workshop on quantum simulations with ultracold atoms ictp...
TRANSCRIPT
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Ultracold dipolar bosonic molecules
Workshop on Quantum Simulations with Ultracold Atoms
ICTP Trieste
Tetsu Takekoshi, 20.07.12
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Hanns-Christoph
Nägerl (PI) Francesa
Ferlaino
Rudi Grimm Tetsu Takekoshi
(Postdoc)
Markus Debatin
(PhD)
Lukas
Reichsöllner
(PhD)
Verena
Pramhaas
(Masters)
Carl Hippler
(Masters)
Michael
Kugler
(PhD)
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• RbCs motivation – a toy for artificial condensed matter systems:
novel many-body states, quantum phase transitions,
strong correlations, many-body transport, etc.
• we have rovibhyper ground state RbCs
• ground state RbCs from a double Mott insulator soon…
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Why degenerate dipolar gases?
Review articles:
T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, Rep. Prog. Phys. 72, 126401 (2009).
M.A. Baranov, Phys. Rep. 464, 71 (2008).
Σ 1 rigid rotor
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Why degenerate dipolar gases?
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Why degenerate dipolar gases?
electric dipoles
magnetic dipoles
Review articles:
T. Lahaye et al., Rep. Prog. Phys. 72, 126401 (2009).
M.A. Baranov, Phys. Rep. 464, 71 (2008).
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0.01 0.1 1 1010
0
101
102
103
104
105
106
NaCs
KCs
SrRbRbCs
166Er
2
166Er
H2O
CFH3
OH
LiCs
NaCl
MgH
NH3
KRb
52Cr
chara
cte
ristic r
ange (
bohr)
dipole moment (debye)
87Rb
𝑎𝑑𝑑 = 𝑚𝑑2
4𝜋𝜖0ℏ2
𝑑
Why degenerate dipolar gases?
Stuttgart
Colorado optical lattice spacing
typical experimental atomic scattering lengths
Dy Urbana / Stanford
dipolar gases
Innsbruck
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0.01 0.1 1 1010
0
101
102
103
104
105
106
NaCs
KCs
SrRbRbCs
166Er
2
166Er
H2O
CFH3
OH
LiCs
NaCl
MgH
NH3
KRb
52Cr
chara
cte
ristic r
ange (
bohr)
dipole moment (debye)
87Rb
𝑎𝑑𝑑 = 𝑚𝑑2
4𝜋𝜖0ℏ2
𝑑
Why degenerate dipolar gases?
optical lattice spacing
typical experimental atomic scattering lengths
Dy
dipolar gases
Innsbruck in progress...
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Identical fermions interact • exotic quantum phases • • novel spectrum of excitations • geometry-dependent interaction • …
-
-
+ +
Novel effects appear – NOT contact interaction!
Long-Range Anisotropy
Why degenerate dipolar gases?
B. Capogrosso-Sansone et al.,
Phys. Rev. Lett. 104, 125301 (2010)
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Carr DeMille Krems and Ye, NJP 11, 055049 (2009).
Why degenerate dipolar gases?
KRb Boulder Cs2 Innsbruck
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Carr DeMille Krems and Ye, NJP 11, 055049 (2009).
Why degenerate dipolar gases?
RbCs Innsbruck RbCs Durham
RbCs Yale G. Pupillo, blue shielding
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Recipe for high phase-space density dimer gas
degenerate atomic gas(es) → Feshbach molecules → coherent ground state transfer
• high phase space density almost guaranteed
• high experimental complexity
• limited to "boring" molecules, i.e. dimers
• successfully applied thus far to
KRb, Cs2, Rb2 ( Σ3 ), …RbCs
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The colliding BEC method
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Coupled channel model
Colliding BEC method much higher signal to noise
C.R. LeSueur, J.M. Hutson, P.S. Julienne, S. Kotochigova, E. Tiemann
model includes LIF-FFT spectroscopic data from Riga, Rio de Janeiro
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Coupled channel model
C.R. LeSueur, J.M. Hutson, P.S. Julienne, S. Kotochigova, E. Tiemann
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Cs |3,3> + 87Rb |1,1>
Coupled channel model
Data for model – Feshbach molecule binding energies through magnetic field modulation
C.R. LeSueur, J.M. Hutson, P.S. Julienne, S. Kotochigova, E. Tiemann
The source of much trouble...
