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Ana Maria Rey
Boulder Summer School, July 19-23 (2021)
âą Brief overview of alkaline earth-atoms and atomic clocksâą Collisions and SU(N) Interactions âą Density shifts in spin polarized gasesâą Probing SU(N) orbital magnetismâą Spin-Orbit Coupling
Dark Timepulse pulse
A TALE OF TWIN ELECTRONS
Fermionic isotopes have nuclear spin I > 0.
Why?: Bosonic isotopes have nuclei with even number of protons and neutrons: I=0Fermionic Isotopes have even number of protons and odd number of
neutrons
Nucleus
I
Electrons
âȘLong lived metastable 3P0 state: 3P 0-1S 0 is a dipole and spin forbidden
transition with a linewidth ~ mHz . â Spectral resolution
Dipole (đœ=0â đœ=0 ) and Spin forbidden (đ=0â đ=1) transition
Unique atomic structure
Strontium: 87 Sr
Once set, it swings during
the entire age of the
universe
1S0 (g)
3P0 (e) Linewidth
Dn0~ mHzn0=5x1014 Hz
lifetime ~ 102 sec
Quality factor: Q=n0/Dn0 >1017
Metastable states
âȘLong lived metastable 3P0 state: 3P 0-1S 0 is a dipole and spin forbidden
transition with a linewidth ~ mHz . â Spectral resolution
âȘ Total electronic angular momentum J=0 â Only nuclear spin I
Unique atomic structure
Strontium: 87 Sr
΀đ đ΀đ đ ΀đ đ ΀đ đ ΀đ đ ΀âđ đ ΀âđ đ ΀âđ đ ΀âđ đ ΀âđ đ
I=9/2 10 different states
ground (g)
excited (e)
(Jun Ye : Presentation)
Why? Because Hyperfine Interactions H = AI â đœ
3P0= 3đ·đ(đ)
+ a 3đ·đ(đ)
+ b 3đ·đ(đ)
+ c 1đ·đ(đ)
How about Strontium: 88 Sr?
Strontium: 87 Sr
(Jun Ye : Presentation)
Counter
Oscillator
Optical
Light Source
Atoms
Oscillator
Counter
N. Ramsey. Nobel
prize 1989
w0g
e
wL
d= (wL- w0 )
Detuning
See handwritten notes
w0g
e
wL
N. Ramsey. Nobel
prize 1989
e
g
d= (wL- w0 )
Detuning
z
xy
e-atom
s
Contrast
Phase
ÎŽ t
z
y
âΩ áđđ„
đ
đ = ΩđĄ
ââđż áđđ§
x
đș = 1
2{0, sin đ, â cos đ} đ â(đ) = âđ
2 sin đđâđđżđ đ đ§(đ) = 1
2 sin đ cos(đżđ)
Pulse area
About 1000 times better than current cesium standard
âą Accuracy approaching 10-18
âą Low stability: only single ion
Neither gain nor loseone second in some 15billion yearsâroughlythe age of the universe.
âą Accuracy at 10-18
âą High stability: 103-4 Many atoms
What happens in the real experiment with many atoms?
S
Large collective spin: Better signal to noise
No interaction: All the spins precess collectively
Clock
΀đ đ ΀đ đ ΀đ đ ΀đ đ ΀đ đ ΀âđ đ ΀âđ đ ΀âđ đ ΀âđ đ ΀âđ đ
đ đ§(đ) = đ”đsin đ cos(đżđ)
đșđ,đ,đ =đ
đ
đ
à·đđđ,đ,đ
Requirement:
â All experience same Rabi frequency
(Resource for spin-orbit coupling)
â All have same detunings: No Doppler&
Stark Shifts: Deep Magic Lattice
â Atomic Collisions?
Interactions:
âą Degraded signal: Even in identical fermionic atoms. In 2008 gave rise to the second largest
uncertainty to the 10 -16 error budget
JILA:G. Campbell et al Science 324, 360 (09)
NIST: N. Lemke et al PRL 103,063001 (09)
w0 wâ w0
Atomic collisions change the frequency. Packing more atoms makes the error worse
More atoms better signal to noise
Need to
understand
interactions
Many-Body
Physics
Spin
orbitalCharge
Nuclear spins
Spin-orbit
coupling
3P
0,1S
0
Strongly correlated materials
đ(r) â đđđđ§ + đ(đ)đđđđ
đGoal: find scattered wave f: scattering amplitude
Solve Schrödinger
ââ2
2đđ»đč2đ â
â2
2đđ»đ2đ + đ(đ)đ=Eđ
r=r1-r2 relative coordinate, m=đ/2: reduced mass
R=(r1+r2)/2 Center of Mass coordinate, đ = 2đ, Total mass
đ đč, đ = ΚđȘđŽ(đ)Κ (đ«) cross section
(r) =Ïâ,đđ â,đ đ, đž đâ,đ (đ, đ)
There is only a phase shift at long range!!
