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Quantum magnetism with few cold atoms
Gerhard Zürn
Graduiertenkolleg Hannover, 17.12.2015
States of matter
• What are the key ingredients for high-Tc supercondutivity?
• Can we „simulate“ some of them?
http://en.wikipedia.org/wiki/High-temperature_superconductivity
The Heisenberg model:
- 2D structure: study phase transition and
phase correlation
picture: T. Fukuhara et al., Nat.Phys. 9, 235–241 (2013)
- effective magnetic interaction
exchange interaction
first order correlation function g1(r)
Related phenomena
M. Ries et al., PRL 114, 230401 (2015)
P. Murthy et al., PRL 115, 010401 (2015)
Spin order on a lattice
For positive J:
1D: (Bethe-Ansatz, DMRG, ...) 2D and large N, anisotropic Jex,…difficult…
Many body system at high temperature Many body system at low temperature
The cold gases approach
- Reduce the complexity of a system as much as possible until only the essential
parts remain!
(e.g. consider only nearest neighbor interaction)
- Tunability of interaction: vary s-wave scattering length a by Feshbach resonances
- Shape of the external confinement
(optical dipole traps, lattices)
A lattice for ultracold atoms
• Shape a lattice potential with light
… …
Jt
A lattice for ultracold atoms
• Shape a lattice potential with light
… …
Tunneling Jt
On-site interaction U
A lattice for ultracold atoms
• For large enough U/Jt : Mott-insulating state is reached,
fixed particle number per site
… …
Tunneling Jt
On-site interaction U
Fermionic Mott insulators
D. Greif et al., arXiv:1511.06366
M. Greiner group, Harvard
Control ordering?
• Energy scale relevant for ordering:
Superexchange
… …
Jt 2/U
Our Idea - Bottom up approach:
Create this fundamental building block
Jex = 4 Jt 2/U
Challenge: low enough entropy to observe long range correlation
…and scale system up
Outline
• Preparation and control of few-fermion systems
• Two fermions in a double well potential
• Finite antiferromagnetic spin chains without a lattice
Outline
• Preparation and control of few-fermion systems
• Two fermions in a double well potential
• Finite antiferromagnetic spin chains without a lattice
Prepare finite, deterministic samples, where all degrees of freedom are controlled
Pauli principle: each single particle state not occupied by more than one Fermion
Control over the quantum states of the trap
Control over particle number
Requirement:
• highly degenerate Fermi gas
• control over the trap depth more precise then level spacing
Requirements for Control
High Fidelity Preparation
• 2-component mixture in reservoir T=250nK
• superimpose microtrap
scattering thermalisation
• switch off reservoir
p0= 0.9999
+ magnetic field gradient in
axial direction
expected degeneracy: T/TF= 0.1
count
E
P(E) 1
Single atom detection
CCD
1-10 atoms can be distiguished with
high fidelity. (> 99% )
1/e-lifetime: 250s
Exposure time 0.5s
0 1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Counts
Flourescence normalized to atom number
CCD
F. Serwane et al., Science 332, 336 (2011)
fluorescence normalized to atom number
2 atoms 8 atoms
F. Serwane et al., Science 332, 336 (2011)
Lifetime in ground state ~ 60s >> 1/w
count the
atoms
High Fidelity Preparation
750 800 850 900
-20
-10
0
10
20
a3
D [
10
3 a
0]
magnetic field [G]
3D
-1
0
1
g1
D [
10
ap
erp ħ
wp
erp]
750 800 850 900
-20
-10
0
10
20
a3
D [
10
3 a
0]
magnetic field [G]
1D
M. Olshanii, PRL 81, 938 (1998)
2
31D 2
3
2 1g =
1 /
D
D
a
ma Ca a
radially strongly confined
aspect ratio: wII / wperp 1:10 1D
Z. Idziaszek and T. Calarco, PRA 74, 022712 (2006)
Interaction in 1D
G. Zürn et al., PRL 110, 135301 (2013)
Few atoms in a harmonic trap
control of the particle number
tunability of the interpart. interaction
imbalanced
balanced
T. Busch et al., Found. Phys. 28, 549 (1998)
-3 -2 -1 0 1 2 3
-2
0
2
E [hwll]
-1/g
[allhw
ll]-1
Measure the interaction energy
vary the number of majority particles
0,0 0,5 1,0 1,5 2,0 2,5 3,00,0
0,2
0,4
0,6
interaction shift [kHz]
Tra
nsfe
rre
d f
ractio
n
0 1 2 3 4 5
0,0
0,5
1,0
1,5
2,0
E
[ħ
wll]
N
Determine: how many is many?