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Rb Cs
Cs |3,3> + 87Rb |1,1>
Coupled channel model
C.R. LeSueur, J.M. Hutson, P.S. Julienne, S. Kotochigova, E. Tiemann
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Rb Cs
Cs |3,3> + 87Rb |1,1>
Coupled channel model
C.R. LeSueur, J.M. Hutson, P.S. Julienne, S. Kotochigova, E. Tiemann
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Johann
Danzl
(now “Dr.”)
Russell
Hart
(Rice)
Manfred
Mark
Rb(Cs) (Rb)Cs
Feshbach molecules
The colliding BEC method
Typically 60k Cs + 150k Rb gives 4000 RbCs (we detect only atoms)
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Johann
Danzl
(now “Dr.”)
Russell
Hart
(Rice)
Manfred
Mark |1
|2
|3
spontaneous emission
Eigenstates with light on:
Ω1
Ω2
Ground state transfer
tanΘ =Ω1
Ω2
K. Bergmann, H. Theuer, and B.W. Shore: Rev. Mod. Phys. 70, 1003 (1998)
dark state 𝐻 𝑡 =
ℏ
2
0 Ω1 0Ω1 0 Ω2
0 Ω2 0
Hamiltonian: |1 |2 |3
|1 |2 |3
Θ goes from 0 to Π adiabatically
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Johann
Danzl
(now “Dr.”)
Russell
Hart
(Rice)
Manfred
Mark
tea
𝐻 𝑡 =ℏ
2
0 Ω1 0Ω1 0 Ω2
0 Ω2 0
|1
|2
|3
spontaneous emission
Hamiltonian:
K. Bergmann, H. Theuer, and B.W. Shore: Rev. Mod. Phys. 70, 1003 (1998)
Ω1
Ω2
tanΘ =Ω1
Ω2
dark state
Θ goes from 0 to Π adiabatically
|1 |2 |3
|1 |2 |3
Eigenstates with light on:
Ground state transfer
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Energies and rotational constants agree well with Docenko et al., PRA 81 042511 (2010). (Riga, Rio de Janiero)
L S
J N
L S
W
Λ = 1
𝑆 = 1, Σ = −1,0,1
Ω = Λ + Σ = 0,1,2
𝑏 Π13
Ground state transfer
Suggested Bergeman et al., PRA 67 050501 (2003). (Riga, Rio de Janiero)
𝐴 Σ+1 − 𝑏 Π0+3
𝐴 Σ+1 − 𝑏 Π0+3 𝐴 Σ+1 − 𝑏 Π0+
3 𝑏 Π13
Feshbach molecule absorption
Hund´s case (a)
dark states (projection)
spin-orbit mixing ΔΩ = 0, ΔJ = 0
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Energies and rotational constants agree well with Docenko et al., PRA 81 042511 (2010). (Riga, Rio de Janiero)
L S
J N
L S
W
Λ = 1
𝑆 = 1, Σ = −1,0,1
Ω = Λ + Σ = 0,1,2
𝑏 Π13
Ground state transfer
Suggested Bergeman et al., PRA 67 050501 (2003). (Riga, Rio de Janiero)
𝐴 Σ+1 − 𝑏 Π0+3
𝐴 Σ+1 − 𝑏 Π0+3 𝐴 Σ+1 − 𝑏 Π0+
3 𝑏 Π13
Feshbach molecule absorption
Hund´s case (a)
STIRAP (adiabatic)
spin-orbit mixing ΔΩ = 0, ΔJ = 0
??? how ??? ??? why ???
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RbCs two-photon STIRAP to v=0, J=0.
(on a thermal RbCs sample at 200-300 nK
trapped in an optical lattice)
single-pass
transfer efficiency ≈ 87%
STIRAP time [μs]
laser 1
laser 2
• we detect only atoms • STIRAP references -- two optical cavities locked to Cs atomic reference laser • estimated relative laser linewidth: 5-10kHz
Ground state transfer
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dominant terms – scalar nuclear dipole-dipole, nuclear Zeeman J. Aldegunde (Salamanca) and Jeremy M. Hutson (Durham)
Cs |3,3> + 87Rb |1,1> in incoming s-wave collision has MF=4, therefore, Feshbach molecules also have MF=4
Ground state transfer
all accessible directly through STIRAP (in theory)
where are we ?!?
