www.quantum-munich.de

Ludwig-Maximilian Universität München Max-Planck Institut für Quantenoptik

Simon Braun Philipp Ronzheimer Michael Schreiber

Sean Hodgman

Immanuel Bloch U.S.

LMU & MPQ München

+Proposal by A. Mosk & A. Rapp, S. Mandt and A. Rosch + discussions with A. Rosch, H. Wagner, W. Zwerger, D. Huse, …

Funding: DARPA (OLE), DFG, EU, Nanosystems Initiative Munich

Bose-Einstein condensate

e.g. 87Rb / 39K atoms

ground states at T=0

Non /weakly-interacting

Parameters:

Densities: 1015

cm-3

Temperatures: nanoKelvin

Number of Atoms: about 105-6

Degenerate Fermi gas

e.g. 40K atoms

Understand and Design Quantum Materials

One of the hardest problems of Quantum Physics in the 21st Century

Technological Relevance – High temperature superconductivity

(Power Transmission)

– Magnetism (Storage, Spintronics...)

– Quantum Computing

YB

CO

hig

h-t

emp

su

per

con

du

cto

r YB

CO

la

ttic

e st

ruct

ure

Well understood?

Mott Insulator:

half filled band

but insulating

Magnetisms:

e.g. Antiferromagnetism

High-Temperature Superconducter

State-of-the-art: < 40 electrons (240 Variables ≙ 1TB) (How much memory is needed for 300 electrons?)

1.1 Petaflops/s 2000 t 3.9 MW

2300 ≙ Number of protons in the universe

Parameter:

• Lattice constant

• Lattice depth

• Density/ doping

• Interaction strength

R. P. Feynman, Int. J. Theo. Phys. (1982) Found. Phys (1986)

A Quantum Simulator is a model system used to simulate the dynamics of

another quantum system.

Complications:

• Disorder

• Lattice defects

• Multiple bands

• phonons

Clean and easy model system

Tunable model system

Implement and study important model Hamiltonians from e.g. solid-state physics: Important Example: Hubbard models

Overlapping two counterpropagating laser beams produce a standing wave:

Dipole potential: Interaction of atoms with far-detuned light, e.g. optical tweezer

red-detuned

repulsive

blue-detuned

attractive

Analogous to periodic Coulomb potential experienced by electrons in solids

Single band picture is sufficient

Bose-Einstein condensate of Potassium 39K atoms 3D optical lattice

+

Restrict to single (lowest) band

expand in localized wannier functions

Tunneling matrix element: Onsite interaction matrix elemtent:

Bose-Hubbard

Hamiltonian

controlled by lattice depth controlled by lattice depth & scattering length a:

𝑎 > 0 repulsive 𝑎 < 0 attractive

Cubic 3D Lattice: Uc = 29.3 J at unity filling

𝐽 ≫ 𝑈: Superfluid 𝐽 ≪ 𝑈: Mott Insulator

†0 ˆ 0

N

N

m

m

a

†ˆ 0

n

m

m

a

• Superfluid • Compressible • Long-range order • Phase coherent • Gapless excitations

• Insulating • Incompressible • No Phase coherence • Gapped

Momentum distribution

lattice depth scattering length

First observation: Greiner et al. Nature 415,39 (2002)

22 Erecoil

0 Erecoil

First observation: Greiner et al. Nature 415,39 (2002)

J. Sherson et al. Nature 467, 68 (2010), see also Greiner group: W.S. Bakr et al. Science (2010)

What´s new?

Earlier work: 85Rb, 133Cs, 52Cr, 7Li, 39K: Lens, Cambridge,

Feshbach resonance: C. D´Errico et al., NJP 9, 223 (2007)

see also: M.J. Mark et.al. PRL 107, 175301 (2011)

Red line: U/J=30

hot cold

…a measure of unordered movement

Is this comparable for arbitrary systems?

How?