A. Wenz et al., Science 342, 457 (2013)
Resolution: ~ 1% of EF
repulsive interaction
Realize systems with magnetic ordering…
Can we prepare an ordered state with similar fidelity and control?
We have very low entropy per particle!
Further requirement: Shape the trapping potential:
Start with the smallest possible system
We have full control of a harmonically trapped system.
Outline
• Preparation and control of few-fermion systems
• Two fermions in a double well potential
• Finite antiferromagnetic spin chains without a lattice
Creating a tunable potential
Acousto-
optic
deflector
Tunneling in a double-well
Jt Preparation Detection scheme
U = 0
• Single-part. tunneling
• Calibration of J, J << w
• Correlated tunneling
U ≠ 0
9
J Preparation Detection scheme
Tunneling in a double-well
Measuring Energies in a double well
Conditional
single-particle tunneling Resonant pair tunneling
Measure amplitudes of
single-particle and pair
tunneling as a function of tilt
U>>J
Preparation of the ground state
Two site Hubbart model - eigenstates
Spectrum of eigenenergies
(=0)
Number statistics for the balanced case depending on the interaction strength:
Single occupancy
Double occupancy
12
Occupation statistics
S. Murmann et al., PRL 114, 080402 (2015)
Single occupancy
Double occupancy
12
Number statistics for the balanced case depending on the interaction strength:
Occupation statistics
S. Murmann et al., PRL 114, 080402 (2015)
Measuring energy differences
Method:
Trap modulation
spectroscopy
-5 0 5
-5
0
5
|a>
|c>
E [J]
U [J]
|b>
U ~
S. Murmann et al., PRL 114, 080402 (2015)
Next…
Assemble antiferromagnetic ground state by merging several double wells
Basic building block is realized:
- S=0 state.
- one particle per site
- control of super-exchange energy
M. Lubasch et al., 107, 165301 (2011)
Outline
• Preparation and control of few-fermion systems
• Two fermions in a double well potential
• Finite antiferromagnetic spin chains without a lattice
A Heisenberg spin chain without a lattice…
…. in a single 1-D tube through strong repulsion!