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Excited state model (help from Romain Vexieu, Anne Crubelier, Oliver Dulieu)
Ground state transfer
𝑏 Π13
3 parameter fit to effective Hamiltonian: aRb =127 MHz x h
aCs =74 MHz x h
overall frequency shift
𝐴 Σ+1 − 𝑏 Π0+3
(1st unambiguous observation of orbital hyperfine in bialkalis)
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Ground state transfer
Ground state model reproduced from Aldegunde et al, PRA 78 033434 (2008). 182G is intermediate Zeeman regime. Stark shifts added for future dipolar expts.
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dominant terms – scalar nuclear dipole-dipole, nuclear Zeeman J. Aldegunde (Salamanca) and Jeremy M. Hutson (Durham)
Cs |3,3> + 87Rb |1,1> in incoming s-wave collision has MF=4, therefore, Feshbach molecules also have MF=4
Ground state transfer
all accessible directly through STIRAP (in theory)
where are we ?!?
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J. Aldegunde (Salamanca) and Jeremy M. Hutson (Durham)
Cs |3,3> + 87Rb |1,1> in incoming s-wave collision has MF=4, therefore, Feshbach molecules also have MF=4
Ground state transfer
all accessible directly through STIRAP (in theory)
18
2G
STI
RA
P
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MF=
5
>90
G lo
wes
t
VH polarization 182G, excited state MF=4.
MF=3
5
2-7
2G
low
est
MF=3
MF=3
Feshbach -> ground 87%
Three parameter fit: Ω1, Ω2, relative laser linewidth. (Ω1 agrees with direct measurement and ab initio calculation.)
MF=3 4 5
Ground state transfer
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VV polarization 182G, excited state M=4
MF=4
7
2-9
0G
low
est
MF=4
MF=2
<5
2G
low
est
Inappropriate ramps used. Data must be retaken.
Prediction from previous fit results
MF=3 4 5
Ground state transfer
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182G
Ground state transfer
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Outline RbCs from a double Mott insulator
Our current estimated phase-space density ~0.01? A new method is necessary to get us to 1.
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We really want something like this! (atom pairs)
Outline RbCs from a double Mott insulator
Noah Bray-Ali and Carl Williams preliminary calculations
J. Freericks numerical simulations starting
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2 superfluids
Outline RbCs from a double Mott insulator
𝑈0𝑅𝑏𝑅𝑏/𝐽𝑅𝑏 << 35 𝑈0
𝐶𝑠𝐶𝑠/𝐽𝐶𝑠 << 35
𝑎𝐶𝑠𝐶𝑠 = 1700 𝑎0 𝑎𝑅𝑏𝑅𝑏 = 100 𝑎0 𝑎𝑅𝑏𝐶𝑠 = tunable (0.2 G wide Feshbach resonance)
𝑈 ∝4𝜋ℎ2𝑎
𝑚
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• Freeze out Cs • Rb superfluid flows onto Cs • Use onsite interactions 𝑈 to prevent 3 particles per lattice site
Outline RbCs from a double Mott insulator
𝑈RbRb ~ +𝑘B × 15 nK (ℎ × 300 Hz)
𝑈CsCs ~ +𝑘B × 400 nK (ℎ × 8000 Hz) 𝑈RbCs = tunable, make negative? Need to prevent site occupation by: Rb Rb Cs → three body loss
want 𝑈RbRb + 𝑈RbCs > 0
∆𝑎𝑅𝑏𝐶𝑠
∆𝐵=
𝑎𝑅𝑏𝐶𝑠
𝑏𝑎𝑐𝑘𝑔𝑟𝑜𝑢𝑛𝑑
∆=
649𝑎0
0.2 𝐺 = 3.3
𝑎0
𝑚𝐺
5−10 mG noise measured in lab → ∆𝑎𝑅𝑏𝐶𝑠 = 17-33𝑎0
near 182G Feshbach resonance
𝑈0𝑅𝑏𝑅𝑏/𝐽𝑅𝑏 =2500 𝑈0
𝐶𝑠𝐶𝑠/𝐽𝐶𝑠 = 4.3
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Outline RbCs from a double Mott insulator J/
h
Mean-field phase diagram of Rb atoms in an optical lattice resonantly interacting with a Cs Mott insulator. J/h is the Rb tunneling rate. (Noah Bray-Ali, Carl Williams)
Rb-Cs Feshbach
resonance
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Now make Feschbach molecules, and do STIRAP (Much simpler version works for Rb2, Cs2)
Lower lattice adiabatically for molecular superfluid
Outline RbCs from a double Mott insulator
B
dipolar lattice physics ! (or melt for RbCs BEC)
make Feshbach molecules
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Dual species SF-MI: same lattice but not yet overlapped
Rb
Cs
150 300 600 900 600 P (mW) 300
V0 = 14 Er
V0 = 30 Er
Rb
Cs
fin
al
latt
ice
po
wer
Outline RbCs from a double Mott insulator
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What if this is not enough for degeneracy?
