Quantum dynamics with ultra cold atoms Nir Davidson Weizmann Institute of Science Billiards BEC I. Grunzweig, Y. Hertzberg, A. Ridinger (M. Andersen, A. Kaplan) Eitan Rowen, Tuesday

R E 1 nm Dynamics inside a molecule: quantum dynamics on nm scale Fsec laser pulse

Is there quantum chaos? Classical chaos: distances between close points grow exponentially Quantum chaos: distance between close states remains constant Asher Peres (1984): distance between same state evolved by close Hamiltonians grows faster for (underlying) classical chaotic dynamics ??? Answer: yes….but also depends on many other things !!! One thing with many names: survival probability = fidelity = Loschmidt echo R. Jalabert and H. Pastawski, PRL 86, 2490 (2001)

PRL 86, 1518 (2001), PRL 87, (2001), PRL (2003) …and effects of soft walls, gravity, curved manifolds, collisions….. Atom-optics billiards: decay of classical time-correlations

Wedge billiards: chaotic and mixed phase space

Criteria for “quantum” to “classical” transition Old: large state number Quantum dynamics with ~10 6 : challenges and solutions: Very weak (and controlled) perturbation –optical traps + very strong selection rules No perturbation from environment - ultra cold atoms Measure mixing – microwave spectroscopy Pure state preparation? - echo New: “mixing” to many states by small perturbation But “no mixing” is hard to get

Pulsed microwave spectroscopy Prepare Atomic Sample → MW-pulse Sequence → Detect Populations Off On cooling and trapping ~10 6 rubidium atoms optical pumping to π-pulse: π/2-pulse: optical transition MW “clock” transition

Ramsey spectroscopy of free atoms H = H int + H ext → Spectroscopy of two-level Atoms π/2 T MW Power Time

Ramsey spectroscopy of trapped atoms  E HF |1,Ψ> |2,Ψ> |1,Ψ> H2H2 H1H1 e -iH 2 t |2,Ψ> e -iH 1 t |1,Ψ> … Microwave pulse General case: Nightmare Short strong pulses: OK (Projection)

Ramsey spectroscopy of single eigenstate π/2 MW Power Time T For small Perturbation:

Ramsey spectroscopy of thermal ensemble π/2 MW Power Time T Averaging over the thermal ensemble destroys the Ramsey fringes For small Perturbation:

Echo spectroscopy (Han 1950) π/2 T MW Power Time π T t=T t=2T NOTE: classically echo should not always work for dynamical system !!!!

Echo spectroscopy π/2 T MW Power Time π T Coherence De-Coherence Ramsey Echo BUT: it works here !!!!

Ramsey Echo Echo vs. Ramsey spectroscopy H2H2 H 1 H2H2 H1H1 H 2

Quantum dynamics in Gaussian trap Coherence De-Coherence Calculation for H.O. T osc /2T osc  E HF

Long-time echo signal Coherence De-Coherence 2-D: 1-D:

Observation of “sidebands” Π-pulse 4π-pulse

Quantum stability in atom-optic billiards ~10 4

Quantum stability in atom-optic billiards ~10 4 D. Cohen, A. Barnett and E. J. Heller, PRE 63, (2001)

Avoid Avoided Crossings

Quantum dynamics in mixed and chaotic phase-space Coherent  Incoherent  Perturbation strength Perturbation-independent decay

Quantum dynamics in perturbation-independent regime

Shape of perturbation is also important

… and even it’s position

No perturbation-independence

Finally: back to Ramsey (=Loschmidt)

Quantum dynamics of extremely high-lying states in billiards: survival probability = Loschmidt echo = fidelity=dephasing? Quantum stability depends on: classical dynamics, type and strength of perturbation, state considered and…. “Applications”: precision spectroscopy (“clocks”) quantum information Conclusions Can many-body quantum dynamics be reversed as well? (“Magic” echo, Pines 1970’s, “polarization” echo, Ernst 1992)

Control classical dynamics (regular, chaotic, mixed…) Quantum dynamics with ~10 6 ???? TzahiArielNir Atom Optics Billiards

Positive (repulsive) laser potentials of various shapes. Standing Wave Trap Beam Z direction frozen by a standing wave Low density  collisions “Hole” in the wall  probe time-correlation function