Sufficient for classical ideal gas…

⟨𝐸𝑘𝑖𝑛⟩ ∝ 𝑘𝐵𝑇

⟨𝐸⟩ ∝ 𝑘𝐵𝑇 Only for quadratic degrees of freedom and classical particles

Temperature defines an ordering relation between systems!

𝑇1 𝑇2 Heat Flow?

Heat always flows from the hotter to the colder system, until both systems have the same temperature

𝑇 Thermal bath

with temperature 𝑇

𝑝𝑖 ∝ 𝑒−

𝐸𝑖𝑘𝐵𝑇

Boltzmann distribution

Canonical Ensemble:

No thermal contact with experimental apparatus

Experimental sequence:

1) Evaporative cooling „semi-open“ system N,S: decreasing

2) Loading into the lattice closed system (with classical control parameters)

No Environment:

No external temperature

Solid State Cold Atoms

In contact with experimental apparatus

Heat and particle exchange with external reservoirs Grand-canonical ensemble

𝜇, 𝑇: defined by Environment

𝑝𝑖 ∝ 𝑒−

𝐸𝑖𝑘𝐵𝑇

Tota

l en

ergy

per

par

ticl

e

∝ 𝒆−𝜷𝑬𝒊

Requirement: Hamiltonian bounded from above: 𝐸

𝑁≤ 𝜖𝑚𝑎𝑥

1

𝑇=𝜕𝑆

𝜕𝐸

𝑆 = −𝑘𝐵 𝑝𝑖 log 𝑝𝑖𝑖

𝑇 → 0+

−𝛽 → −∞

𝑇 → ±∞

−𝛽 → 0

𝑇 → 0−

−𝛽 → +∞

1. Heat, Heat, Heat, … Challenging/Impossible: Above 𝑇 = ∞ entropy decrease again (like cooling). 2. „Flip“ the energy axis:

"𝐻 ⇒ −𝐻 "

Population inversion: Basis of Laser steady state but no equilibrium state

U tunable via Feshbach resonance in 39K

V can be inverted due to blue-detuned optical lattice

limited to 𝑻 > 𝟎 limited to 𝑻 < 𝟎

Proposal by: Allard P. Mosk PRL 95, 040403 (2005) A. Rapp, S. Mandt and A. Rosch PRL 105, 220405 (2010)

Superfluid weak IA U>0 T>0

Mott Insulator strong IA U>0 T>0

Mott Insulator atomic limit

U>0

Mott Insulator strong IA U<0 T<0

Mott Insulator atomic limit

U<0

Superfluid weak IA U<0 T<0

U→-U V0→-V0

„H -H“

JU

UJ

J0

Stable Bose gas at upper band edge

Entropy production: Low

Happens mainly outside of MI PRL 105, 220405 (2010)

U→-U V0→-V0

Latt

ice

Dep

th

No Flip

Flip: U→-U V0→-V0

Science 339, 52 (2013)

Science 339, 52 (2013)

Blue: 𝑇𝑓𝑖𝑡 = 2.7𝐽

𝑘𝐵

Red: 𝑇𝑓𝑖𝑡 = −2.2𝐽

𝑘𝐵

Kinetic energy distribution is well fitted by Bose-Einstein distribution. System is (locally) in thermal state.

Critical temperature in 2D: • homogeneous, interacting

|𝑇𝐵𝐾𝑇| ≈ 1.8𝐽

𝑘𝐵

• trapped, non-interacting

𝑇𝑐 = 3.4 2𝐽

𝑘𝐵

Science 339, 52 (2013)

E.A. Donley et al. Nature 412, 295 (2001) C.A. Sackett et al. PRL 82, 876 (1999) J.M. Gerton et al. Nature 408, 692 (2000)

In harmonic trap:

For a<0 a large BEC undergoes a fast collapse Bose-Nova

Stable attractive BECs only for small atom numbers

(85Rb, 𝑎 ≈ −150 𝑎0, JILA)

Negative Temperature at U<0 as stable as positive T at U>0

Anti-trapping potential V0→-V0

is important!

