PASI lectures, Ushuaia, Argentina, October 2000
C.O. Reinhold
In collaboration with
THEORY: S. Yoshida, P. Kristofel, and J. Burgdörfer
EXPERIMENT: M. Frey, B. Tannian, C. Stokely, and B. Dunning
Lecture 1: Introduction, basic concepts.
Tailoring Rydberg wavepackets
Lecture 2: The kicked Rydberg atom
Why Rydberg atoms ?: (i.e. atoms in highly excited states)
- unique arena to study classical quantum correspondence
- high degree of control (experimentally) using current
electromagnetic pulses
Short designer
pulses
Coherent control.
Non-linear dynamics.
PROBLEM: Rydberg Atom in a classical timedependent electric field F(t)
H H
H = H atom + r . F(t)
p2
=
+ V(r)
2
H atom
Atomic units:
D =1,
Hydrogen: V(r) = -1/r
Alkali: V(r) = -1/r + short
range potential
me=1, e=1, rBohr=1, pBohr=1
Atom in a given n level
F(t)
En=-1/(2n2)
Torb= 2πn3 = 2π/(En+1-En)
Forb = 1/<r2>= n-4
DESIRED PULSES:
Short pulses:
duration < Torb
Strong pulses: field strength > Forb
n
Torb
Forb
1
0.15 fs
5 109 V/cm
20
1.2 ps
3 104 V/cm
400
10
0.2
ns
V/cm
Laser/microwave pulse
F(t)
Half-cycle pulse (HCP)
Fp
Tp
t
t
Collisional pulse at a large impact parameter
q
v
b
perpendicular field
Fp=q/b
parallel field
2
Tp=2b/v
t
t
Designer pulses
Field
Single pulse
time
Field
Two pulses
time
Field
Trains of pulses
time
1993 Sub picosecond half-cycle pulses
Jones et al, Noordam et al, Bucksbaum et al
HCP
Tp = 0.5ps
ν ~ 1 THz
Orbital period
of H(n=15)
1996 Nanosecond "designer" pulses
Field (V/cm)
Dunning et al, T p>0.5ns
0.6
0.4
0.2
0.0
5
2ns
0.8
0.4
109ns
0.0
10 15 20 25 30 100
time (ns)
Tp ~ 10ns (ν ~ 100MHz),
200
300
time (ns)
Orbital period of H(n=400)
400
Typical experiment
Actual field in an experiment
F(t)
ns
µs
Prepare the Rydberg
atom using a long
laser pulse
µs
time
Manipulate the
Analyze the
atom with a
product with a
designer pulse F(t) ramped field
Typical result for a single HCP on K(388p)
Survival probability
ionization threshold
How many atoms
were not ionized
by F(t) ?
1.0
0.8
0.6
Theory
0.4
0.2
0.0
0.0
Experiment
0.2
0.4
Fp / Forb
0.6
Typical calculations
2D time-dependent
Lz=Const. ==> problem
H = Hatom + z . F(t) ;
State of the electron
QUANTUM
| Ψ(t)ñ
Wavefunction
CLASSICAL
f(r,p,t)
Probability density in phase space
Dynamics
Classical Liouville equation.
Solvable using a Monte Carlo
approach ==> Hamilton equations.
Schrödinger equation.
Solvable using an expansion
in a finite basis set ==>
finite set of coupled equations
Final
Initial
Initial state
f(r,p,0) = Const δ[En-Hatom(r,p)]
x
χ(l<L<l+1) x χ
(m<Lz<m+1)
| Ψ(0)ñ = |n,l,mñ
Ionization by a single pulse
10% ionization thresholds for Na(nd), H(nd), K(np)
∆p/p orb
10
Jones et al (1993) *2.5
Dunning et al (1996-99)
1
quantum & classical
0.1
0.01
0.1
1
10
Tp/T orb
Tp >> Torb
equivalent to static
field ionization
Tp << Torb
equivalent to a sudden
impulse or ''kick''
ò
∆p = - dt F(t)
Pif = | áψf | exp(i∆p.r) | ψi ñ |
Threshold: ∆p ~ porb = n-1
2
Threshold F p ~Forb = n-4
Classical scaling invariance for H(n)
scaled variables
2
r
p
r 0 = r/rorb = r/n
p 0 = p/porb = p n
t
t 0 = t/Torb = t/2πn
F
F 0 = F/Forb = Fn
3
4
Classical dynamics is only a function of scaled variables.
