Agenda Brief overview of dilute ultra-cold gases Quantum dynamics of atomic matter waves The Bose Hubbard and Hubbard models

Quantum degenerate Bose gas High temperature T: Thermal velocity v Density d-3 “billiard balls” Low temperature T: De Broglie wavelength lDB=h/mv~T-1/2 “Wave packets” T=Tcrit : Bose Einstein Condensation De Broglie wavelength lDB=d “Matter wave overlap” T=0 : Pure Bose Einstein Condensate “Giant matter wave ” Ketterle

BEC in dilute ultracold gases In 1995 (70 years after Einstein’s prediction) teams in Colorado and Massachusetts achieved BEC in super-cold gas.This feat earned those scientists the 2001 Nobel Prize in physics. S. Bose, 1924 A. Einstein, 1925 E. Cornell C. Wieman W. Ketterle Light Atoms Using Rb and Na atoms

How about Fermions? At T<Tf ~Tc fermions form a degenerate Fermi gas 𝐸 𝐹 ~ ℏ 2 2𝑚 3 𝜋 2 𝑛 2/3 1999: 40 K JILA, Debbie Jin group

To cool the atoms down we need to apply lasers in all three directions Laser cooling An atom with velocity V is illuminated with a laser with appropriate frequency Trapping and cooling: MOT Atom To cool the atoms down we need to apply lasers in all three directions laser Atoms absorb light and reduce their speed Slower atom Laser cooling is not enough to cool down the atoms to quantum degeneracy and other tecniques are required As the atoms slow down the gas is cooled down

Evaporative cooling Temperatures down to 10-100 nanoK Each atom behaves as a bar magnet This process is similar to what happens with your cup of coffee. The hottest molecules escape from the cup as vapor In a magnetic field atoms can be trapped: magnetic field~coffee cup By changing the magnetic field the hot atoms escape and only the ones that are cold enough remain trapped 2001 BEC

How to see a BEC ? T>Tc T<Tc T=Tc Time of flight images t=0 Turn off trapping potentials Light Probe Image T>Tc T<Tc T=Tc MPI,Munich

Anisotropic

Characterizes low energy collisions Scattering in quantum mechanics (1) Particles behave like waves (T → 0) − ℏ 2 2𝜇 𝛻 2 Ψ+𝑉(𝑟) Ψ=EΨ There is only a phase shift at long range!! Phase shift at low energy proportional to “scattering length” 𝑎 Characterizes low energy collisions

Basic Scattering in quantum mechanics Goal: find scattered wave f: scattering amplitude ∝ 𝑒 𝑖𝑘𝑧 +𝑓(Ω) 𝑒 𝑖𝑘𝑟 𝑟 (r) = ℓ 𝑅 ℓ (𝑟,𝐸)𝑃 ℓ ( cos 𝜃) . l = 0, 1, 2… s-, p-, waves, … There is only a phase shift at long range!! 𝑅 ℓ (𝑟→∞)→ 1 𝑘𝑟 sin 𝑘𝑟−ℓ 𝜋 2 − 𝛿 ℓ

Basic quantum scattering: s-wave cross section atoms/s scattered flux into d plane wave flux (atoms/cm2/s) Solve Schrödinger equation for each l 𝛿 ℓ Quantum statistics matter Pauli Exclusion principle Get phase shift 𝛿 ℓ Identical bosons: even Identical fermions: odd Non-identical species: all 𝛿 ℓ 𝑘 → 𝐴 ℓ ( 𝐴 ℓ 𝑘) 2ℓ 𝑘→0 𝐴 0 =a “scattering length” Characterize s-wave collisions 𝐴 1 =b3 “scattering volume ” Characterize p-wave collisions

Pseudo-Potential • Two interaction potentials V and V’ are equivalent if they have the same scattering length • So: after measuring a for the real system, we can model with a very simple potential. • Actually, to avoid divergences you need Huang and Yang, Phys. Rev. 105, 767 (1957) Also E. Fermi (1936), Breit (1947), Blatt and Weisskopf (1952)

Weakly interacting gases Quantum phenomena on a macroscopic scale. Ultra cold gases are dilute a: Scattering Length n: Density Cold gases have almost 100% condensate fraction. In contrast to other superfluids like liquid Helium which have at most 10%

Many body Bosonic systems Field operator Many body Hamiltonian Equation of Motion In general Ψ ′ =0

Gross Pitaevskii Equation Dilute ultracold bosonic atoms are easy to model 1. Short range interactions 2. At T=0 all share the same macroscopic wave function Ψ Φ =Φ Φ Gross-Pitaevskii Equation We can understand the many-body system by a single non linear equation

BEC and quantum coherence A BEC is a coherent collection of atomic deBroglie waves laser light is a coherent collection of electromagnetic waves laser

Quantum coherence at macroscopic level Vortices JILA 2002 MIT (1997). Coherence Weakly interacting Bose Gas Superflow Non Linear optics MIT (1997). NIST (1999).