Non-Equilibrium Physics with Quantum Gases David Weiss Yang Wang Laura Adams Cheng Tang Lin Xia Aishwarya Kumar Josh Wilson Teng Zhang Tsung-Yao Wu Neel Malvania NSF, ARO, DARPA,
Outline Intro: cold atoms as isolated quantum systems Mean field experiments at various T Lightly sampled Gases with correlations 1D gases with δ-interactions: thermalization Time scales, Integrability, Correlations, Thermalization
small complications: intensity, current, position fluctuations; background gas collisions; spontaneous emission Cold gas experiments Cold gases are metastable trapped gas inelastic 3-body (typically) metal collisions coating the This is usually unimportant vacuum chamber Any study of dynamics has to be concerned with timescales Cold gases are very well isolated quantum systems light trapping p E, U=-p. E U AC Intensity magnetic trapping U = -B
S-wave interactions can be accounted for with the Huang pseudo-potential 2 4 3 V r = a r m Long range behavior correct Enforces boundary condition The Mean Field R 1 a r r a 0 = = This leads to the Gross-Pitaevskii equation (non-linear S.E.) = 2 2 2 4 2 0 V r a = E = 2m m N The effects of collisions are in the mean field term. There is nothing irreversible about it! The evolution is integrable, with excitations of the only degrees of freedom 1
Collapse and Revival Prepare atoms in a superposition of number states at each lattice site Hansch/Bloch These collisions are coherent
3D BEC Integrable Evolution Dalibard breathing mode Q~2000 dipole mode mean-field frequency scale thermal collision rate Ketterle for short enough time scales
BEC Formation Esslinger The coherence grows faster than the N BEC
Some Mean Field Non-Eq. Expts. Solitons Tkachenko oscillations Josephson Oscillations Hulet Magnetic dipole+ mean field Stamper-Kurn Cornell Oberthaler Also: Cold neutral plasmas Rydberg blockaded gases Cold molecules
Coupling strength In a Bose gas, the ratio of two energies,, governs the extent of correlations in a quantum gas: 2 4 a Eint = n3 D m Mean field energy 2 2 kf EK = 2m Kinetic energy k F = E E int K 2 = interparticle spacing For low, it is less energetically costly for single particle wavefunctions to overlap than be separated Mean field theory & the G-P equation apply: weak coupling For high, it is less energetically costly for wavefunctions to avoid each other strong coupling As increases, dynamics becomes a quantum many-body problem
Significance of correlations Weak correlations allow long range phase coherence, and macroscopic wavefunction phenomena Eg., interference, superfluidity, vortices Semiclassical Ketterle Bloch Cornell Strongly correlated systems are much harder to calculate, especially out of equilibrium Quantum for bosons at high density in 3D, in optical lattices, or in reduced dimensions Esslinger
Single atom dynamics Mobile spin impurities MISF coherence grows fast Greiner Bloch
1D Bose gases with variable point-like interactions Elliot Lieb and Werner Liniger, 1963: Exact solutions for 1D Bose gases with arbitrary (z) interactions N 2 2 A Bethe ansatz approach H 1D = - + g1dδ zi z 2 j yields solutions j=1 2m z i<j j parameterized by m g1d = Lieb & Liniger, Phys Rev 130 1605 (1963) 2 n 1D Wavefunctions and all other (local and non-local) properties are exactly calculable. Maxim Olshanii, 1998: Adaptation to real atoms 4a3D Ca = 2 1- an1d a 3D -1 a 3D = 3D scattering length a = transverse oscillator length C 1.46 a
>>1 Tonks-Girardeau gas The Lieb-Liniger limits kinetic energy dominates large g 1D low density <<1 mean field theory (Thomas-Fermi gas) mean field energy dominates small g 1D high density g 2 0 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.3 1 3 10 Weiss eff Integrable systems have N constants of motion they cannot thermalize p a, p b, p c p a, p b, p c (no diffractive collisions)
Experimental 1D gases Lattices Chips For 1D:all energies < ħω ; negligible tunneling
1D Evolution in a Harmonic Trap ms 0 Position (μm) -500 0 500 40 μm 1 st cycle average Weiss 5 15 195 ms 10 30 390 ms Quantum Newton s Cradles
Optical thickness Optical thickness Optical thickness 1 st cycle average 15 distribution 40 distribution Lattice depth: 63 E r Steady-state Momentum Distributions Evolution without grating pulses Project the evolution Optical Thickness A B C Position Position Position After dephasing (prethermalization), the 1D gases reach a steady state that is not thermal equilibrium Generalized Gibbs Ensemble- Rigol/Olshanii Each atom continues to oscillate with its original amplitude Lower limit: thousands of 2-body collisions without thermalization
What happens in 3D? Thermalization is known to occur in ~3 collisions. 0 2 4 9 How does thermalization begin in a slightly non-integrable systems? Will it always eventually thermalize?
Summary A lot of non-equilibrium physics can be studied with cold atoms. Mean field integrability is fragile stronger correlations Classical Semi-Classical Quantum mechanics statistical mechanics When integrability is built into the interaction Hamiltonian, the system is robust against thermalization. But how robust? From many diverse phenomena, perhaps universal behavior can be identified