quantum chaos and its potential for robust quantum control in ...

Order from disorder: quantum chaos
and its potential for robust quantum control
Andreas Buchleitner
Quantum optics and statistics
www.quantum.uni-freiburg.de
Institute of Physics, Albert Ludwigs University of Freiburg
Nonlinear Dynamics
in Quantum Systems
Tainan, 31st August 2010

Nonlinear Dynamics
in Quantum Systems

In Collaboration with . . .
funded by DFG, DAAD, AvH and VolkswagenStiftung

Vision: Quantum Control through Complexity
complex dynamics generates new and robust quantum phenomena
similar questions in apparently remote areas
[L. Kurtz, LMU München 2001]
[The Economist, Quantum Dreams, 10/3/2001]

Many particle interactions
cold atoms
[N. Bohr, 1936]
compound nuclei
[Wigner, Random Matrix Theory]
[Greiner et al., 2002]

Disorder
In
energy
E 2
E
E 1
0
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position
Anderson localization
metal-insulator transition in
disordered solids [Anderson, 1958]
. . . and in many other systems
with (quasi)randomness
[Fishman et al., 1982; Galvez et al., 1988, Arndt et al.,
1991; Krug et al., 2003; Störzer et al., 2006; Aspect et
al., 2008]

Dynamical Chaos
– e.g., atomic hydrogen in a static magnetic field ‖ẑ –
near integrable
chaotic
5
5
z
z
0
0
−5
−2 −1 0 1 2
ρ
−5
−2 −1 0 1 2
ρ
[Delande & Gay, 1986]

Order/Control from Disorder/Chaos
I Wave packets, nonlinear resonances
II Nondispersive wave packets – examples
III Web-states in non-KAM systems

Wave packets in the harmonic oscillator

[Schrödinger, 1926]

Collaps and revivals in the Coulomb problem
a (radial) Rydberg wave packet (at n 0 = 72). . .
• exhibits a collaps/dispersion,
for classical reasons, after
T collapse ≃
n 0
3π(∆n) 2 × T Kepler
• and revives, due to the
discreteness of the quantum
spectrum, at
T revival ≃ n 0
3 × T Kepler
[Yeazell & Stroud, 1991]
• with possible fractional
revivals in between . . .

The cause of dispersion in an unharmonic potential
It’s the phase space, . . . !
• time evolution of the canonical angle
˙θ = ∂H 0
∂I
= Ω(I), θ(t) = Ω(I)t
with I-dependent frequency Ω(I)
X → action I, Y → angle θ
open circles: t = 0
filled circles: t = T(I = 1)
[Mallalieu & Stroud, 1991]
• Hence, to prevent the wave packet from
spreading, we have to change the phase
space structure – e.g., by an external field!

Near resonant driving of the Rydberg/Kepler dynamics
e
• planetary Kepler motion vs.
Rydberg spectrum
• correspondence principle: classical
round trip frequency T −1
Kepler approx.
equal to quantum level spacing, i.e.
E n0 − E n0 ±1 ≃ dE n
dn = dH 0
dI = Ω(I)
→ large n 0 . . . high density of states!
→ tiny microwave photon energies ω . . . near-resonant, strong coupling!
H(t) = ⃗p2
2 − 1 r + ⃗r ⃗F(t), H(t) = H(t + T), T = 2π/ω

An elliptic island is born
I/n 0
1.10
1.05
1.00
0.95
Hamiltonian (1D, for simplicity) in
action-angle variables
H = H 0 + λ
m=+∞
∑
m=−∞
V m (I)cos(mθ − ωt)
0.90
0 π 2π
θ
For ω ≃ Ω, the term m = 1 dominates and allows for the (time independent)
secular approximation, Θ = θ − ωt, I 0 ≡ I/n 0 = 1, ⃗r F(t) ⃗ = Fz cos(ωt)
H sec ∼ H 0 (I) + λV 1 (I)cos(Θ) ∼ ∂2 H 0
∂I 2 | I 0
(I − I 0 ) 2 + λV 1 (I 0 ) cos(Θ)
I.e., it’s almost a pendulum . . . [Berman & Zaslavsky, Phys. Lett. A 61, 295 (1977)]

Classical phase space of driven Rydberg dynamics
– Poincaré section at ωt = 0 –
(spanned by canonical action I and angle θ)
near integrable
chaotic
1.10
1.4
1.05
1.2
I/n 0
1.00
I/n 0
1.0
0.95
0.8
0.90
0 π 2π
θ
0.6
The elliptic island confines the classical angle!
(mode locking)
0 π 2π
θ
Kolmogorov-Arnold-Moser (KAM) theorem:
island extremely robust against perturbations/experimental imperfections!