(KRb currently at 1.5TF, PSD 0.1)
Outline RbCs from a double Mott insulator
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Piotr S. Zuchowski and Jeremy M. Hutson PRA 81, 060703 (2010).
What if this is not enough for degeneracy?
3D Evaporative cooling of ground state RbCs with no electric field?
Outline RbCs from a double Mott insulator
LiNa, LiK, LiRb, LiCs and KRb
NaK, NaRb, NaCs, KCs and RbCs
all
dimer-dimer
Three body loss?
+ +
C6 = 140000ao calculated for RbCs Kotochigova NJP 12, 073041 (2010).
• Spin changing collisions driven by collision anisotropy
• Centrifugal barrier ~ µ-3/2C6-1/2
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Evaporative cooling of molecules:
ground state molecules are precious.
perhaps one can use atoms instead?
Outline RbCs from a double Mott insulator
dimer-monomer collisions: one will be forbidden
+
+
-
-
+ -
-
all
all
OR
Rb2
RbCs
Cs2
𝑣 = 0
𝑣 = 0
𝑣 = 0
183 cm-1
154 cm-1
Use Cs atoms as a coolant
RbCs
Cs
x1.6
• requires high Cs-Cs, RbCs-Cs thermalization rates
• requires low Cs-Cs-Cs, RbCs-RbCs-RbCs, RbCs-RbCs-Cs, RbCs-Cs-Cs three body recombination rates (may need to goto 20G!)
+
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• we have rovibhyper ground state RbCs (PSD 0.01)
• our toy is almost finished !
Coming soon
• RbCs from a double Mott insulator
• The fun stuff…
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Organizers: Guido Pupillo (chair), Francesca Ferlaino, Hanns-Christoph
Nägerl
UNIVERSITY OF
INNSBRUCK
€SF Conference on
Cold and Ultracold Molecules
http://www.esf.org/index.php?id=9144
University Center Obergurgl (“near” Innsbruck) November 18-23, 2012
More to come…
• Direct and indirect cooling techniques • Controlled quantum chemistry • Ultracold molecules for tests of quantum physics • Molecular quantum gases • Frontiers in molecular quantum control • …
![Page 47: Workshop on Quantum Simulations with Ultracold Atoms ICTP ...andreatr/WORKSHOP_QUANTUM_SIMULATIONS… · tea 𝐻 = ℏ 2 0 Ω1 0 Ω1 0 Ω2 0 Ω2 ... laser 1 laser 2 • we detect](https://reader035.vdocuments.net/reader035/viewer/2022071220/605bc81cc5f99e566964aee1/html5/thumbnails/47.jpg)
Johann
Danzl
(now “Dr.”)
Russell
Hart
(Rice)
Manfred
Mark
![Page 48: Workshop on Quantum Simulations with Ultracold Atoms ICTP ...andreatr/WORKSHOP_QUANTUM_SIMULATIONS… · tea 𝐻 = ℏ 2 0 Ω1 0 Ω1 0 Ω2 0 Ω2 ... laser 1 laser 2 • we detect](https://reader035.vdocuments.net/reader035/viewer/2022071220/605bc81cc5f99e566964aee1/html5/thumbnails/48.jpg)
The colliding BEC method
Cs BEC requires optical traps. Optical BEC of Rb makes dual species apparatus simpler.