Departures from scaling invariance ==> quantum dynamics
Uncertainty principle
σp σr > 1 (atomic units)
σp0 σr0 > n-1
0
n
∞
"scaled Planck constant" = n
-1
Ionization probability
Correspondence principle for one kick
10
0
10
-1
10
-2
quantum
n=1
10
-3
n=5
classical scaling
invariant curve
n=10
10
-4
10
-1
10
0
10
∆p0
classical suppression
Quantum result
Classical limit
λ = 1/∆p << rorb=n
∆p0 >> n
-1
2
1
Producing and probing coherent states
PUMP
FA(t)
FB(t)
Hfree
PROBE
Coherent state (wavepacket)
| Ψ(t)ñ =
Σ
ak | φkñ e
-i εk t
Hfree | φkñ = εk | φkñ
áΨ(t)|O| Ψ(t)ñ =
Σ
ak a*n áφn|O| φkñ e -i (εk-εn) t
quantum beating frequencies: ωkn = (εk - εn)
Probing the time evolution
H(t) = Hfree + z FB(t)
Probe duration < (2π/ωkn)
RADIAL WAVEPACKETS
PRODUCED BY SHORT LASER PULSES
Stroud (UR), Noordam(FOM), Bucksbaum(UM) et al (1980s)
PUMP
Hfree=Hatom
PROBE
| Ψ(t)ñ
n=39,40,41
PUMP
| φiñ
PROBE
ionizes the atom
when r is small
Recent pump-probe schemes using designer pulses
Dunning et al, Jones et al
Pumps
kick
field step
train of kicks
Probes
probing the momentum
probing the z coordinate
Coherent state produced by a sudden kick
Probability density
10
2
á∆Eñ = (∆p) /2
Ionization
after a
kick with
∆p0=0.5
1
before
the kick,
K(417p)
0.1
-1.5
-1.0
-0.5
0.0
0.5
E0=E / |E n|
á pz ñ t=0 = ∆p
á pz ñ t =
Σ
Ann' e -i (E n'-En) t
E(n+1) -En ≈ n
-3
: classical orbital frequency
Correspondence principle for the time evolution
Short times: classical-quantum correspondence
0.5
∆p0 = -0.5
ápz0ñ
0.3
H(100s)
0.1
-0.1
classical
quantum
-0.3
-0.5
0
2
4
6
8
0.5
quantum revival
ápz0ñ
0.3
0.1
-0.1
-0.3
-0.5
0
10
20
30
40
50
60
t0=t/Torb
Long times: classical dephasing / quantum revivals
The correspondence "break" time
Why is there one ?
Classical: continuous energy spectrum
Quantum: discrete energy spectrum
revival ==> time evolution phases exp(-iEn) can be in phase
Why is there break time long for Rydberg atoms ?
narrow width spectrum ==> harmonic spectrum
E n + δn
2
ö
æ 3δn
δn
æ
ö
− E n = δnω n ç1 −
+ 2ç ÷ ÷
ç
2n
è n ø ÷ø
è
classical orbital frequency ωn = n-3, Torb=2π/ ωn
t dephasing =
4n
T
2 orb
3δn
n
t revival = Torb
3
Probing the momentum with a sudden kick
-
2
Before the kick, t=t
p
__
+ V(r)
Hatom =
2
+
Immediately after the kick, t=t
^ 2
(_________
p + _∆p z ) + V(r)
Hatom =
2
Classical energy transfer
2
( ∆p )
_____
+ pz ∆p
∆E = ∆Hatom =
2
Whether or not the atom is ionized depends on the value ofz p
Quantum result
+
| Ψ(t )ñ = e
i z ∆p
-
| Ψ(t )ñ
á pz ñ + = á pz ñ - ∆p
t
t
á Hatom ñ + - á Hatom ñt- = ∆p á pz ñ - + (∆p)2/2
t
t
Pump/probe experiment for K(417p) Dunning et al (1996)
1.0
0.3
0.9
experiment
0.8
0.2
ápzñ
0.1
0.7
0.0
-0.1
theory
-0.2
F(t)
0.5
delay
0.4
-0.3
-0.4
0.3
1.0
0.9
0.8
0.7
0.6
0.5
F(t)
Survival probability
0.6
0.4
0.3
0
10
20
30
Delay time (ns)
40
50
Pump/probe experiments for Rb(390p) Dunning et al (2000)
-0.2
delay
0.0
0.8
0.6
0.6
0.1
experiment
0
10
20
0.2
30
40
0.80
0.2
áz0ñ
delay
0.75
0.1
áz0ñ
Survival probability
0.8
-0.1
ápz0ñ
ápzñ
1.0
0.70
0.0
0.65
-0.1
experiment
0.60
0
10
20
Delay time (ns)
30
-0.2
40
Probing the z coordinate with a field step
Before the field step
H = Hatom
After the field step
FDC
H = Hatom + z FDC