Synchronicity of regular and complex spectral structures
KAM torus in “chaotic sea”
J
π
π/2
0
-π/2
-π
-π -π/2 0 π/2 π
θ
regular level structure embedded
into irregular level dynamics –
much alike regular island
embedded into chaotic phase
space
eigenvalues anchored to the
island “go straight”!
3
Energy (10 −4 a.u.)
−1.372
−1.382
−1.392
−1.402
RESCALED ENERGY
2
1
0
−1
−2
−3
0.03 0.032 0.034 0.036 0.038 0.04
F 0
[Zakrzewski et al., 1997]
−1.412
0 0.02 0.04 0.06
0 0.02 0.04 0.06
F

Wrap-up I
Nonlinear/elliptic resonance islands . . .
• are a generic feature of periodically driven, bounded systems
• are robust against perturbations – due to KAM
• manifest as solitonic solutions in the quantum spectrum
[A.J. Lichtenberg, M.A. Lieberman, Regular and Stochastic Motion, Springer
1983; E. Schrödinger, Naturwissenschaften 14, 664 (1926); G.P. Berman, G.M.
Zaslavsky, Phys. Lett. A 61, 295 (1977); J. Zakrzewski et al., Z. Phys. B 103,
115 (1997)]

Rescaling of
by
Digression – Classical scaling laws
H = p2
2 − 1 z
+ Fz cos(ωt)
λH = (pλ1/2 ) 2
− 1
2 (zλ −1 ) + Fλ2 zλ −1 cos(ωλ 3/2 tλ −3/2 )
leaves Hamiltonian equations of motion form-invariant, but not the commutator:
[p,z] → λ −1/2 [p,z] = iλ −1/2 ,
nor the action
S =
∮
pdq → λ −1/2 ∮
pdq .
Scaled quantities, with λ = n 2 :
H sc = Hn 2 , p sc = pn, z sc = zn −2 , F sc = Fn 4 ,ω sc = ωn 3 , S sc = Sn −1 , sc = n −1 .
Thus can argue in terms of fixed classical phase space structure!

Order/Control from Disorder/Chaos
I Wave packets, nonlinear resonances
II Nondispersive wave packets – examples
III Web-states in non-KAM systems

Islands of stability: nondispersive electronic wave packets
along a 1D Rydberg orbit
phase space projection of a Floquet eigenstate – 2π/ω-periodic!!!
I u (a)
(c)
!t=0 !t=2 !t= !t=32 !t=2
(b) (d) F = 0.035/n 4 , ω = 1.0/n 3 , n = 57
no dispersion – lifetimes Γ −1
ǫ ≃ 10 6 × 2π ω
[–, 1993, D. Delande & –, 1994]
θ u
8.0
8.0
6.0
6.0
4.0
4.0
2.0
2.0
Floquet state in configuration space
electronic density
0.0
0.0 0.3 0.6 0.9
0.0 0.3 0.6 0.9 0.0 0.3 0.6 0.9
radial distance (104atomic units)
0.0
0.0 0.3 0.6 0.9 0.0 0.3 0.6 0.9

Experimental evidence
position-sensitive detection
experiments [Maeda et al., 2004-09; Mestayer et al., 2009]
experimental life time
≥ 15000 Kepler orbits!
linear polarization ‖ẑ and parallel static electric field,
ωt = 0, π/2, 3π/4, π,
n = 60, ρ, z = ±10000 a.u.;
[–, Delande & Zakrzewski, Phys. Rep. 368, 409 (2002)]