Evap .
Cs |3,3> or 87Rb |1,1>
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The colliding BEC method
Cs BEC requires optical traps. Optical BEC of Rb makes dual species apparatus simpler.
Evap .
Cs |3,3> or 87Rb |1,1>
Our dream: like 87Rb 85Rb mixture
S.B. Papp, J.M. Pino, C.E. Wieman PRL 101 040402 (2008).
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The colliding BEC method
Making Feshbach molecules requires a high phase space density mixture.
mixture problems bad, but not insurmountable really bad mixture problems
Rb + Cs
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Starting point: all-optical Cs BEC Making Feshbach molecules requires a high phase space density mixture
mixture problems bad, but not insurmountable
really bad mixture problems
Evap .
Rb + Cs
The colliding BEC method
Cs MOT size ato
m n
um
ber
in d
imple
Rb
Cs
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Traps are deeper for Rb than for Cs. (evaporative heat load mostly on Rb)
high Rb/Cs thermalization Bad for simultaneous evaporation Good for Cs cooling! (efficient) Bad luck #1
The colliding BEC method
Important for Rb-Cs mixtures:
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Traps are deeper for Rb than for Cs. (evaporative heat load mostly on Rb)
Large interspecies background scattering length (aRbCs~649a0 from coupled channel model)
Ratio of „good“ to „bad“ collisions = 4πaRbCs
2 /(KRbRbCsnRb)
Large three-body recombination rates For example KRbRbCs~|aRbCs|
4≈10-24 cm6s-1 (measured)
The colliding BEC method
Important for Rb-Cs mixtures:
high Rb/Cs thermalization Bad for simultaneous evaporation Good for Cs cooling! (efficient)
Bad luck #2
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The colliding BEC method
• Raman cooling • reservoir (spin filter) • separate dimples
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The colliding BEC method
100,000 atoms
20,000 atoms Cs |3,3>
Rb |1,1>
dimple power
Inte
grat
ed d
en
sity
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The colliding BEC method
Durham group PRA 84 011603 (2011).
S.B. Papp, J.M. Pino, C.E. Wieman PRL 101 040402 (2008).
𝑎𝑅𝑏𝑅𝑏
−𝑎𝐶𝑠𝐶𝑠
𝑎𝑅𝑏𝐶𝑠
2 - 1 > 0 miscible
Combine and magnetoassociate quickly
Bad luck #3
![Page 57: Workshop on Quantum Simulations with Ultracold Atoms ICTP ...andreatr/WORKSHOP_QUANTUM_SIMULATIONS… · tea 𝐻 = ℏ 2 0 Ω1 0 Ω1 0 Ω2 0 Ω2 ... laser 1 laser 2 • we detect](https://reader035.vdocuments.net/reader035/viewer/2022071220/605bc81cc5f99e566964aee1/html5/thumbnails/57.jpg)
The colliding BEC method
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Outline RbCs from a double Mott insulator
Rb superfluid-Mott insulator transition
Superfluid state J>>U
• delocalised
• poissonian distribution
• phase coherence
• interference pattern
Image after expansion – matter wave interference
![Page 59: Workshop on Quantum Simulations with Ultracold Atoms ICTP ...andreatr/WORKSHOP_QUANTUM_SIMULATIONS… · tea 𝐻 = ℏ 2 0 Ω1 0 Ω1 0 Ω2 0 Ω2 ... laser 1 laser 2 • we detect](https://reader035.vdocuments.net/reader035/viewer/2022071220/605bc81cc5f99e566964aee1/html5/thumbnails/59.jpg)
Outline RbCs from a double Mott insulator
Rb superfluid-Mott insulator transition
Mott insulator state J<<U
• localized atoms
• no phase coherence
• no interference pattern
• fixed atom number per site
![Page 60: Workshop on Quantum Simulations with Ultracold Atoms ICTP ...andreatr/WORKSHOP_QUANTUM_SIMULATIONS… · tea 𝐻 = ℏ 2 0 Ω1 0 Ω1 0 Ω2 0 Ω2 ... laser 1 laser 2 • we detect](https://reader035.vdocuments.net/reader035/viewer/2022071220/605bc81cc5f99e566964aee1/html5/thumbnails/60.jpg)
Outline RbCs from a double Mott insulator
Rb superfluid-Mott insulator transition
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310 315
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
22000
24000A
tom
s r
em
ain
ing
B (Gauss)
Rb
Cs
Coupled channel model
Data for model – Feshbach resonances
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Outline RbCs from a double Mott insulator
Increasing the lattice depth Phase transition from the superfluid BEC to a localized Mott-insulator state
Proposal: P. Zoller et al., 1998
This should also work the other way round!!!
