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Quantum dynamics with ultra cold atoms Nir Davidson Weizmann Institute of Science I. Grunzweig, Y. Hertzberg, A. Ridinger (M. Andersen, A. Kaplan) Billiards n 10 10 6 8 0.1 BEC n 1 0.2 0.3 0.4 0.5 0.6 0.7 Eitan Rowen, Tuesday Dynamics inside a molecule: quantum dynamics on nm scale E 1 nm Fsec laser pulse R Is there quantum chaos? • Classical chaos: distances between close points grow exponentially • Quantum chaos: distance between close states remains constant H n2 exp i H exp i n1 n2 | n1 Asher Peres (1984): distance between same state evolved by close Hamiltonians grows faster for (underlying) classical chaotic dynamics ??? H H n exp i 2 exp i 1 n n |n 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) Atom-optics billiards: decay of classical time-correlations Fraction of Surviving Atoms 10 0 -1 10 -2 10 -3 10 -4 10 0.00 0.05 0.10 0.15 Time [sec] …and effects of soft walls, gravity, curved manifolds, collisions….. PRL 86, 1518 (2001), PRL 87, 274101(2001), PRL 90 023001 (2003) Wedge billiards: chaotic and mixed phase space Criteria for “quantum” to “classical” transition Old: large state number n 1 New: “mixing” to many states by small perturbation But “no mixing” is hard to get n n' n 1 n n' nn' E / E n 106 Quantum dynamics with <n>~106: challenges and solutions: • Very weak (and controlled) perturbation –optical traps + very strong selection rules • No perturbation from environment - ultra cold atoms • Measure n 106 106 1 mixing – microwave spectroscopy • Pure state preparation? - echo Pulsed microwave spectroscopy Prepare Atomic Sample → MW-pulse Sequence → Detect Populations On • cooling and trapping ~106 rubidium atoms • optical pumping to 1 Off π-pulse: 1 i 2 1 i 1 2 2 2 π/2-pulse: 1 3 optical transition MW “clock” transition 2 (5S1/ 2 , F 3, mF 0) 1 (5S1/ 2 , F 2, mF 0) Ramsey spectroscopy of free atoms MW Power H = Hint + Hext → Spectroscopy of two-level Atoms 1 π/2 1 ie 1 i 2 / 2 π/2 T iT 2 / 2 Time 1 e 1 1 e 2 / 2 iT 1 P2 1 cos ΔT 2 iT Ramsey spectroscopy of trapped atoms H H1 1 1 H2 2 2 H int H ext 2 EHF General case: Nightmare Short strong pulses: OK (Projection) 1 Vopt I /( L A ) Microwave pulse |2,Ψ> |1,Ψ> |1,Ψ> H2 H1 e-iH2t|2,Ψ> Microwave pulse <Ψ| eiH1te-iH2t|Ψ>… e-iH1t|1,Ψ> MW Power Ramsey spectroscopy of single eigenstate T π/2 π/2 Time P2 n 1 1 n(t 0) n(t T ) cos nT 2 For small Perturbation: n n' nn' n(t 0) n(t T ) 1 MW Power Ramsey spectroscopy of thermal ensemble T π/2 π/2 Time P2 n 1 1 n(t 0) n(t T ) cos nT 2 For small Perturbation: n n' nn' n(t 0) n(t T ) 1 Averaging over the thermal ensemble destroys the Ramsey fringes MW Power Echo spectroscopy (Han 1950) π/2 T π T π/2 Time t=T t=2T NOTE: classically echo should not always work for dynamical system !!!! MW Power Echo spectroscopy π/2 T π T π/2 Time De-Coherence Ramsey Echo Coherence BUT: it works here !!!! Echo vs. Ramsey spectroscopy H1 H2 n exp i exp i n Ramsey H2 H1 H H H H n exp i 1 exp i 2 exp i 1 exp i 2 n H2 H1 H1 H2 Echo Quantum dynamics in Gaussian trap Tosc/2 De-Coherence Tosc Calculation for H.O. =1.5 nm n' n δn,n' =3.2 nm Coherence =10 nm 2 EHF 1 Long-time echo signal De-Coherence n n' nn' E E / E n 1 P2 1 n' n n 2 n n' nn' E E / E n Coherence n n' nn' E / E 1/ n •2-D: E / E 10 6 •1-D:E / E 10 3 4 Observation of “sidebands” Π-pulse 4π-pulse Quantum stability in atom-optic billiards <n>~104 Quantum stability in atom-optic billiards <n>~104 D. Cohen, A. Barnett and E. J. Heller, PRE 63, 046207 (2001) Avoid Avoided Crossings Quantum dynamics in mixed and chaotic phase-space Incoherent Coherent Perturbation-independent decay Perturbation strength Quantum dynamics in perturbation-independent regime 0,4 P2 Chaotic Mixed 0,2 0,0 0,000 0,005 Time between pulses (s) 0,010 Shape of perturbation is also important … and even it’s position No perturbation-independence Finally: back to Ramsey (=Loschmidt) Conclusions •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 Can many-body quantum dynamics be reversed as well? (“Magic” echo, Pines 1970’s, “polarization” echo, Ernst 1992) Atom Optics Billiards •Control classical dynamics (regular, chaotic, mixed…) •Quantum dynamics with <n>~106 ???? Tzahi Ariel Nir Atom Optics Billiards Positive (repulsive) laser potentials of various shapes. Standing Wave Trap Beam • Low density collisions • Z direction frozen by a standing wave • “Hole” in the wall probe time-correlation function