Fermionization
Non-interacting Repulsive
G. Zuern et al., PRL 108, 075303 (2012)
A spin chain without a lattice
2
Distinguishable fermions with
strong repulsive interactions
„Fermionization“
Identical fermions:
Pauli principle
→ Mapping on a Heisenberg Hamiltonian - Theory by Frank Deuretzbacher
Deuretzbacher et al., PRA 90, 013611 (2014)
Volosniev et al., Nature Comm 5, 5300 (2014)
A Wigner-crystal-like state:
−1 𝑔1𝐷 𝑎||ℏ𝜔||−1
5
Non-interacting
Gharashi, Blume, PRL 111, 045302 (2013)
Lindgren et al., New J. Phys. 16 063003 (2014)
Bugnion, Conduit, PRA 87, 060502 (2013)
Non-interacting Repulsive Attractive Non-interacting
Realization of spin chain
5
Fermionization
−1 𝑔1𝐷 𝑎||ℏ𝜔||−1
Non-interacting
Non-interacting Repulsive Attractive Non-interacting
Realization of spin chain
Gharashi, Blume, PRL 111, 045302 (2013)
Lindgren et al., New J. Phys. 16 063003 (2014)
Bugnion, Conduit, PRA 87, 060502 (2013)
5
−1 𝑔1𝐷 𝑎||ℏ𝜔||−1
Non-interacting Fermionization
Non-interacting Repulsive Attractive Non-interacting
Realization of spin chain
Gharashi, Blume, PRL 111, 045302 (2013)
Lindgren et al., New J. Phys. 16 063003 (2014)
Bugnion, Conduit, PRA 87, 060502 (2013)
Distinguish states by:
• Spin densities
• Level occupation
5
Non-interacting Antiferromagnet
Ferromagnet
Fermionization
Non-interacting Repulsive Attractive Non-interacting
Realization of spin chain
Gharashi, Blume, PRL 111, 045302 (2013)
Lindgren et al., New J. Phys. 16 063003 (2014)
Bugnion, Conduit, PRA 87, 060502 (2013)
At resonance: Spin orientation of rightmost particle allows for identification of state
Non-interacting system
−1 𝑔1𝐷 𝑎||ℏ𝜔||−1
Ramp on interaction strength
Measurement of spin orientation
Measurement of spin orientation
Non-interacting system
Spill of one atom
Ramp on interaction strength
„Minority tunneling“
−1 𝑔1𝐷 𝑎||ℏ𝜔||−1
„Majority tunneling“
Measurement of spin orientation
spin orientation determines state
Theory by Frank Deuretzbacher, Luis Santos, Johannes Bjerlin, Stephie Reimann
S. Murmann et al., PRL 115, 215301 (2015)
Note: data points in gray can be explained by CoM-Rel motion coupling
AFM
Measurement of occupation probabilities
Spill technique to measure
occupation numbers
Remove majority
component
with resonant light
AFM has “most
symmetric” spatial
wavefunction
many levels required to
describe the “kink”
FM
IM
AFM
exp. data
S. Murmann et al., PRL 115, 215301 (2015)
We can prepare an AFM spin chain!
−1 𝑔1𝐷 𝑎||ℏ𝜔||−1
Important: We don‘t really need to resolve this energy scale in our experiment!
spin symmetry is fixed by initial preparation
S. Murmann et al., PRL 115, 215301 (2015)
Can we scale it up ?
It will be extremely challenging to produce longer spin chains
• But: Can we couple many individual spin chains ?
𝐽𝐸𝑥 1
𝐽𝐸𝑥 2
… …
Current work in progress…
Vincent Klinkhamer
Andrea Bergschneider
Selim Jochim
Thank you for your attention!
Funding:
Andre Wenz
Dhruv Kedar
Martin Ries
Mathias Neidig
Puneet Murthy
Simon Murmann
Michael Bakircioglu Justin
Niedermeyer
Luca Bayha
In collaboration with:
Frank Deuretzbacher, Luis Santos
Leibniz Universität Hannover
Johannes Bjerlin, Stephanie Reimann
Lund University
Combine 2D- and lattice geometry
We have another experiment:
• 2D gas with a lattice!
𝐽𝐸𝑥 1
𝐽𝐸𝑥 2 ?
M. Ries et al., PRL 114, 230401 (2015)
P. Murthy et al., PRL 115, 010401 (2015)
I. Boettcher et al. arXiv:1509.03610, PRL, in press
Precise energy measurements
„bare“ RF – transition
RF – transition with interaction
48
RF photon+ ΔE
RF photon
Radio Frequency spectroscopy
CoM – Rel. motion coupling
2-1 sample, detect majority
cubic and quartic terms in the potential
Creating imbalanced samples
Eigenstates
Antiferromagnet
Ferromagnet
Intermediate
- 3J - J
0 0 0 0 0
0 0
Tunneling Model
Tunneling rate:
Overlap of spin chains
Gamma-factor
Probabilty to tunnel from |i> |f>
AOD setup
Light intensity distribution