as
– Proof: Consider the polynomial
Unique up to phase and magnitude of spinors
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Meaning of the roots
unchanged if we normalize
stereographic projection to plane
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Some Examples
Spin s coherent state: all 2s points coincide
– i.e. a rotation of
Regular polygon at the equator
For s=1 includes polar state (“headless vector”)
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Ordering in spin-2 condensates
– Barnett, Turner, Demler PRL 2006
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Directions of moment in ferromagnet 2 variables
Same goes for director n in polar state with spin 1
For nematic and tetrahedral states of spin 2
– Need to specify a full rotation matrix R starting from ref. state – Order parameter manifold in fact forH
stabilizer subgroup of ref. state (global topology)
Order parameter dynamics
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Order parameter manifold
General parametrization
– representation of rotation, reference spinor Constellation may be unchanged under some
subgroup (stabilizer or isotropy subgroup)
– Ferromagnet
– Polar (spin 1)
– Tetrahedral state (spin 2)
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Metric on the order parameter manifold
Consider two states and
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Conjugate variables
< S >=0 on g.s. manifold of “nematic” states (c.f. ferromagnet)
– In ferromagnet transverse spin deviations are canonically conjugate
– In nematic states conjugate variables lie off the manifold
– these deviations have a “stiffness”
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Parameterizing conjugate variables
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Eliminate l
Or in terms of
– Identical action describes non-collinear magnets!
Final form of spinwave Lagrangian
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Summary
Magnetism in ultracold atomic gases
– Offers not just a new setting but new magnetic states
Dynamics of Bose ferromagnets; dipolar forces – Understanding (quasi-)equilibrium states requires more work
More exotic magnets and their dynamics
– Relation to the dynamics of non-collinear Néel states
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Polar condensates and a paradox
For c 2 >0 minimize <F>. Pick a quantization (z ) axis
Problem: for a more general state
seek ground state of
Must be a singlet
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The parable of the chair
Chair has (rotational) Hamiltonian
States very unlike the chair we see
Tiny energy differences swamped by perturbations
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Symmetry breaking in atomic gases
The same goes for
From excitations on top of singlet ground state, wecan build a state
at little cost, with a definite axis (but still <F>=0)
Spontaneous symmetry breaking