â2
2đ
1
đ2đ2 đ
đđ+ đ2 â
â(â+1)
đ2đ â,đ= đ(đ)đ â,đ
â2đ2
2đ= đž
Solve Schrödinger equation for each l to get đżâ, đ¶â,đ
đâ =(đ2đđżâ â 1)
2đđđ Ξ =
â
(2â + 1)đâ( cos đ) đâ
đ â,đ(đ â â) âđ¶â,đđđ
sin đđ â âđ
2â đżâ
Angular momentum is quantized:
đđđ(đ.đą)
Van der Waals-C6/r6
Chemical bonding
Centrifugal barrier
l=1
l=0 r
đđ đ = đ(đ) +âđâ(â+đ)
đđđđ
l = 0, 1, 2⊠s-, p-, waves, âŠ
đżâđâ đŽâ(đŽâđ)
2â
(1) At ultra-cold temperatures đ â 0 l =0 collisions dominate!!
đŽ0 = đ âscattering lengthâ Characterize s-wave collisions
đŽ1 = đ3 âscattering volume â Characterize p-wave collisions
s-wave , l=0,
Spatially symmetric
p-wave , l=1
Spatially anti-symmetric
r(a0)
Pauli Exclusion principle
But Identical fermions
No low energy (l =0)collisions:
Only (l =1, p-wave): cost energy
~30 mK
Identical bosons: evenIdentical fermions: odd
(2) Quantum statistics matter:
I like Fermions
âą Two interaction potentials V and Vps are equivalent if they have the same scattering length
âą So: after measuring đ, đ for the real system, we can model with a very simple potential.
Huang and Yang, Phys. Rev. 105, 767 (1957)E. Fermi Ricerca Scientifica, 7: 13â52 (1936) Breit Phys. Rev. 71, 215 (1947), Blatt and Weisskopf Theoretical Nuclear Physics (Wiley, New York, 1952), pp. 74â75Idziaszek PRA 79, 062701 (2009)
đ đđđ đâČ =4đâ2
đż3đ
â
2â + 1 đŽâ2â+1đâđâČâđâ( đ â đ
âČ)
đż3
Vps
For s-wave collisions
Note, it is suppressed at low energies
đ đœđđâ=đ đâČ =
4đâ2
đż3đđ
For p-wave collisions đ đœđđâ=đ đâČ =
12đâ2
đż3đđ3(đ â đâČ)
Be careful with regularization to avoid divergencies
See uploaded by Anjun Chu notes for details
See handwritten notes
Electrons independent of nuclear spin â collisions independent of nuclear spin.(except via Fermi statistics).
I
J=0
M. Cazalilla and A. M Rey Reports on Progress in Physics 77, 124401 (2014)
collide
See handwritten notes
Electrons independent of nuclear spin â collisions independent of nuclear spin.(except via Fermi statistics).
Up to N=2I+1=10SU(N=2I+1) symmetry:
Nuclear spin independent scattering parameters
No spin changing collisions.
I
J=0
M. Cazalilla and A. M Rey Reports on Progress in Physics 77, 124401 (2014)
collide
đđđ± ~ đđđ
± 3
~đđđâ
2
đż
Fermionic Field operator ΚđŒđ đč , ΚđœđâČâ (đčâČ) = đż(đč â đčâČ)đżđŒđœđżđ,đâČ
đŒ, đœ = đ, đ
đ = 9/2
Rey et al Annals of Physics 340, 311(2014)
đŁđŒ,đœ =6
Rey et al Annals of Physics 340, 311(2014)
1S0 (g)3P0 (e) Weak interactions simplify physics
Thermal
wR ~500 Hz,
E(n1x
,n1y
)
E(n2x
,n2y
)
E(n4x
,n4y
)
Interaction Energy~ Hz
Energy lattice:âŠâŠâŠâŠbut frozen motional levels
NO mode changing collisions
Hamiltonian can be reduced to a Spin model with long range couplings
n1
n2
Rey et al PRL (2009), Gibble PRL (2009), Yu et al PRL (2010)
đđđâ
đđđ+ â đđđ
3
đđđ+ â đđđ
3
đđđ+ â đđđ
3
Only p-wave interactions relevant. No coupling to singlet
What happens if Ω1 â Ω2?
