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
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
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)
Earlier work: 85Rb, 133Cs, 52Cr, 7Li, 39K: Lens, Cambridge,
Feshbach resonance: C. D´Errico et al., NJP 9, 223 (2007)
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
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 "
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)
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
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