Reduction of lattice depth: Phase transition from a localized Mott-insulator state of molecules to a molecular BEC („mBEC“) Proposal: P. Zoller et al., 2002
mBEC
atomic BEC
atomic Mott insulator
molecular Mott insulator
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1/1064nm = 9398 cm-1
Lang et al., Faraday Dis. 142, 271 (2009).
Romain Vexieu, Nadia Bouloufa, Oliver Dulieu
Ground state transfer
Luck!!
magic wavelength
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Coupled channel model
Initial medium range potential taken from Fourier transform spectroscopy study (Riga) O. Docenko et al., PRA 83, 052519 (2011). level energies accurate to ~ 1GHz x h
Short range
Long range
spin-spin and 2nd order spin-orbit (avoided crossing strengths)
C. Ruth LeSueur, Jeremy M. Hutson (Durham) Paul S. Julienne (JQI, NIST, UMD) Svetlana Kotochigova (Temple) Eberhard Tiemann (Hannover)
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broader than expected from calculations (Vexieau, Bouloufa, Dulieu)
= 𝜇 ⋅ 𝐸/ℏ
Ground state transfer
𝐴 Σ+1 − 𝑏 Π0+3 𝐴 Σ+1 − 𝑏 Π0+
3 𝐴 Σ+1 − 𝑏 Π0+3
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Johann
Danzl
(now “Dr.”)
Russell
Hart
(Rice)
Manfred
Mark
𝐻 𝑡 =ℏ
2
0 Ω1 0Ω1 0 Ω2
0 Ω2 0
|1
|2
|3
spont. em.
𝑡𝑎𝑛Θ =Ω1
Ω2
Eigenstates with light on:
time
K. Bergmann, H. Theuer, and B.W. Shore: Rev.
Mod. Phys. 70, 1003 (1998)
Ω1
Ω2
Ground state transfer
Ground state spectroscopy:
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Johann
Danzl
(now “Dr.”)
Russell
Hart
(Rice)
Manfred
Mark
𝐻 𝑡 =ℏ
2
0 Ω1 0Ω1 0 Ω2
0 Ω2 0
|1
|2
|3
spont. em.
𝑡𝑎𝑛Θ =Ω1
Ω2
Eigenstates with light on:
time
K. Bergmann, H. Theuer, and B.W. Shore: Rev.
Mod. Phys. 70, 1003 (1998)
Ω1
Ω2
Ground state transfer
Ground state spectroscopy:
projection 𝑐𝑜𝑠Θ onto dark state
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𝑃1→𝑎0→1 = 𝑃1→𝑎0𝑃𝑎0→1 = cos4 Θ =Ω1
4
Ω12 + Ω2
2 2
Johann
Danzl
(now “Dr.”)
Russell
Hart
(Rice)
Manfred
Mark
𝐻 𝑡 =ℏ
2
0 Ω1 0Ω1 0 Ω2
0 Ω2 0
|1
|2
|3
spont. em.
𝑡𝑎𝑛Θ =Ω1
Ω2
Eigenstates with light on:
time
K. Bergmann, H. Theuer, and B.W. Shore: Rev.
Mod. Phys. 70, 1003 (1998)
Ω1
Ω2
Ground state transfer
Ground state spectroscopy:
projection 𝑐𝑜𝑠Θ onto |1 >
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Johann
Danzl
(now “Dr.”)
Manfred
Mark
|1
|2
|3
spont. em.