g
e
È Û§đĄâ È Û§đĄ+ È Û§đĄ0 È Û§đ
n1
n2
Rey et al PRL (2009), Gibble PRL (2009), Yu et al PRL (2010)
đđđâ
đđđ+ â đđđ
3
đđđ+ â đđđ
3
đđđ+ â đđđ
3
What happens if Ω1 â Ω2 = ÎΩ â 0?
g
e
È Û§đĄâ È Û§đĄ+ È Û§đĄ0 È Û§đ
D-D
+
D-+-
D+
=
-
+
eg
eg
ee
gg
PS
U
V
V
V
H
02/2/
02/2/
2/2/0
2/2/0
d
d Initially thought to bethe mechanism leadingto density shifts butwas ruled out later
Atoms as a quantum magnet: frozen in energy space
M. Martin et al, Science 341, 632 (2013), Rey et al Annals of Physics 340, 311(2014)
Delocalized modes: Long range interactions
đ» â
Rey et al Annals of Physics 340, 311(2014)
đ đœđđâ=đ đ â
đ3
Volume
â2đ2
2đâ
đ3
Area đâđđ đđ”đ â
đ3
đâđđ
â đđ
(đâđđ )2đđ”đ
đđ”đ
đŁđŒ,đœ =6
Temperature Independent
Long Range!!!
Atoms as a quantum magnet: frozen in energy space
M. Martin et al, Science 341, 632 (2013), Rey et al Annals of Physics 340, 311(2014)
đ» â
àŽ„đ and àŽ„đȘ : Mean P-wave Interaction parameters
đ» â âđż áđđ§ + àŽ„đ ( áđđ§)2 + àŽ„đȘ đ áđđ§
áđđŒ =
đ=1
đ”
áđđđđŒ
Collective spin model
àŽ„đȘ = (àŽ„đœ đđ+
-àŽ„đœ đđ+
)/2
àŽ„đ = (àŽ„đœđđ+ -2àŽ„đœ đđ
++ àŽ„đœ đđ
+)/2
Delocalized modes: Long range interactions
âđż áđđ§ + àŽ„đ ( áđđ§)2 + àŽ„đ¶ đ áđđ§ â âđż áđđ§ + 2ÏÌ áđđ§ áđđ§ + àŽ„đ¶ đ áđđ§
đż â đż â 2ÏÌ áđđ§ + àŽ„đ¶ đDensity Shift
Spin precesses at a rate that depends on atom number and
excitation fraction
Controlled by pulse area đ
Excitation fraction: (1-đđšđŹđœ )/đ
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Theory vs experiment
âđż ~đ” àŽ„đȘ â àŽ„đ đđšđŹđœ
2
-2
4
6
4
0
-4
-6
-8
âą q controlled by first pulse
âą Determine p-wave interaction parameters
âą Operate sweet spot: no density shift
M. Martin et al, Science 341, 632 (2013), Rey et al Annals of Physics 340, 311(2014)
Lemke et al PRL 107, 103902 (2011) Ludlow et al, Phys. Rev. A 84, 052724 (2011)In Yb:
Contrast Phase
đ â(đ) = đ â(đ) đâđ(đż+âđż)đ
M. Martin et al, Science 341, 632 (2013), Rey et al Annals of Physics 340, 311(2014)
âą At the mean field level interaction only affect the precession rate.
ButâŠ.. in the experiments there are many pancakes with different atom number.
Mean-field interactions causes the pancakes with more atoms to precess faster.
