QUANTUM-RESONANCE RATCHETS: THEORY AND EXPERIMENT

1. B”H QUANTUM-RESONANCE RATCHETS:THEORY AND EXPERIMENT I. Dana (Bar-Ilan University) Theory: ID and V. Roitberg, PRE 76, 015201(R) (2007). Experiment: ID, V. Ramareddy, I. Talukdar, and G.S. Summy, PRL 100, 024103 (2008).

2. Classical Hamiltonian Ratchets General concept of “Ratchet”: Translationally-invariant system in which a directed transport can be established without a biased force (average force = 0). Usually, the directed transport is due to the breaking of a spatial/temporal symmetry. E.g., molecular “motors” in biological systems with dissipation and external noise [see, e.g., R.D. Astumian and P. Hänggi (2002)]. Classical Hamiltonian Ratchet [S. Flach et al. (2000), T. Dittrich et al. (2000, 2001)]: No dissipation and external noise is replaced by deterministic chaos.

3. Hamiltonian-Ratchet Maps: pt+1 = pt + f(xt), xt+1 = xt + pt+1 mod(2π), f(x + 2π, t) = f(x), f(x) = 0. Momentum Current (acceleration) of phase-space regionA: IA = |A| limt→∞ ptA /t f(x) = 0 ICHAOS + IISLANDS = 0

4. Asymmetric case:f(-x) ≠ - f(x), ICHAOS = - IISLANDS ≠ 0 Symmetric case:f(-x) = - f(x), ICHAOS = - IISLANDS = 0 ICHAOS = 0 for fully chaotic system.

5. Generalized Quantum Kicked Rotor Quantum Resonances (QRs): Rationalτ/(2π) = l/q, with a band quasienergy spectrum. Purely quantum ballistic motion: QR Ratchets: For asymmetric ,e.g., a ratchet acceleration may arise at QR, even for fully chaotic classical system(Iclassical = 0).

6. Quantum Kicked Particle: Translational invariance implies conservation ofquasimomentum, 0 ≤  < 1, in time-evolution of Bloch wave with 2π-periodic At fixed, andx→ θ: “-rotor”. General QR Conditions [for integers l, q, r, g, with coprime (l, q) ]:

7. Exactly solvable case of main QRs:τ = 2πl. Resonant quasimomenta: = r,g = r/(lg) – 1/2 mod(1). For general potentialand initial wave packet one finds, for arbitraryand defining τ= πl(1 + 2),

8. For resonant = r,g , a QR-ratchet acceleration is obtained: with ratchet coefficient R ≠ 0for generic potentials and wave packets.

9. Atom-Optics Experimental Realization of QR-Ratchets Potential: , with symmetry center at. Initial wave packet:, symmetric under time reversal and inversion around symmetry center at . Ratchet acceleration for resonant with coefficient

10. For a BEC with quasimomenta (initial momenta) Gaussian distributed with small widthΔ << 1around some given,the average QR-ratchet behavior for arbitrary is exactly given by: For resonant : Experimental values:l =1, only resonant = 0.5,Δ  0.1,γ0=0.

15. CONCLUSIONS: • QR-Ratchet: Purely quantum momentum current (acceleration) for resonant quasimomenta  . • QR-ratchet effects can emerge also for symmetric potentials and initial wave packets if, e.g., their symmetry centers do not coincide. Results are totally unaffected by potential high harmonics for simple initial wave packets. • Consideration of arbitrary : Indispensable for taking into account the small but finite quasimomentum width of the BEC, leading to a suppression of the QR-ratchet acceleration. Pronounced ratchet effect near resonant  . • Work in progress: Experimental realization of QR-ratchets in the free-falling frame of quantum kicked particle under “gravity” (linear potential).