𝐻 𝑡 =ℏ
2
0 Ω1 0Ω1 2Δ1 Ω2
0 Ω2 2(Δ1 − Δ2)
Δ2 Δ1
Ground state transfer
Ground state spectroscopy:
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Johann
Danzl
(now “Dr.”)
Manfred
Mark
|1
|2
|3
spont. em.
𝐻 𝑡 =ℏ
2
0 Ω1 0Ω1 2Δ1 Ω2
0 Ω2 2(Δ1 − Δ2)
Δ2 Δ1
Ground state transfer
• Have mapped out ground state v=0 N=0,2. • Rotational constants agree very well with
Fourier transform spectroscopy experiments
Ground state spectroscopy:
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iso J N s n Trm(cm-1) <R>(Ang) Fr(A) Fr(b0) Fr(b1) Fr(b2) (cm-1)
2 1 52 2 27 10019.87 4.69195 0.23704 0.76296 0 0 6208.30
2 1 53 3 24 10023.59 4.51642 0 0 1 0 6212.01
2 1 54 2 28 10057.53 4.73666 0.29303 0.70697 0 0 6245.95
2 1 55 1 0 10060.47 5.04516 0.89831 0.10169 0 0 6248.89
2 1 56 3 25 10070.94 4.525 0 0 1 0 6259.36
2 1 57 2 29 10094.18 4.76101 0.31243 0.68757 0 0 6282.60
2 1 58 1 1 10108.55 5.0381 0.88108 0.11892 0 0 6296.98
2 1 59 3 26 10118.18 4.53366 0 0 1 0 6306.60
2 1 60 2 30 10130.05 4.78599 0.33485 0.66515 0 0 6318.47
2 1 61 1 2 10157.9 5.03051 0.85921 0.14079 0 0 6346.32
2 1 62 3 27 10165.3 4.54241 0.00001 0.00001 0.99998 0 6353.72
2 1 63 2 31 10165.42 4.80475 0.34841 0.65157 0.00001 0 6353.84
2 1 64 2 32 10200.1 4.84594 0.40076 0.59924 0 0 6388.52
2 1 65 1 3 10208.36 5.00023 0.79685 0.20315 0 0 6396.78
2 1 66 3 28 10212.3 4.55122 0 0 1 0 6400.73
2 1 67 2 33 10234.73 4.85737 0.39963 0.60037 0 0 6423.16
2 1 68 1 4 10259.03 4.99905 0.78569 0.21395 0.00036 0 6447.45
2 1 69 3 29 10259.19 4.56026 0.00029 0.00007 0.99964 0 6447.62
2 1 70 2 34 10269.07 4.87554 0.41192 0.58808 0 0 6457.50
2 1 71 2 35 10303.22 4.89033 0.41885 0.58115 0 0 6491.65
2 1 72 3 30 10305.96 4.56905 0 0 1 0 6494.39
2 1 73 1 5 10309.96 4.99505 0.76648 0.23352 0 0 6498.38
2 1 74 2 36 10337.24 4.91178 0.43766 0.56234 0 0 6525.67
2 1 75 3 31 10352.62 4.57806 0 0 1 0 6541.04
2 1 76 1 6 10360.75 4.97269 0.70972 0.29028 0 0 6549.18
2 1 77 2 37 10371.33 4.94569 0.48286 0.51714 0 0 6559.76
2 1 78 3 32 10399.15 4.58713 0 0 1 0 6587.58
2 1 79 2 38 10404.19 4.90554 0.4046 0.5954 0 0 6592.62
2 1 80 1 7 10412.55 5.02509 0.77545 0.22455 0 0 6600.97
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![Page 74: Workshop on Quantum Simulations with Ultracold Atoms ICTP ...andreatr/WORKSHOP_QUANTUM_SIMULATIONS… · tea 𝐻 = ℏ 2 0 Ω1 0 Ω1 0 Ω2 0 Ω2 ... laser 1 laser 2 • we detect](https://reader035.vdocuments.net/reader035/viewer/2022071220/605bc81cc5f99e566964aee1/html5/thumbnails/74.jpg)
Supercavities!
• sub-Hz laser stabilities possible (cavities themselves are the reference)
• limited by acoustics
• finesse ~200000
• narrow linewidth diode lasers also being built
Ground state transfer