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, in press (http://arxiv.org/abs/0706.0871 )

B”H

Classical Hamiltonian RatchetsClassical Hamiltonian Ratchets

General concept of “Ratchet”: Translationally-invariant system in which a directed transport can beestablished 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.

Hamiltonian-Ratchet Maps: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 region A:

IA = |A| limt→∞ ptA /t

f(x) = 0 ICHAOS + IISLANDS = 0

Symmetric case: f(-x) = - f(x),

ICHAOS = - IISLANDS = 0

Asymmetric case: f(-x) ≠ - f(x),

ICHAOS = - IISLANDS ≠ 0

ICHAOS = 0 for fully chaotic system.

Generalized Quantum Kicked RotorGeneralized Quantum Kicked Rotor

2 2ˆ .t

N t

2ˆˆ ( ) ( ' ), ( 2 ) ( ).

2 t

NH kV t t V V

Quantum Resonances (QRs)Quantum Resonances (QRs): : RationalRational τ/(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).

( )V ˆ ,

tN t

( ) ( ),V V

( 1, 1) :M

Quantum Kicked ParticleQuantum Kicked Particle:

2ˆˆ ˆ( ) ( ' ), ( 2 ) ( ).2 t

pH kV x t t V x V x

ˆp̂ N

Translational invariance implies conservation of quasimomentum ,

0 ≤ < 1, in time-evolution of Bloch wave with

2π-periodic At fixed , and x → θ: “-rotor”.

General QR Conditions [for integers l, q, r, g, with coprime (l, q) ]:

,, mod(1).2 2r g

l r gq

q gl

exp( ) ,( )i x x

( ).x

Exactly solvable case of main QRs: τ = 2πl.

Resonant quasimomenta: = r,g = r/(lg) – 1/2 mod(1).

For general potential and initial wave packet

20

1( ) exp( ), | ( ) | ( ) exp( ),

2mm m

V V im C m im

0 ( ), ( )V

one finds, for arbitrary and defining τ = πl(1 + 2),

00

sin( / 2)ˆ ˆ ( ) exp[ ( 1) / 2].sin( / 2)m

tm

m tN N ik mV C m im t

m

For resonant = r,g , a QR-ratchet acceleration is obtained:

with ratchet coefficient

0

ˆ ˆ ,t

N N Rt

0

2 Im[ ( )].jgj

R k j V C jg

R ≠ 0 for generic potentials and wave packets.

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 .

( ) cos( )V

0 0( ) 1 exp[ ( )]i

0

0sin( ).2

kR

00

sin( / 2)ˆ ˆ sin[ ( 1) / 2].2 sin( / 2)t

tkN N t

Ratchet acceleration for resonant with coefficient

For a BEC with quasimomenta (initial momenta) Gaussian

distributed with small width Δ << 1 around some given , the

average QR-ratchet behavior for arbitrary is exactly given by:

00

1

2ˆ ˆ sin exp 2 .2t

s

tkN N s l s

For resonant :

00

1

2ˆ ˆ sin exp 2 .2t

s

tkN N l s

Experimental values: l = 1, only resonant = 0.5, Δ 0.1, γ0 = 0 .

87Experimental Configuration: BEC of Rb atoms initially prepared in a

superposition state and exposed to a pulsed optical potential moving

relative to the BEC with adjustable "velocity" (quasimomentum)

Mean momentum vs. for 1.4, 5, and resonant 0.5.

Best theoretical fits for 0.056

k t

Mean momentum vs. for 1.4, / 2, and resonant 0.5.

Best theoretical fits for 0.056

t k

Mean-momentum change vs. for 1.4, 5, and

(a) / 2, (b) / 2. Best theoretical fits for 0.056

k t

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).