1. Selim Jochim, Universität Heidelberg
Ultracold fermions: A bottom-up approach
Selim Jochim, Universität Heidelberg

2. A quick advertisement:
Our 2-D Fermi gas experiment

3. Momentum Distribution Imaging
High T
Low T
Temperature
Macroscopic occupation of low-momentum states x y kx ky
T/4 = 25ms
in-situ density distribution n(x,y)
momentum distribution ñ(kx,ky)
P. Murthy et al., PRA 90, (2014)

4. Phase Diagram normal phase exp.: Tc/TF condensed phase
Non-Gaussian
fraction
normal phase
exp.: Tc/TF
condensed phase
Measured phase diagram: x-axis normalized interaction strength in 2D relevant parameter ln(kf a2d) where kf is the fermi wavevector ( related to the sqrt of density)
Y-axis temperature normalized by the central density ( in units of temperatur Tf=2pi hbar^2 n0/2mk_b)
Colorscale gives fraction of non gausan quasi condensate fraction.
Black line position where we expect the phase transition from above analysis. Again see quasicondensate fraction non zero before transition
Critical temperature on bosnic side ( as expected) realtively high and constant
White line ( weakly interacting) bosonic theory phase transition : good agreement exp data little bit higher but within systematic and statstical errors agreement
Bosnic theory works well until ln kf a zero also expected as in 2D there is a bound state for all interaction strengths ( on 3D reonance still 2D bosons pressent).
Compare to theory: matches relatively well. Bump in the center expected as one has to match the two limiting cases.
Central part significantly higher than theroy expection for true 2D but maybe due to residual influence of 3rd dimension ( biger to the BCS side for our experiment) also predicted by some theorists that in crossing to 3D critical temperature might be higher
On Bcs side not cold enqough to measure significant quasi condensate fraction can not determine critical temperature there but expect to go down ( also seen from upper bound from our data) towards bcs limit
bosonic
fermionic
M. Ries et al., PRL 114, (2015)
see also viewpoint: P. Pieri, Physics 8, 53 (2015)

5. Investigate the phase coherence of these “condensates”

6. Phase correlations in 2D
𝑔 1,trap (𝑟)= ℱ𝒯 𝑛 trap (𝒌)
=ℱ𝒯( )
Extract correlation function
from momentum distribution
Tc/TF = 0.129
BKT: 𝑔 1 𝑟 ∼ 1 𝑟 𝜂
consistent with BKT superfluid
We are able to extract
η(T, ln(kF a2D))
P. Murthy et al., PRL 115, (2015)

7. This talk: Experiments with few particles
Discrete systems: Work at „T=0“

8. Our approach to prepare few atoms
Fermi-Dirac dist. E
~100µm 1 n
p0=
2-component mixture in reservoir
superimpose microtrap (~1.8 µm waist)
F. Serwane et al., Science 332, 336 (2011) 8

9. + magnetic field gradient in axial direction
Our approach
switch off reservoir
p0=
+ magnetic field gradient in axial direction
F. Serwane et al., Science 332, 336 (2011) 9

10. Spilling the atoms ….
We can control the atom number with exceptional precision
(including spin degree of freedom)
Note aspect ratio 1:10: 1-D situation
So far: Interactions tuned to zero …
F. Serwane et al., Science 332, 336 (2011)

11. Realize multiple wells …
….. with similar fidelity and control?
S. Murmann, A. Bergschneider et al., Phys. Rev. Lett. 114, (2015)
See also viewpoint: Regal and Kaufman, Physics 8, 16 (2015)

12. The multiwell setup Light intensity distribution
S. Murmann, A. Bergschneider et al., Phys. Rev. Lett. 114, (2015)

13. A tunable double well J $\Ket{\Psi(t=0)}=\Ket{L}_1\Ket{L}_2=\Ket{LL}$
$\Ket{\chi}=\frac{1}{\sqrt{2}}(\Ket{\uparrow}_1\Ket{\downarrow}_2-\Ket{\downarrow}_1\Ket{\uparrow}_2)= \frac{1}{\sqrt{2}}(\Ket{\uparrow\downarrow}-\Ket{\downarrow\uparrow})$

14. switch off left well before counting atoms
A tunable double well
Interactions switched off:
switch off left well before counting atoms
well |𝐿⟩
well |𝑅⟩ J
$\Ket{\Psi(t)}=\Ket{\psi(t)}_1\Ket{\psi(t)}_2$
$\Ket{\psi(t)}_1=\frac{1}{\sqrt{2}}(\Ket{L}_1+\Ket{R}_1)+\frac{1}{\sqrt{2}}(\Ket{L}_1-\Ket{R}_1)e^{-i\Delta E t/\hbar}$

15. Two interacting atoms J U Interaction leads to entanglement: well |𝐿⟩
$\Ket{\Psi(t)}\neq\Ket{\psi(t)}_1\Ket{\psi(t)}_2$

16. Preparing the ground state
If we ramp on the second well slowly enough, the system will remain in its ground state:
An isolated singulett
S. Murmann, A. Bergschneider et al., Phys. Rev. Lett. 114, (2015)

17. How to scale it up?
Preparation of ground states in separated double wells
Combination to larger system
Can this process be done adiabatically ?
Can it be extended to larger systems ?
Motivated by: D. Greif et al., Science 340, (2013) (ETH Zürich)

