110, 190401 (2013) Evidence for a Quantum-to-Classical...

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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 Schneble Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA BEC in a pulsed, incommensurate optical lattice Wavelengths λ 1 λ 2 T τ N - 1 N s 1 s 2 1 2 (E R = h 2 /2Mλ 1 2 ) (λ 1 = ηλ 2 ; η ~ 1.36) λ 1 = 1064 nm λ 2 = 782 nm V 1,2 = s 1,2 E R Depths Ĥ = - ћ 2 2 z 2M s 2 E R τ 2 + cos ( 2 η k 1 z ) δ ( t - j T ) j = 1 N Σ [ ] s 1 E R τ 2 cos ( 2 k 1 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 p mn = (m + ηn) 2 ћk 1 momentum-mode-dependent quasi-energies (phase-accruals) pseudo-random for κ ≠ 4π and η 2 κ ≠ 4π ( â m,n-1 â m,n + â m,n+1 â m,n ) m,n Σ K 2 κ φ V 2 = discrete-time tunneling in momentum-space between modes separated by 2ћk 1 ( â m-1,n â m,n + â m+1,n â m,n ) m,n Σ K 1 κ φ V 1 = δ ( t - j T ) j = 1 N Σ Ĥ = Ĥ 0 + ћ ( φ V 1 + φ V 2 ) ^ ^ tunneling between modes separated by 2ηћk 1 s 1,2 E R τ 2ћ K 1,2 κ = controls strength of tunneling m,n ћ T m,n Σ κm 2 + η 2 κn 2 + 2ηκmn 2 Ĥ 0 = { 4 E R T h κ 4π = controls depth of “disorder” cross-dimensional coupling 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 10 p / ħk 1 10 -1 10 -2 10 -3 10 -4 |ψ(p)| 2 N = 40 K 1, 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 10 p / ħk 1 K 1, 2 = { 0 , 4.6 } N = 40 z y Dynamical localization due to pseudo-random quasi-energies η 2 κ/4π = 0.54 Single-lattice n 2 η 2 κ / 4π mod[1] 0 n 0 -2 -4 -6 2 4 6 Coupling leads to classical diffusion (T = 36 µs) Two-lattice κ/4π = 0.29 expt. sim. K 1 = K 2 Energy change (in E R ) 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 N 40 0 20 60 80 100 σ p 5 0 -5 5 0 -5 0 -10 10 m n n 20 15 10 5 0 a b 1 .5 0 P m,n N = 100 N = 100 η 2 κ/4π = 1.86 (T = 124 µs) Resonant, ballistic momentum spreading along modes of lattice 1(with K 1 = 1.6) is suppressed by strong off-resonant kicking with K 2 N 0 5 10 15 20 3 2 1 0 3 2 1 0 3 2 1 0 σ p K 2 0 5 10 15 ∆σ p / N .05 0 K 2 = 0 K 2 = 8 K 2 = 16 σ p σ p K 1 = 1.6 num. sim. w/ finite size num. sim. × 1/2 (due to finite size) κ/4π 1 ηκmn ε / N Agrees well with time-independent Floquet analysis of unit-step evolution operator Dynamics P m,n without coupling term with coupling term K 1,2 = 5.8 N = 100 a [b]: K 1,2 = {1.6,12.8} [a]: K 1,2 = {1.6,0} Simulated long-time dynamics Simulated long-time dynamics momentum width energy

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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