Whether or not the atom is ionized depends on the value of z
V(r) + z F DC
H > Ebarrier
Ebarrier
Using a field step to produce Stark wavepackets
H0 = Hatom + z FDC
DC pump
Short rise time ==> coherent superposition of Stark states
| Ψ(t)ñ =
Σ
áφs|Ψ(0)ñ | φsñ e
-i εk t
H0 | φsñ = εs | φsñ
Hydrogen:
εs = εn,k = -
1
2n
2
3
+ FDC n k
2
εs - εs' ≈ j ωorb + k ωStark
nearest neighbor spacing: ωStark = 3 FDC n
Stark dynamics
Field
j1,2 = ( L ± n A ) / 2
ω = 2 ωStark
L: angular momentum
A: Runge-Lenz vector
j1
A
j2
1st
Field
2nd
3rd
Angular momentum wavepacket
H(17d) at t=0
Quantum
t/Torb
Classical
t/Torb
/
Probing Stark beats with a short half-cycle pulse
HCP probe
Field
delay
DC pump
time
ωStark
Survival probability
0.9
Energy
ωorb
ωStark
Field
0.7
0.5
classical simulation
1.0
0.8
0.6
0.4
experiment K(388p)
0
100
200
300
400
Delay time (ns)
FDC = 5mV/cm, FHCP= ± 80mV/cm
Stark map near n=100 of K(m=0)
Diagram of extreme Stark states near n=400
450
n=
45
0
field
ionization
threshold
410
n=(-2E)
1/2
430
FULL MIXING
chaotic spectrum
390
n=
39
9,4
00
,40
1
370
0
n=35
350
0.00
0.01
0.02
0.03
F0
0.04
0.05
0.06
0.07
Raw nearest neighbor spacing density near n=100
Spectral density
Hydrogen:
Poisson spectrum
Potassium:
Wigner spectrum
∆E/(3FDC)
Step field ==> weighted spectrum
weight= overlap with state prior to the step
0.25
0.20
Hydrogen
0.15
Spectral density
0.10
0.05
0.00
0.15
Potassium
0.10
0.05
0.000
50
100
∆E/(3F DC )
150
Coherent states produced by a train of pulses
PROBE
PUMP
Field
delay time
Survival probability
time
100 pulses with scaled frequency ν 0
0.30
0.35
ν0=0.7
0.28
0.26
0.31
0.24
0.29
0.22
0.27
0.20
0.25
0
p0
5
Initial
Energy
level
10
15
ν0=1.3
0.33
20
0
5
10
15
Delay time (ns)
ν0=0.7
q0
ν0=1.3
20
Tailoring Rydberg wavefunctions
Classical quantum correspondence
High n, large perturbations, short times
Producing and probing coherent states
Recent interesting studies
Information storage and retrieval with Rydberg atoms
Ahn, Weinacht, and Bucksbaum, Science (2000)
Rydberg antihydrogen production using a fast field step
Wesdorp, Robicheaux, Noordam, Phys. Rev. Lett. 84, 3799 (2000
The kicked Rydberg atom
H(n)
Half-cycle pulse (HCP)
Torb= 2πn
F(t)
Forb= n
3
-4
Tp
Tp << Torb
sudden momentum transfer or ''kick''
∆p = -
ò dt F(t)
The kicked atom: linear superposition of HCPs
H = Hatom + z F(t)
unidirectional
F(t)
T
alternating
F(t)
time
ν = 1/T
Simple driven systems
Ideal testing ground for quantum systems that may
become chaotic in their classical limit
Experimental realizations
1980s-90s Rydberg atoms + microwave field
Koch et al (Stony Brook), Bayfield (Pittsburgh)
1994 Atoms in a trap subject to a modulated
standing wave
Raizen et al (Austin)
1997 Rydberg electrons subject to trains of
half-cycle pulses
Dunning et al (Rice)
The 3D periodically kicked atom (unidirectional pulses)
H = Hatom + z F(t)
2 _
__
1 _ z ∆p
p
__
≈
r
2
Σ
δ(t-kT)
k
Lz = Const ==> 2 1/2 degrees of freedom
time dependent
The 1D periodically kicked atom
2 _
__
1 _ q ∆p
__
p
H=
δ(t-kT)
q
2
k
1 1/2 degrees of freedom
Σ
Time evolution ==> simple discrete map
Classical or quantum
evolution operator
U(kT,0) =
Π
k
Free
U(Hatom,T)
-iHatomT
e
Kick
U(∆p)
iz∆p
e
(quantum)
How many atoms survive after many pulses ?
Experiments and classical simulations
0.6
3D, n=390, 50 pulses
1D, n=50
1
0.4
finite
width
pulses
100 δ kicks
experiment
0.3
0.2
0.0
1
1000 δ kicks
0.1
delta kicks
0.1
Survival probability
Survival probability
0.5
0.1
10
1
Scaled frequency, ν0 = ν/νorb
0
3D, n=390
10
-1
10
ν0 = 1.3
0.7
Classical
stabilization !!
0.6
-2
10
2.1
1.9
-3
10
0
10
1
2
3
4
5
6
10 10 10 10 10 10
Number of pulses, N
10
Isolating stable orbits
stroboscopic Poincare
section
t=k/νp (k=1,2,...)
ρ=Const.
pρ=Const.
Consecutive intersections of a stable
orbit with a plane yields a closed loop.
2
Hatom =
2
pz + p ρ
2
+
2
Lz
2
2ρ
1
-
2
2 1/2
(ρ + z )
Section for ∆p=0, ρ ~ 0, p ρ ~ 0, L z~0
2
2
pz
1
Hatom ~ __ - ___
2
|z|
Hatom = 0
pz
1
0
Lz = Const ~ 0
Hatom = -2
Hatom = -0.5
-1
-2
0.0
0.5
1.0
1.5
z
2.0
2.5
3.0
Section just before a kick, ρ=0, pρ=0, for ν0=1.3
2
( ∆p )
_____
+ p z ∆p = 0
∆E =
2
pz=-∆p/2
x
Most stable periodic orbit
∆p
x
Fixed
point
Correspondence between the 1D and 3D models
Structures in the classical survival probability
Survival probability
1D, n=50
1
100 kicks
1000 kicks
0.1
0.1
1
10
Scaled frequency
ν0=1.3
ν0=0.7
p0
q0
p 02
1
Initial energy level E 0 = − 0.5 =
−
2
q0
Classical-quantum correspondence ?
1D kicked atom
∆p
1D ==> accurate quantum
calculations can be
be performed for large n
1/q
Survival probability
10
1D H(n=50) atom after 200 kicks
0
Classical
Quantum
10
-1
10
-1
10
0
Scaled frequency
10
1
How can we compare classical and quantum
dynamics in phase space ?
Husimi distributions:
Quantum analog of classical phase space distributions:
Husimi disctributions of Floquet states
Quantum analog of Poincare sections
Usual quantum description of the electron
- State of the electron: |Ψ(t)ñ
- Dynamics: Schrödinger Equation
i
d|Ψ(t)ñ
dt
= H Ψ
| (t)ñ
- Boundary condition
|Ψ(0)ñ = |Φi(t)ñ : initial state
Alternative quantum description of the electron
State of the electron:
Wigner function
Weyl transform of the density matrix
1
fw (q,p,t) =
π
ò
2ipy
dy áq-y|Ψ(t)ñ áΨ(t)|q+yñ e
fw (q,p,t): real quasi probability density
(not positive definite)
áAñ =
ò
dqdp fw(q,p,t) A(q,p)
2
|áq|Ψ(t)ñ|
=
Dynamics:
ò
dp fw(q,p,t)
quantum Liouville equation
dfw
= L qm fw
dt
Classical limit:
L qm ≈ Lcl
Probability density
Comparing classical and quantum densities
0.25
0.20
quantum
0.15
classical
0.10
0.05
0.00
0.0
0.5
1.0
1.5
2.0
p
2.5
3.0
Direct comparison with Wigner functions is difficult
Husimi distribution:
Gaussian smoothed Wigner function
2
fh (q,p,t) = | ágq,p|Ψ(t)ñ| > 0
2
-(q-q') /a
gq,p(q') = C e
e
ipq'
minimum uncertainty Gaussian wavepacket
Husimi distribution of the 1D n=50 level
PQM(q,p)
0.05
0.03
0.01
-0.01
-0.03
2
-0.05
1
0
p0
2.0
2.5
1.5
-1
1.0
-2
0.5
0.0
q0
classical
-0.5 =
2
p
0
__
2
_ __
1
q0
-1/2
PCL ~ (p0)
probability maximizes
at the outer turning point
Classical phase space vs quantum Husimi distributions
Floquet analysis in a finite basis set
Soft boudary conditions
outgoing flux
(norm not conserved)
finite Hilbert
space
Floquet states:
U(T,0) |φkñ = e
|φkñ
complex quasi eigenenergy
period-one
evolution operator
|Ψ(NT)ñ =
-iεkT
Σk
ck e
-iεkNT
|φkñ
Stable Floquet states: Im(εk) = 0
Husimi distributions of stable Floquet states for ν0=1.3
quantum
classical
quantum
Periodic orbits and fixed points
stable
unstable
Any trace of unstable periodic orbits in
quantum mechanics ?
