Download Theory of Spin-Orbit-Coupled Cold Atomic Systems

Theory of Spin-Orbit-Coupled
Cold Atomic Systems
Victor Galitski
Joint Quantum Institute and Physics Department
University of Maryland
Acknowledgements: Work supported by NSF-PFC & ARO-MURI
Collaborators
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Outline
• Synthetic spin-orbit coupling schemes, old and new and many more
• A zoo of spin-orbit-coupled BECs: (mostly) theory
• How to create vortices in spin-orbit-coupled BECs?
• Topological optical lattices & lattice quantum Hall states
• Practical applications: Quantum gravimetry and interferometry
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Atom in a r-dependent laser field. Tripod scheme
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Two equivalent descriptions
To get the effective Hamiltonian...
1. Diagonalize Ĥa−l via a unitary rotation, Ĥa−l → Û †(r)Ĥa−lÛ (r)
2. Project the result onto the dark subspace
�
�
− �2 ∇2
†
Ĥeff = P̂darkÛ (r)
Û (r)P̂dark
2m
• Picture I: Particle in a non-Abelian “gauge field”
�2
1 �
i
Ĥeff =
−i�∇ − A (r)σ̂i
2m
• Picture II: Spin-orbit coupled particle
p2
Ĥeff (p) =
+ b(p) · σ̂
2m
�
�
For the tripod scheme in 2D: b(p) = 0, vpx, v �py
SU(3) “spin”-orbit coupling
A different choice of phases in the tetragonal scheme leads to a Hamiltonian that can NOT be spanned by Pauli matrices alone, but may be
spanned by 3 × 3 Gell-Mann matrices
�
�
k λ̂
λ̂i, λ̂j = ifij
k
Gell-mann matrices are generators of su(3).
su(3) spin-orbit coupled system:
8
�
p2
Ĥsu(3) =
+
bi(p)λ̂i
2m i=1
No analogue in condensed matter (or any other matter)...
G. Boyd, B. Anderson, & V. Galitski, tbp
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How to create vortices in a BEC?
How to create vortices in a BEC?
How to create vortices in a BEC?
• Rotation: If an ordinary BEC is stirred
by a laser “spoon” or the anisotropic
trap is rotated:
•– Rotation:
If an ordinary
is stirred by a laser “spoon” or the
The Hamiltonian
in theBEC
rotating
anisotropic
trap is rotated:
frame is time-independent:
– The Hamiltonian in the rotating frame is time-independent:
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HRF = H0 − Ω · L
H
=H −Ω·L
– Equilibrium stat. mechanics RF
apply 0
Equilibrium
stat. mechanics apply
– –
Vortex
lattice appears
– Vortexmagnetic
lattice appears
• Synthetic
field for neutral
• Synthetic
field formagnetic
neutral atoms leads to an effective magatoms
leads magnetic
to an effective
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netic
for states
dressed states
field
for field
dressed
Rotating the trap generally does NOT work for spin-orbit BECs, as there
is no rotating frame where the Hamiltonian is time-independent (nonequilibrium physics, heating). However, to combine synthetic spin-orbit
coupling with synthetic magnetic field should work.
ing scheme creates an effective gauge field
Heff
p2
=
+ vpxσ̂z + Ωσ̂x + δ(y)σ̂z
2m
Spin-orbit coupling + spatially-dependent detuning
troducing
spatially-dependent
detuning
δ(r) δ(
inr)I.inSpielman’s
exist• Introducing
spatially-dependent
detuning
I. Spielman’s
existg scheme
an effective
gaugegauge
field field
ing creates
scheme creates
an effective
Heff
2
p2
p
= vpxσ̂
+z vp
σ̂z +
Ωσ̂δ(y)σ̂
x + δ(y)σ̂
= Heff +
+xΩσ̂
x+
z z
2m
2m
• The combination of the SOC and synthetic gauge field yields two
main effects:
1. Spatial separation of the left- and right-movers
2. Synthetic mag. field for each component and vortex nucleation
• The combination of the SOC and synthetic gauge field yields two
main effects:
he combination of the SOC and synthetic gauge field yields two
1. Spatial separation of the left- and right-movers
ain effects:
2. Synthetic mag. field for each component and vortex nucleation
Spatial separation of the left- and right-movers
Parity effect in vortex nucleation
Due to symmetry of the effective gauge field with respect to reflection
about the y = 0 axis and almost spin-independent interactions, an interesting parity effect is observed (in GPE simulations):
the number of vortices is the same in both components.
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Synthetic gauge fields
Lattice formed by
3 standing waves
Trap
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Steven Chu
From Steven Chu group web-site at Stanford:
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Existing gravimeters do not probe time-dependent gravity/acceleration,
ȧ(t). But it is possible with synthetic gauge fields [B. Anderson, J.Taylor,
& VG, Phys. Rev. A 83, 031602(R) (2011)].
“Vector potential:”
An(p) = i �un(p)|∂pun(p)�
Dual “magnetic field:” Bdual
(p) = ∇p × An(p)
n
Trajectories
for “spin-up”
Semiclassical Eqs. of motion
(Luttinger,
50’s; and
Niu, Haldane, 2000’s):
“spin-down” differ depending
∂En(p)
ṙon
=time-dependent
+ Bdual
(p) × ṗ
n gravity!
∂p
ṗ = −eE + eB × ṙ
Summary of the recent progress
• Dressed states with cold atoms potentially host a much richer variety of spin-orbit structures than that available in solids (Rashba,
Dresselhaus, “Weyl,” su(3)-SOC, ...)
• Synthetic spin-structures may be practically useful for quantum interferometry
• Abelian spin-orbit BECs have already been observed. Theory predicts macroscopically-entangled states in non-Abelian SO-BECs
(staying tuned for new experiments...)
• Synthetic SOC + synthetic magnetic field = new vortex structures
• SO coupling on a lattice may lead to lattice quantum Hall effect
• Future work: Non-equilibrium physics + synthetic fields. Cold atoms
may be ideal candidates to realize Floquet topological insulators.