Transcript Power Point

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, in press (http://arxiv.org/abs/0706.0871 )
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.
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→∞ ptA /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 Rotor ( M  1,

ˆ2
N
Hˆ 
 kV ( )   (t ' t ),
2
t 
 1) :
V (  2 )  V ( ).
Quantum Resonances (QRs): Rational τ/(2π) = l/q, with a band
quasienergy spectrum. Purely quantum ballistic motion:
2
ˆ
N
QR Ratchets: For asymmetric V ( ), e.g., V ( )  V ( ),
a ratchet acceleration may arise at QR,
Nˆ
t
 t,
even for fully chaotic classical system (Iclassical = 0).
t .
2
t
Quantum Kicked Particle:
2

ˆ
p
Hˆ 
 kV ( xˆ )   (t ' t ), V ( x  2 )  V ( x).
2
t 
Translational invariance implies conservation of quasimomentum ,
0 ≤  < 1, in time-evolution of Bloch wave exp(i  x) ( x), with
2π-periodic  ( x). At fixed , p̂  Nˆ   and x → θ: “-rotor”.
General QR Conditions [for integers l, q, r, g, with coprime (l, q) ]:
l

 ,
2 q
  r , g
r gq
 
mod(1).
gl 2
Exactly solvable case of main QRs: τ = 2πl.
Resonant quasimomenta:  = r,g = r/(lg) – 1/2 mod(1).
For general potential V ( ) and initial wave packet
V ( )  Vm exp(im ),
m
1
| 0 ( ) | 
2
2
 0 ( ),
 C (m) exp(im ),
m
one finds, for arbitrary  and defining τ = πl(1 + 2),
Nˆ
t
 Nˆ
0
 ik  mVm C (m)
m0
sin(m  t / 2)
sin(m  / 2)
exp[im  (t  1) / 2].
For resonant  = r,g , a QR-ratchet acceleration is obtained:
Nˆ
t
 Nˆ
0
 Rt ,
with ratchet coefficient
R  2k  j Im[V jg C ( jg )].
j 0
R ≠ 0 for generic potentials and wave packets.
Atom-Optics Experimental Realization of QR-Ratchets
Potential: V ( )  cos(   ) , with symmetry center at
 0 ( )  1  exp[i( 0   )] , symmetric under
Initial wave packet:
time reversal and inversion around symmetry center at
Nˆ
t
 Nˆ
 .
0
   0.
k sin(  t / 2)

sin[ 0      (t  1) / 2].
2 sin(  / 2)
Ratchet acceleration for resonant  with coefficient
k
R  sin( 0   ).
2
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:

Nˆ
t
 Nˆ
0

k t
2

  sin   0      s  exp 2  l  s  .



2 s 1
For resonant  :
 Nˆ
t
 Nˆ
0

t
k
2

 sin   0     exp 2  l  s  .



2
s 1
Experimental values: l = 1, only resonant  = 0.5, Δ  0.1, γ0 = 0 .
Experimental Configuration: BEC of 87 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 k  1.4, t  5, and resonant   0.5.
Best theoretical fits for   0.056
Mean momentum vs. t for k  1.4,    / 2, and resonant   0.5.
Best theoretical fits for   0.056
Mean-momentum change vs.  for k  1.4, t  5, and
(a)    / 2, (b)    / 2. Best theoretical fits for   0.056
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).