Science 339, 52 (2013)

Landau-Lifshitz, Vol5

Stability criterium

𝑝

𝑇> 0

𝑇 > 0 positive pressure

𝑇 < 0 negative pressure Negative Temperature stabilizes attractive Bose gas

No Bose-Nova

Science 339, 52 (2013)

𝑇 > 0

𝑇 < 0

Friction: entropy increases

Medium heats up Particle slows down

Anti-Friction: entropy increases

Medium cools down Particle accelerates

particle spectrum is assumed to be unbound

(but direction is randomized in long-term limit)

Carnot Efficiency:

Δ𝑊 = Δ𝐸2 − Δ𝐸1 𝜂 =Δ𝑊

Δ𝐸2≤ 1

Cold reservoir 𝑇1 > 0: 𝑆1, 𝐸1

Hot reservoir 𝑇2 > 0: 𝑆2, 𝐸2

Δ𝑆

Δ𝐸1

Δ𝐸2

Δ𝑊

Concentrate on hot reservoir only:

Cold reservoir = environment (comes for free)

Δ𝐸2

𝑇 → 0+

−𝛽 → −∞

𝑇 → ±∞

−𝛽 → 0

𝑇 → 0−

−𝛽 → +∞

Carnot Efficiency:

Δ𝑊 = Δ𝐸2 + Δ𝐸1 𝜂 =Δ𝑊

Δ𝐸2> 1

Cold reservoir 𝑇1 > 0: 𝑆1, 𝐸1

Hot reservoir 𝑇2 < 0: 𝑆2, 𝐸2

ΔE1

Δ𝑆

Δ𝐸2

Opposite entropy flow

Δ𝑊

Note: No quasi-static process can connect T<0 and T>0 Carnot-Engine is impossible

Cold reservoir 𝑇1 > 0

|𝑒

|𝑔

Hot reservoir 𝑇2 ≈ −0

|𝑒

|𝑔

|𝑒

|𝑔

Engine

classical 𝜋 - pulse

Many two-level systems • without spontaneous emission • thermalization via collisions

e.g. hyperfine states

E S

E S

Classical light field becomes amplified Work is extracted

Cold reservoir 𝑇1 > 0: 𝑆1, 𝐸1

Hot reservoir 𝑇2 > 0: 𝑆2, 𝐸2

Δ𝐸1

Δ𝑆

ΔE2

Δ𝑊

Cold reservoir 𝑇1 > 0: 𝑆1, 𝐸1

Hot reservoir 𝑇2 < 0: 𝑆2, 𝐸2

Δ𝐸1

Δ𝑆

Δ𝐸2

Opposite entropy flow

Δ𝑊

𝑇1 > 0, 𝑇2 > 0 𝑇1 > 0, 𝑇2 < 0

𝜂 =Δ𝑊

Δ𝐸2≤ 1 𝜂 =

Δ𝑊

Δ𝐸2> 1

Energy and Entropy are globally conserved! No violation of thermodynamic laws No solution to energy problem!

Bipartite lattices: (e.g. simple cubic, hexagonal)

attractive model at T<0 = repulsive model at T>0

e.g. simulate attractive SU(3) model with 173Yb A. Rapp PRA 85, 043612 (2012)

Non-Bipartite lattices: (e.g. triangular, Kagome) New many-body systems

e.g. stabilize Bosons in flat band of Kagome lattice (Stamper-Kurn setup)

T>0 only T>0 and T<0 T<0 only

T>0 and T<0 in principle on equal footing But Most Hamiltonians are asymmetric in practice often only T>0 relevant

Bose Gas at negative absolute Temperatures

- realized for U<0, V0<0

Thermodynamically stable state with T<0, p<0

No collapse!

Enables above unity Carnot efficiency

𝜂 =Δ𝑊

Δ𝐸2> 1 but No perpetuum mobile

(E, S still conserved!)

New many-body systems:

- e.g. in Kagome lattice, attractive SU(3) in 173Yb,

Science 339, 52 (2013) Ulrich Schneider LMU & MPQ Munich