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Semiclassical Propagation of Wavepackets

Raúl O. VallejosCBPF, Rio de Janeiro

UFF, 28 de outubro de 2010

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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

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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

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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”:

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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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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

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Solution Littlejohn, JSP92

0),(,,

tqS

ttq

q

SH Hamilton-Jacobi

2,),( tqAtq

0),(v,),(

tqtq

qtq

t

q

Sp

tqpHp

tq

),,(),(v

transport

equation

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Geometrical Interpretation

Lagrangian manifold

q

qSqpp

)(00

qq

qqp ),(p

p

0L

/)(

00)()0,(

qiSeqAtq Initial wavefunction

q

dqpqS

0

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Solving the Hamilton-Jacobi equation

qq

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

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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

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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//),(

),(),(

qq

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)

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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

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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

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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 !