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Long quantum transitions due to unstable semiclassical dynamics
Dmitry Levkov, Alexander Panin
Institute for Nuclear Research RAS levkov@ms2.inr.ac.ru

August 22, 2009

D. Levkov, A. Panin (INR RAS)

Long quantum transitions

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Introduction

Why transition time?
Experimental signature! Long transitions long­living intermediate "states":
V(X)

Quasistationary state
Yamamoto, Miyamoto, Hayashi, 1998 Peskin, Galperin, Nitzan, 2002

X

D. Levkov, A. Panin (INR RAS)

Long quantum transitions

August 22, 2009

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Introduction

Why transition time?
Experimental signature! Long transitions long­living intermediate "states":
V(X)

Quasistationary state
Yamamoto, Miyamoto, Hayashi, 1998 Peskin, Galperin, Nitzan, 2002

X V(X)

Classically unstable "state" (Not a quantum state at all)
Takahashi, Ikeda, 2006

E = V0 V0 X

D. Levkov, A. Panin (INR RAS)

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Introduction

Sur prise
Classically unstable "states" form in the generic processes of multidimensional tunneling
V(X) V0 y X

D. Levkov, A. Panin (INR RAS)

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Introduction

Sur prise
Classically unstable "states" form in the generic processes of multidimensional tunneling
V(X) V0 y X

Examples -- tunneling in regular quantum systems (N = 2) electroweak theory (N = ) scalar field theory (N = )
D. Levkov, A. Panin (INR RAS) Long quantum transitions

Bonini, Cohen, Rebbi, Rubakov, 1999 Takahashi, Ikeda, 2003

systems with dynamical chaos (N = 1.5, 2)

Bezrukov, DL, Rebbi, Rubakov, Tinyakov, 2003 DL, Sibiryakov, 2005
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The model

The model
S= dt (X 2 + y 2 - 2 y 2 )/2 - exp(-(X + y )2 /2) |E , E
y

U(X,y) y

X

P (E , Ey ) 0
D. Levkov, A. Panin (INR RAS)

Semiclassical description
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Traversal time

Traversal time definition
t=0
i
X
n

V(X)
o

t=
ut

j()
LX

-L 0 Normalized probability distribution:
( ) = j ( )/Pto Average time of passing Time dispersion
2 t

D. Levkov, A. Panin (INR RAS)

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Semiclassical method

Semiclassical method
Counts time spent by trajectory in the region X < L:


Tint [X ] =
0

dt (L - X (t )) ,

Faddeev­Popov unity


1=
0

d ( - Tint [X (t )])

o 0

ut

=

[dX (t ) dy (t )] · ei

S/

· in (Xin , yin )

saddle­point approximation for all integrals!

D. Levkov, A. Panin (INR RAS)

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Semiclassical method

Semiclassical expression

Ptot = A · e-

F/

( ) = A( ) · e-

F ( )/

Semiclassical trajectory with additional constraint Tint [X ] =

D. Levkov, A. Panin (INR RAS)

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August 22, 2009

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Results

Results
direct
0.4 0.3

via unstable "state"
0.15 0.1 0.05 0

()

0.2 0.1 0 30 32 34 36

()

30

35

40

45

50





Gaussian = N · e-
( - )2 /2t2

Gumbel type I distribution · exp - - · e- - log
2 O ( 0 )

O( 0)
2 O ( )

D. Levkov, A. Panin (INR RAS)

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Results

Results

40



35 30

h = 10-1 h = 10-2 h = 10-3

1.3

p

c,1

1.5

1.6

p
D. Levkov, A. Panin (INR RAS)

0
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Long quantum transitions


Conclusions

Conclusions

Quantum traversal time is an observable of quantum transition. Unstable semiclassical motions correspond to long quantum transitions.

D. Levkov, A. Panin (INR RAS)

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Conclusions

Conclusions

Quantum traversal time is an observable of quantum transition. Unstable semiclassical motions correspond to long quantum transitions.

Thank you for attention!

D. Levkov, A. Panin (INR RAS)

Long quantum transitions

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