Evidence for a Quantum-to-Classical Transition in a Pair of Coupled Quantum Rotors
Phys. Rev. Lett. 110, 190401 (2013)
Bryce Gadway, Jeremy Reeves, Ludwig Krinner, and Dominik SchnebleDepartment of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA
BEC in a pulsed, incommensurate optical lattice
Wavelengthsλ1λ2
T
τ
N - 1 Ns1
s2
1 2
(ER = h2/2Mλ12)
(λ1 = ηλ2 ; η ~ 1.36)λ1 = 1064 nm
λ2 = 782 nm
V1,2 = s1,2 ER
Depths
Ĥ = - ћ2 ∂ 2z
2M
s2 ER τ
2+ cos ( 2 η k1 z ) δ ( t - j T )
j = 1
N
Σ[ ] s1 ER τ
2cos ( 2 k1 z ) +
For short δ-like pulses, the system is described by
For irrational η, with no intersection between sets of states coupled by the two lattices, we can write an effective 2D Hamiltonian in the basis of plane-wave
states m , n with momentum pmn = (m + ηn) 2 ћk1
momentum-mode-dependentquasi-energies (phase-accruals)
pseudo-random forκ ≠ 4π and η2κ ≠ 4π
( â†m,n-1 âm,n + â†
m,n+1 âm,n ) m,nΣK2
κφV2 =
discrete-time tunnelingin momentum-space
between modes separated by 2ћk1
( â†m-1,n âm,n + â†
m+1,n âm,n ) m,nΣK1
κφV1 =
δ ( t - j T )j = 1
N
Σ Ĥ = Ĥ0 + ћ ( φV1 + φV2 )^ ^
tunneling between modesseparated by 2ηћk1
s1,2 ER τ2ћ
K1,2
κ= controls strength
of tunneling
�m,n ћ
T m,nΣ κm2 + η2κn2 + 2ηκmn
2 Ĥ0 = {
4 ER T
h
κ4π
=
controls depthof “disorder”
cross-dimensionalcoupling term
ηκmn
• quantum-to-classical transition in simple closed system with unitary, time-reversible dynamics
Motivation
We study the dynamical response of matter-waves to a pulsed incommensurate lattice.
A quantum-to-classical transition is predicted to occur in this system [S. Adachi, M. Toda, K. Ikeda, PRL 61, 659 (1988)]
This system realizes a variant of the δ-kicked rotor model [Casati, et al. (1979) ; Moore, et al. (1995)] , that of coupled kicked quantum rotors
• classical physics emerges naturally without coupling to reservoir & decoherence
• new light on nature of localization in 2D systems
Off-Resonant Kicking (κ / 4π ≠ 0)
-10 -5 0 5 10p / ħk1
10-1
10-2
10-3
10-4
|ψ(p
)|2
N = 40
K1, 2 = { 2.3, 3.7 }
N = 1
N = 40
N = 1
N = 40
10-1
10-2
10-3
10-4
|ψ(p
)| 2
-10 -5 0 5 10p / ħk1
K1, 2 = { 0 , 4.6 }
N = 40
zy
Dynamical localization due topseudo-random quasi-energies
η2κ/4π = 0.54
Single-lattice
n2 η2 κ
/ 4π
mod
[1]
0
n0-2-4-6 2 4 6
Coupling leads toclassical diffusion
(T = 36 µs)
Two-lattice
κ/4π = 0.29
expt. sim.
K1 = K2
Energy change (in ER) over 40 kicks
On-Resonant Kicking (κ / 4π ≈ 0)
Coupling destroys resonance for modes n ≠ 0. However, nearly 1/2 time spent in n = 0 subspace
Suppression of transport reminiscent of Kapitza pendulum / ponderomotive potentials
N400 20 60 80 100
σp
5
0
-5
5
0
-5
0-10 10m
n
n
20
15
10
5
0
a
b
1
.5
0
Pm,n
N = 100
N = 100
η2κ/4π = 1.86(T = 124 µs)
Resonant, ballistic momentum spreading along modes of lattice 1(with K1 = 1.6) is suppressed by strong off-resonant kicking with K2
N0 5 10 15 20
321032103210
σ p
K2
0 5 10 15
∆σp /
∆N .05
0
K2 = 0
K2 = 8
K2 = 16
σ pσ p
K1 = 1.6num. sim.w/ finite size
num. sim. × 1/2(due to finite size)
κ/4π ≈ 1
ηκmn
∆ ε /
∆N
Agrees well withtime-independentFloquet analysisof unit-step evolution operator
Dynamics
Pm,nwithout coupling termwith coupling term
K1,2 = 5.8 N = 100
a
[b]: K1,2 = {1.6,12.8}
[a]: K1,2 = {1.6,0}
Simulated long-time dynamics
Simulated long-time dynamics
mo
men
tum
wid
then
erg
y
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