Arbitrary control in 3D – wave packet on circular trajectory
control through circularly polarizated e-m field, ρ,z = ±10000 a.u.
[Bia̷lynicki-Birula, Kaliński & Eberly, 1994; Brunello, Uzer & Farelly, 1996; Zakrzewski, Delande, -, 1995]
EXPERIMENTS: [Maeda et al., 2009; Mestayer et al, 2009]
isovalue plots of the electronic density

Wave packet on elliptic trajectory
linearly polarizated control field, rotational symmetry around z-axis,
snapshots at different phases of Kepler cycle
ρ, z = ±3500 a.u.
[-, Sacha, Delande & Zakrzewski, 1998]
isovalue plots of the electronic density

Clear spectroscopic signature
Floquet spectroscopy of the dressed atom
4
Square dipole (10 −7 a.u.)
3
2
1
0
−1.36 −1.35 −1.34
Energy (10 −4 a.u.)
[-, Delande & Zakrzewski, 1996,2002]

Driven helium
Tuneling
H = p2 1 + p 2 2
2
− Z r 1
− Z r 2
+ 1
r 12
+ F(x 1 + x 2 ) cosωt
e
e
The frozen planet configuration
e
Z
e
Scattering
[Corkum, 1993]
lives on a well defined adiabatic potential.
V eff
/ N −2
0.0
−0.2
0 5 10
z 1
/ N 2
Does atomic structure matter?

Nondispersive wave packets in driven (2D) helium
phase space
(projection along field polarization axis)
y
1
0
configuration space
0.11
p 1
0.00
-1
0 2 4 6
x
−0.11
0.11
projective phase space
p 1
0.00
0.4
a
b
−0.11
0 200 0 200 0
200
ωt = 0 . ..π/2 . ..π (left to right)
life times approx. several hundred field cycles
x
1
p
1
0.0
-0.4
0 10 20
x
1
0 10 20
[Schlagheck & –, 1998, 2003]
[J. Madroñero, PhD thesis, LMU Munich (2004); Madroñero & –, PRA 2008]

A grain of salt – chaos-assisted tunneling
Example: atomic hydrogen 1D
10 −6
10 −7
ionisation rate
10 −8
10 −9
10 −10
10 −11
10 −12
10 −13
10 −14
10 −15
10 −16
spontaneous emission
effective decay rate
10 30 50 70 90 110 130
principal quantum number n 0
• tunneling rate
Γ ǫ ∼ exp(−S sc n 0 /)
• spontaneous emission M ∼ n −α
0
• fluctuating decay rates:
chaos-assisted ionization
[Hornberger & –, 1998; Delande & Zakrzewski, 1998]
[Zakrzewski, Delande & –, 1998]

Solitonic levels in many-particle problems
E
4
2
0
−2
−4
0.02
Bose-Hubbard Hamiltonian under static tilt
(
H = − J B ∑ )
2 l a† l+1 a l + h.c. + W B
2
∑l n l(n l − 1) + Fd ∑ l ln l
• parametric spectral evolution
H(F)|ψ(F)〉 = E(F)|ψ(F)〉
• inset: reorganization
to form – localized – “solitonic level”
occupation
1.5
−0.02
0 0.01
0 0.1 0.2 0.3
F
N = 3, L = 11, J B = 0.038, W B = 0.032
1.
0.5
1 2 3 4 5 6 7 8 9 10 11
lattice site

Entanglement generation vs. decoherence
1.75
1.5
C k
C k
1.25
1
0.75
0.5
0.25
0
1.75
1.5
1.25
1
0.75
0.5
0.25
0
1
0
0
1
0 4 8 12 16 20
t
• classically chaotic dynamics encoded in
k = 5 (top), k = 8 (bottom) qubits
• initial conditions in elliptic island (blue), in
chaotic domain (red)
• without (open symbols) and with (filled
symbols) scaled diffusive noise
! elliptic island screens entanglement against
noise-induced decay – the larger k, the
better
[García-Mata et al., PRL 2007]

