quantum magnetism with few cold atoms - rtg1729€¦ · quantum magnetism with few cold atoms...

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