1D
âą Atom number decay also leads to decay of the amplitude
Signal adds â contrast
decay due to dephasing
Contrast Phase
đ â(đ) = đ â(đ) đâđ(đż+âđż)đ
Ramsey fringe decay vs. the spin tipping angle
Shift
Excitation fraction
To eliminate the effect of decay we normalize the amplitude
with atom numberN
orm
aliz
ed A
mp
litu
de
time (ms)
Failure of mean field theory
M. Martin et al, Science 341, 632 (2013)
Symbols: Exp data lines: mean field
2p/9 p/3 p/2 2p/3
Contrast: đđ„ 2 + đđŠ 2
M. Martin et al, Science 341, 632 (2013), Rey et al Annals of Physics 340, 311(2014)
No
rmal
ized
Co
ntr
ast
Quantum correlations induce faster decay of the amplitude
See: Polkovnikov Annals of Physics 325,1790 (2010), J. Schachenmayer et al PRX 5, 011022 (2015)
We can solve for the full many body solution using the Truncated Wigner Approximation:
Average over random initial conditions that account for quantum noise distribution
áđđ„đŠ =N/2 áđđŠ,đ§ =0
Coherent Spin states È Û§â + È Û§â
2
â áđđŠ,đ§ = đ4
z
xy
Many-Body
Physics
Spin
orbitalCharge
Nuclear spins
Spin-orbit
coupling
3P
0,1S
0
Strongly correlated materials
Fermionic Field operator
đđđ
đ , đđâČđâČâ (đâČ) = đż(đ â đâČ)đżđ,đâČđżđ,đâČ
đ, đâČ = đ, đ
đâČ, đ
Ground-ground
Excited-Excited
Direct
Exchange g
e
9/2 7/2
G. Cappellini et al PRL 113, 120402 (2014) && F. Scazza et al Nature Physics 10, pages 779 (2014)
đđđ„ =đđđ+ â đđđ
â
2Exchange
đ(0) = È Û§đ â, đ â =È Û§+ + È Û§â
2
At Îđ” = 0
Eigenstates of Interaction
+ =È Û§đđ +È Û§đđ
2â
È Û§ââ âÈ Û§ââ
2
â =È Û§đđ âÈ Û§đđ
2â
È Û§ââ +È Û§ââ
2
đđđ± â
4đâ2
đđđđ±
Lattice site
G. Cappellini et al PRL 113, 120402 (2014) && F. Scazza et al Nature Physics 10, pages 779 (2014)
đđđ„ =đđđ+ â đđđ
â
2Exchange
đ(0) = È Û§đ â, đ â =È Û§+ + È Û§â
2
At Îđ” = 0
Eigenstates of Interaction
+ =È Û§đđ +È Û§đđ
2â
È Û§ââ âÈ Û§ââ
2
â =È Û§đđ âÈ Û§đđ
2â
È Û§ââ +È Û§ââ
2
đđđ± â
4đâ2
đđđđ±
Lattice site
Orbital Feshbach ResonancePRL 115, 135301( 2015)PRL 115, 265302 (2015)PRL 115, 265301 (2015)
Depends on magnetic field
Ultra-stable clock laser
Atoms trapped in array of disk-shaped pancake
spin (nuclear), 10 flavors:
orbital
(electronic)
ClockB-field Spectators
Statistical mixture
Zhang et al Science, 345,1467 (2014)
Spectators generate a density shift
âą SU(N): density shift only
depends on the total
number of spectators not
on its distribution ΀đ đ
΀đ đ ΀đ đ ΀đ đ ΀đ đ ΀âđ đ ΀âđ đ ΀âđ đ ΀âđ đ ΀âđ đ
đđđ°~đđ°àŽ„đȘ â đđšđŹđœ àŽ„đ
đđ°
Interrogated
Interrogated atoms p-wave shift
đđđș = àŽ„đČđđș
ÎÎœ = ÎđđŒ + Îđđ
đđș : đđ©đđđđđđšđ«đŹ
Both s and p-wave
contributions
Zhang et al Science, 345,1467 (2014)
Îđđ
đđđ°
t t2t1
Normalized to 4000
interrogated atoms
Independent of distribution or
temperature at the 3% level
Slope depends
only on p-wave
Zhang et al Science, 345,1467 (2014)
Use: Relation between p and s-wave through Van-der-Waals coefficient
Channel S-wave(ao) P-wave(ao) Determination
gg 96.2(1) 74(2)
[S-wave] Two-photon photo-
associative
[P-wave] Analytic relation
eg+ 169(8) -169(23)
[S-wave] Analytic relation
[P-wave] Density shift in a
polarized sample
eg- 68(22) âđđâđđ+đđđ
[S-wave] Density shift in a spin
mixture at different
temperatures [P-wave]
Analytic relation
ee
(elastic)176(11) -119(18)
[S-wave] Analytic relation
[P-wave] Density shift in a
polarized sample
ee
(inelastic)đđđ = 46(19)
à·©đđđ =
125(15)Two-body loss measurement
Z. Idziaszek, P. S. Julienne, Phys. Rev. Lett. 104, 113202 (2010).