Wrap-up II
Nondispersive wave packets . . .
• anchored to nonlinear resonances in classical mixed, regular-chaotic
phase space
• allow to shuttle around, and to store, quantum states in phase space
• screen entanglement against decoherence
[–, D. Delande & J. Zakrzewski, Phys. Rep. 368, 409 (2002); I. Garcia-Mata et
al., Phys. Rev. Lett. 98, 120504 (2007)]

Order/Control from Disorder/Chaos
I Wave packets, nonlinear resonances
II Nondispersive wave packets – examples
III Web-states in non-KAM systems

Kicked HO: phase space structure and level dynamics
φ
6
4
2
v
−360 −180 0 180 360
360
180
0
−180
−360
u
H = p2
2m + mν2 x 2
2
+ Acos(kx) ∑ n
δ(t − nτ)
• e.g., cold ions/atoms in harmonic
trap
φ
0
0.2 0.3 0.4 0.5
6
4
2
η
v
−40 −20 0 20 40
40
20
0
−20
−40
u
• Eigenphases φ of one-cycle √ Floquet
propagator vs. η = k
2mν ∼ √
• q = 2π
ντ
= 5 (top), q = 6 (bottom),
K = Ak2
mν = 2.0
• insets: classical trajectory after 40000
kicks – note the web!
0
0.2 0.3
η
0.4 0.5

Web-states anticross localized eigenstates
Husimi representations of anticrossing eigenstates at
η = 0.459 (left), η = 0.469 (right)
• at η = 0.464, the localized initial
state |ψ 0 〉 = |0〉 has maximal overlap
with the extended web-state (top
left and bottom right)
• enhanced diffusion due to
anticrossing with web-states!
60
1.45
Mean Energy
40
20
1.35 φ
b
c
1.25
0.455 a 0.465 0.475
η
c
b
classical
0
0 200 400 600
Number of kicks
a
• alike metal-insulator transition
[Fromhold et al., 2004]

An experimental tool for controlled heating
The Lamb-Dicke parameter is easily tuned in experiments on trapped cold atoms or ions
150
6
Mean Energy
100
4
φ
• enhanced mean energy always
associated with strong coupling to
web-states!
• probe of local spectral structure
50
2
[A.R.R. Carvalho & –, 2004]
0
0.2 0.3 0.4 0.5
η
0

What to take away . . .
• nondispersive wave packets are generic (since nonlinear resonances are)!
• eigenstates of virtually any periodically driven quantum system
• they are robust against technical noise, due to KAM
• screen many-particle entanglement against noise (not shown)
• quantum control also in non-KAM systems, e.g. through web-states
Literature (selection):
• G.P. Berman & G.M. Zaslavsky, Phys. Lett. A 61, 295 (1977); H. Klar, Z. Phys. D 11, 45 (1989); I. Bia̷lynicki-Birula, M. Kaliński,
and J.H. Eberly, Phys. Rev. Lett. 73, 1777 (1994); D. Delande & –, Adv. At. Mol. Opt. Phys. 34, 85 (1994); D. Farelly, E. Lee, and
T. Uzer, Phys. Rev. Lett. 75, 972 (1995); –, D. Delande, & J. Zakrzewski, Phys. Rep. 368, 409 (2002); D.G. Arbo et al., Phys. Rev.
A 67, 63401 (2003); J. Madroñero, PhD thesis, LMU München (2004); F.B. Dunning et al., Adv. At. Mol. Opt. Phys. 52, 49 (2005),
Phys. Rev. A 78, 045501 (2009) ; J. Madroñero et al., Adv. At. Mol. Opt. Phys. 53, 33 (2006); J. Madroñero & –, Phys. Rev. A
77, 053402 (2008); H. Maeda & T.F. Gallagher, Phys. Rev. Lett. 92, 133004 (2004), Science 307, 1757 (2005), Phys. Rev. Lett.
102, 103001 (2009); J.J. Mestayer et al., Phys. Rev. A 79, 033417 (2009).
• T.M. Fromhold et al., Nature 428, 726 (2004); A.R.R. de Carvalho & –, Phys. Rev. Lett. 93, 204101 (2004).
• I. Garcia-Mata et al., Phys. Rev. Lett. 98, 120504 (2007).