18. First steps towards magnetic ordering
Realize a Heisenberg spin chain through strong repulsion
Lots of input from theory:
Dörte Blume, Ebrahim Gharashi, N. Zinner, G. Conduit, J. Levinsen, M. Parish, P. Massignan, C. Greene, F. Deuretzbacher

19. Interacting 6Li atoms in 1D
Assume zero range potential in 1D + harmonic confinement
𝑉 𝑥1−𝑥2 =𝑔1𝐷 𝛿 𝑥1−𝑥2
Tune 𝑔 1𝐷 with confinement induced resonance near Feshbach resonance:
Our system: Lithium-6 atoms with 𝜔 2𝜋 ~15kHz transverse confinement
M. Olshanii, PRL 81, (1998)

20. Energy of 2 atoms in a harmonic trap
Relative energy of two contact-interacting atoms:
𝑉 𝑥 = 1 2 𝜇 𝜔 2 𝑥 2 + 𝑔 1𝐷 𝛿(𝑥)
repulsive
attractive
B-field
T. Busch et al., Foundations of Physics 28, 549 (1998)

21. Energy of 2 atoms in a harmonic trap
Relative energy of two contact-interacting atoms:
𝑉 𝑥 = 1 2 𝜇 𝜔 2 𝑥 2 + 𝑔 1𝐷 𝛿(𝑥)
repulsive
attractive
B-field
T. Busch et al., Foundations of Physics 28, 549 (1998)

22. Energy of 2 atoms in a harmonic trap
Relative energy of two contact-interacting atoms:
𝑉 𝑥 = 1 2 𝜇 𝜔 2 𝑥 2 + 𝑔 1𝐷 𝛿(𝑥)
fermionization
repulsive
attractive
B-field
G. Zürn et al., PRL 108, (2012)
T. Busch et al., Foundations of Physics 28, 549 (1998)

23. Energy of more than two atoms?
repulsive
attractive
B-field

24. Energy of more than two atoms
Fermionization
Energy ℏ 𝜔 ||
Non-interacting
S=1/2
− 1 𝑔 1𝐷 𝑎 || ℏ 𝜔 || −1
Gharashi, Blume, PRL 111, (2013)
Lindgren et al., New J. Phys (2014)
Bugnion, Conduit, PRA 87, (2013)

25. Realization of a spin chain
Fermionization
Non-interacting
S=3/2
Energy ℏ 𝜔 ||
S=1/2
S=1/2
− 1 𝑔 1𝐷 𝑎 || ℏ 𝜔 || −1
Gharashi, Blume, PRL 111, (2013)
Lindgren et al., New J. Phys (2014)
Bugnion, Conduit, PRA 87, (2013)

26. Realization of a spin chain
Antiferromagnet
Ferromagnet
Fermionization
Non-interacting
S=3/2
Energy ℏ 𝜔 ||
S=1
Distinguish states by:
Spin densities
Level occupation
S=1/2
− 1 𝑔 1𝐷 𝑎 || ℏ 𝜔 || −1
Gharashi, Blume, PRL 111, (2013)
Lindgren et al., New J. Phys (2014)
Bugnion, Conduit, PRA 87, (2013)

27. Measurement of spin orientation
Non-interacting system
Ramp on interaction strongth
Spin chain
Energy ℏ 𝜔 ||
− 1 𝑔 1𝐷 𝑎 || ℏ 𝜔 || −1

28. Measurement of spin orientation
Non-interacting system
Ramp on interaction strength
Spill of one atom
Spin chain
„Minority tunneling“
„Majority tunneling“
Energy ℏ 𝜔 ||
Remove minority atom
N = 2
N = 1
− 1 𝑔 1𝐷 𝑎 || ℏ 𝜔 || −1

29. Measurement of spin orientation
At resonance: Spin orientation of rightmost particle allows identification of state
Theory by Frank Deuretzbacher et al.

30. Measurement of occupation probabilities
Remove majority component
with resonant light
Spill technique to measure
occupation numbers 8

31. We can prepare an AFM spin chain!9

32. Can we scale it up??
Approach 2:
Can we induce suitable correlations by spilling atoms? ?
𝐽 𝑇𝑢𝑛𝑛𝑒𝑙
𝐽 𝑆𝑝𝑖𝑙𝑙

33. Summary
We studied the phase diagram and coherence properties of a 2-D Fermi gas and
prepare and manipulate isolated mesoscopic systems with extremely good fidelity in flexible trapping geometries
We prepared antiferromagnetic spin chains in 1D tubes
PRL 114, (2015)
PRL 115, (2015) J
PRL 114, (2015)
PRL 108, (2012)
S. Murmann et al.,
arxiv:

34. ? Outlook Can we scale up our systems? or 𝐽 𝑇𝑢𝑛𝑛𝑒𝑙 𝐽 𝑆𝑝𝑖𝑙𝑙
See Andrea Bergschneider‘s poster

35. Thank you for your attention!
Thomas Lompe
(-> MIT)
Thank you for your attention!
Mathias Neidig
Simon Murmann
Andrea Bergschneider
Luca Bayha
Dhruv Kedar
Martin Ries
Michael Bakircioglu
Vincent Klinkhamer
Andre
Wenz
Gerhard Zürn
Justin
Niedermeyer
Puneet Murthy
Funding: