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Semiclassical Propagation of Wavepackets
Raúl O. VallejosCBPF, Rio de Janeiro
UFF, 28 de outubro de 2010
2
Collaborators
Raphael NP Maia
Fernando NicacioFabricio Toscano (UFRJ)
Diego Wisniacki (Buenos Aires)
Roman Schubert (Bristol)
Alfredo Ozorio de Almeida (CBPF)
Olivier Brodier (Tours)
Kaled Dechoum
Antonio Zelaquett
3
Plan of the talk
Introduction
WKB for wavepackets (heuristic)
WKB (rigorous)
Semiclassical propagation of Wigner functions
Perspectives
4
Introduction
5
Gaussian wavepackets in semiclassical regimes
Environment induced decoherence
Zurek, Paz, Habib, Bhattacharya,..., Davidovich
Continuous quantum measurements
Sundaram, Jacobs, ...
Quantum-to-classical transition
ARR Carvalho
6
One example from EIDDuffing oscillator
Monteoliva-Paz
PRL00, PRE01
+ diffusive reservoir
mixed mostly chaotic
Stroboscopic sections
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Decoherence rate for wavepackets
Monteoliva-Paz
PRL00, PRE01
Lyapunov
exponent
8
Zurek-Paz explanation
9
Wavefunction structure – Closed system
Monteoliva-Paz
PRE01
Wigner functions
10
Diffusive reservoir
11
Goal
0),,()2
0),()1
DtpqW
Dtq
Theory for “quantum filaments”:
12
Theories
Complex TDWKB (Heller, ..., de Aguiar)
Variety of methods in chemical physics(Miller, Heller, Herman-Kluk, Kay, Grossmann,Pollack, Shalashilin, ...)
Real TDWKB ?
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13
TDWKB theory
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WKB
Van Vleck, PNAS28Dirac’s book
Maslov, 60’s-70’s
Berry-Balazs, JPA79
Littlejohn, JSP92
),(),(),,ˆ( tqt
itqtqpH
/)(
00)()0,(
qiSeqAtq
/)()(),(
qiS
tteqAtq
Initial wavefunction
WKB form
A(q), S(q) real
vary slowlyAt a later (short) time …
Problem: find At (q), St (q)
primitive WKB
TDWKB
15
Solution Littlejohn, JSP92
0),(,,
tqS
ttq
q
SH Hamilton-Jacobi
2,),( tqAtq
0),(v,),(
tqtq
qtq
t
q
Sp
tqpHp
tq
),,(),(v
transport
equation
16
Geometrical Interpretation
Lagrangian manifold
q
qSqpp
)(00
qqp ),(p
p
0L
/)(
00)()0,(
qiSeqAtq Initial wavefunction
q
dqpqS
0
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Solving the Hamilton-Jacobi equation
qp ,
p
p
L
qp ,
q
L
q
dqptqS
,
new phase
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Continuity equation
q
p
L
q
L
qdq qdq
q
qdtqAqdtqA 22,,
new amplitude
2/1
,,qd
qdtqAtqA
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Caustics
multiple branch WKB
2/1
,,qd
qdtqAtqA
b
itqiS
bbbetqAtq
2//),(),(),(
qp ,1
qp ,2
q 1q2q
L
L
q
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Applying TDWKB to Gaussian wavepackets
/)(
0
4/
00
22
)()(qiSq eqAeq
amplitude “not smooth”!
discarded term is large2
2 )()1(
q
qAvsO
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End of Introduction
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Proposition
1. In closed chaotic systems Gaussian wavepackets eventually
evolve into WKB states:
3. Even if TDWKB fails to propagate Gaussian wavepackets !
2. Construction (exact)
b
itqiS
bbbetqAtq
2//),(),(),(
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Kicked Harmonic Oscillator
6/T
Berman-Zaslavsky
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Classical Dynamics (Liouville)
1n 3n
5n 4n
2n
Toscano, de Matos Filho, Davidovich, PRE05
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Classical vs. Quantum
1n 2n 3n 4n 5n
Wigner functions
Toscano, de Matos Filho, Davidovich, PRE05 26
Observation
Chaotic dynamics stretches wavepackets (nonlinearly).
After a certain time (log ) a wavepacket becomes a
smooth primitive WKB state.
From then on it can be propagated with TDWKB.
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Amplitude: exact propagation
/)()(),(
qiS
tteqAtq
0
1
2
3 4
22 4/
0 )( qeq
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/)()(),(
qiS
tteqAtq
q
S
Phase: exact propagation
0
1
2
3
unstable
manifold
22 4/
0 )( qeq
negative half
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Phase: exact propagation
q
S
0
1
2
34
unstable
manifold /)()(),(
qiS
tteqAtq
22 4/
0 )( qeq
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Recipe
Propagate during a short time either
> numerically,
> using the linear dynamics (if satisfactory),
> complex TDWKB,
> etc
Resume propagation with (real) TDWKB
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Wigner function ),( qpW
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Semiclassical (TDWKB) Wigner function
b
itqiS
bbbetqAtq
2//),(),(),(
/)2/()2/(
1),(
ipeqqdqpW
stationary phase
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A
0vv
),( qpx
Semiclassical Wigner function
close to the manifold
one chord
34
Caustics
Berry 77
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Quantum vs. WKB – Wigner section
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Quantum vs. WKB – Wigner section
full line = exact;
black circles = primitive WKBopen circles = Airy transitional approximation
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Example II
3838
Wavepacket dynamics in the quartic oscillator
Evolution of a coherent state with a Kerr-type Hamiltonian:
One degree of freedom mechanical oscillator, or Single mode of radiation field
3939
Initial stage – Snapshots – Wigner
4040
Delocalization
4141
Fractional revivals
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Theory and experiments
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Cat state generation
It is pointed out here that a coherent state propagating through an amplitude dispersive medium will, under suitable conditions, evolve into a quantum superposition of two coherent states 180 degrees out of phase with each other.
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45 4646
Extracting the manifold
q
p q /)(
111)(),(qiS
eqAtq
Primitive WKB form:A(q’), S(q’) real,slowly varying
q
qSqp
)(
support manifold
4747
TWKB evolution
b
itqiS
bbbetqAtq2//),(
),(),(
16/revTt
4848
Comparison revTt 4
)14,0(, 00 pq
1,, mline: exact, dots: WKB
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Decoherence
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Decoherence
WDWif pp
TTtDgdtthen ˆˆˆ),;()(ˆ0
Gaussian channel
2R
phase space translation
(Glauber)average over random translations
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Continuous time
Use Lie-Trotter decomposition:
WDWHW ppMB ,
)()(
,tWeedttW MBpp HdtDdt
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Stochastic unravelling
Unitary dynamics – dt
Random phase space translation – dt
Iterate
Repeat for another set of random
translations
Average over translations
ensemble of WKB states
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Extensions
Describe initial stretching within WKB
Develop general semiclassical scheme forLindblad equation
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Correcting WKB
),(),(2
),(2
tqVtqtqt
i
/)()(),(
qiS
tteqAtq
),( tqS
AASi
ASiVAASt
AiA
t
S
222
1 22
VSt
S
2
2
1:0
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Transport equation for
AASi
ASit
Ai
22
AHH 10
tqA ,
010 ATTA 10101 TTHTit
T
classical
transport
quantum
dispersion
curvilinear
Laplacian56
Transport equation
010 ATTA 1001
2TTT
i
t
T
Comments:
tT
WKBT
,1
1
1
1
Idea: approximate T1 by a Gaussian unitary !?
(Local linearization of the transport dynamics.
In 1D, Laplacian with time dependent factor.)
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Numerical example(KHO)
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Towards a general semiclassical scheme for Lindblad equation
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WKB for mixed propagator
),,(, tKdtW
Wigner
function
characteristic
function
(Gaussian)
mixed
propagator
WKB theory in double phase-space !?
(Ozorio-Brodier)
60
Unitary case –Example (Kerr)
0.1
0,14
020
40
60
80
100
0
0.5
1
1.5
2
-10 -5 0 5 10
-4
-2
0
2
4
tK ,,
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Caustics
0 50 100 150 200
0
20
40
60
80
100
-10 -5 0 5 10
-4
-2
0
2
4
quantum
amplitude
classical
62
Question
0 50 100 150 200
0
20
40
60
80
100
How does the caustic size scale with the relevant parameters?
?),( tr
Conclusions
63 64
References
Semiclassical evolution of Gaussian wavepackets,
R. N. P. Maia, F. Nicacio, R. O. Vallejos, F. Toscano,
Phys. Rev. Lett. 100, 184102 (2008).
Semiclassical description of wavepacket revival,
F. Toscano, R. O. Vallejos, D. Wisniacki
Phys. Rev. E 80, 046218 (2009)
How do wavepackets spread?
R. Schubert, F. Toscano, R. O. Vallejos
preprint, J. Phys. A (2011)?
Semiclassical propagation of Wigner functions,
A. M. Ozorio de Almeida, O. Brodier, F. Toscano, R. O. Vallejos
2011?
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Appendix
66
Explanation
2
2
1ˆ
/ˆˆ
ni
tHi eeU
phase
ti
HO
tnitni eeeU 4/ˆ2
!
ˆ2ˆ
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II
tnieU2ˆˆ
project onto HO basis
m
kt 2
revival times
2ˆ2
,ˆ
nk
mi
km eU
k
m
nk
mi
m ec1
ˆ2sum of rotations !