Quantum localization for high frequencies
Survival probability
1.0
n=50, ν0=16.8, ∆p0=-0.3
breaktime
0.9
quantum
0.8
0.7
0.6
0.5
0.4
1
classical
10
100
Number of kicks
∆p
Husimi distribution of a stable Floquet state for ν0=16.8
P(q,p)
-2.5
-1.5
-0.5
0.0
0.5
q0
1.0
0.5
1.5
1.5
unstable periodic orbit
2.5
2.0
Stable Floquet states ==> scarred states
Heller (1980s)
p0
Why does the survival probability oscillate ?
Localization in the continuum
E=0
Correspondence for high frequencies, ν0=16.8
and strong perturbations, ∆p0=-0.3
Initial
level
Classical
phase space
D
Husimi distributions of stable Floquet states
X: classical fixed points
: Unstable periodic orbits
Lack of correspondence for weak perturbations, ∆p0=+0.01
n=50
ν0=16.8
Initial
level
Classical
phase space
Husimi distributions of stable Floquet states
D
X: classical fixed points
What do periodic orbits resemble ?
Stark orbit
Coulomb
orbit
F(t) = å ∆p δ(t - kT) = Fav + 2Fav å cos(2πmνt)
m ≥1
k
Fav=∆p/T = average field
H(t) = HStark − 2qFav å cos(2πmνt)
m≥1
HStark = H atom − qFav
Localization in energy space
Types of diffusion
Resonant diffusion
Classical diffusion
What stops diffusion ?
è Non-linear level spacing
è Quantum uncertainty
Occupation probability
after 600 kicks
Localization widths
Dipole coupling and driving frequency
Energy excursion within the periodic orbit, ∆Eorbit
Small ∆p è ∆Eorbit<<ω
ω
Direct excitation probabilities by all harmonics
X Time averaged numerical result
Dipole coupling
Correspondence for the localization width
Occupation probability
after 600 kicks
Large ∆p è ∆Eorbit> ω
∆Eorbit
ω
Direct excitation probabilities by all harmonics
Dipole coupling
The kicked Rydberg atom
Classical quantum
Correspondence
Long times
- short times
- high n
- large perturbations
Classical
quantum
Outlook
- Localization in 3D
- Adding noise and other trains of pulses
- Any train of pulses leading to long lived coherent states ?
Cross-disciplinary
Transport of atoms through solids
Transmission of fast atoms through solids
The randomly
kicked atom
STATE OF THE SYSTEM: reduced density matrix ρ(t)
DYNAMICS: quantum Liouville equation
i dρ
dt
= [Hatom,ρ] + R ρ
p2
Hatom = 2 + V(r)
dissipative interaction
with the environment
Screened
ion field
CLASSICAL TRANSPORT:
STATE OF THE SYSTEM: ρcl(r,p,t)
DYNAMICS: classical Liouville equation
Microscopic interaction
Vint
First order transition probability per unit time
H
dP
∝ ò d∆p
dt
Transition amplitude
for a given ∆p
f |e
HH
i∆p. r
H
~
| i Vint (∆p)
Fourier transform: Probability
distribution of kicks
Similarity to quantum optics
Microscopic approach: Langevin equation
ρ(t) =
1
N
N
.
Σ
u=1
|Ψu(t)ñáΨ u(t)|
quantum trajectory
Langevin equation: undeterministic Schrödinger equation
H = Hatom
r.
Σ
k
Random multiple elastic and inelastic
collisions with particles in the solid.
∆pk δ(t-tk)