Quantum tunneling in nonintegrable systems: beyond the leading order - - PowerPoint PPT Presentation

Quantum tunneling in nonintegrable systems: beyond the leading order semiclassical description

Akira Shudo Department of Physics, Tokyo Metropolitan University

collaboration with Y Hanada and K S Ikeda

Introduction

Dynamical tunneling

• No semiclassical formula
• Energy domain approach based on trance formula

No semiclassical formula for mixed systems (cf. hyperbolic : Gutzwiller, completely integrable : Berry-Tabor)

• Time domain approach based on Van-Vleck Gutzwiller

works well within the leading order semiclassical approximation (cf. recent advances in theory of complex dynamical systems by Bedford and Smillie) but depends on initial and final states, or representations

• Here, not long-time, but just a single step semiclassical analysis

as close as possible to the energy domain by adjusting initial and final states

q

q p

classically allowed

A

A

L :        q p        →        q + ω p + K cos q        q

= q

q p

L(A R2) L(A) (complexified)

A (complexified)

L(A) A ∅ for any A and A.

Completely integrable model

In the real plane In the complex plane

L(A) A′ = ∅ if A′ is outside the classically allowed region.

• −∞
• 1-step propagator

p| ˆ U | p = ∞

−∞

dq exp

• − i
• F(q; p, p)
• where

F(q; p, p) := T(p) + V(q) + q(p − p) Saddle point condition ∂F(q; p, p) ∂q = 0 Re q q Im q real solution complex solution q p q Re q Im q turning point q q q q q p classically forbidden q q q p q Re classically allowed Re q Im q classically forbidden q q q p q Re

Completely integrable model

Langrangian manifold a set of saddle points

2 4 6 8 10 12 14

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

Manifold around the turning point

Completely integrable model

Im q p

Locally, the behavior around the turning point is described by ΨK p = ∞

−∞

expi ΦK t; p dt, where ΦK t; p = tK+2 +

K

• m=1

xmtm with K = 1, that is the Airy function.

Re p

q p

• log |p| ˆ

U |p|2

tunneling tunneling

tunneling real

p (Schematic) (Schematic)

Completely integrable model

q

q p

L :        q p        →        q + ω p + K cos q        A

A

A = { (q, p) ∈ C2 | I(q, p) = Ia ∈ R } A = { (q, p) ∈ C2 | I(q, p) = Ib ∈ R } L(A) A = ∅ for any A and A. q

= q

q p

Completely integrable model

I q p I log |I| ˆ U |I|2

Completely integrable model

1.0 0.5 0.5 1.0 1.1 1.2 1.3 1.4 T(p)

) p

β = 5 β = 10 β = 100

Map with discontinuity

Map: S1 :        q p        →        q + τT(p) p − τV(q + τT(p))        where T(p) = s 2(p − d)2 + ω(p − d)

• θβ(p − d)
• θβ(p) ≡ 1

2

• tanh(βp) + 1

V(q) = K cos(2πq)

q p q p q p q p q p q p

• β = 0.1

1 β = 2 2 β = ∞

1.0 0.5 0.5 1.0 1.1 1.2 1.3 1.4 T(p)

) p

β = 5 β = 10 β = 100

Map with discontinuity

Map: S1 :        q p        →        q + τT(p) p − τV(q + τT(p))        where T(p) = s 2(p − d)2 + ω(p − d)

• θβ(p − d)
• θβ(p) ≡ 1

2

• tanh(βp) + 1

V(q) = K cos(2πq)

q p q p q p q p q p q p

• β = 0.1

1 β = 2 2 β = ∞

β = 1

• 35
• 30
• 25
• 20
• 15
• 10
• 5
• 0.2

0.2 0.4 0.6 0.8

• 35
• 30
• 25
• 20
• 15
• 10
• 5
• 0.2

0.2 0.4 0.6 0.8

1 β = 50 discontinuous line

Here |I denotes the eigenfunction of the integrable map L: U0| I = e− i

E| I

where U0 = e− i

ωpe− i K sin q

1-step time evolution: I|U|I0 where U = e− i

T(p)e− i V(q)

log | I| U |I0 |2 | I

Anomalous tail in the action representation

initial state I|I0

Semiclassical analysis for discontinuous for discontinuous limit (β = ∞)

1-step propagator in the action representation I| ˆ U |I = ∞

−∞

dq ∞

−∞

dq ∞

−∞

dp exp

• − i
• F(q, p, q; I, I)
• where

F(q, p, q; I, I) := S(I, q) − S(I, q) − p(q − q) + T(p) + V(q) Since T(p) has a discontinuity at p = d,

• dq
• dp
• dq =
• dq

d

−∞

dp + +∞

d

dp dq Since T(p) has a discontinuity at p = d,

• dq
• dp
• dq =
• dq

d

−∞

dp + +∞

d

dp dq

|I | e− i

T1(p) | 0|2

log |ψm|U|ψ0|2

10-20 10-15 10-10 10-5 100

• 0.5

0.5 1

5 10 100 200

semiclassical (edge contribution) exact

Semiclassical analysis for discontinuous for large but finite β

with an edge p d, 1-step propagator in the action representation I| ˆ U |I = ∞

−∞

dq ∞

−∞

dq ∞

−∞

dp exp

• − i
• F(q, p, q; I, I)
• where

F(q, p, q; I, I) := S(I, q) − S(I, q) − p(q − q) + T(p) + V(q) − − − for the present map T(p) = s 2(p − d)2 + ω(p − d)

• θβ(p − d)

V(q) = K cos(2πq) S(I, q) = Iq + K sin q

Semiclassical analysis for discontinuous for large but finite β

with an edge p d,       ⇐ ⇒ 1-step propagator in the action representation I| ˆ U |I = ∞

−∞

dq ∞

−∞

dq ∞

−∞

dp exp

• − i
• F(q, p, q; I, I)
• where

F(q, p, q; I, I) := S(I, q) − S(I, q) − p(q − q) + T(p) + V(q) − − − for the present map T(p) = s 2(p − d)2 + ω(p − d)

• θβ(p − d)

V(q) = K cos(2πq) S(I, q) = Iq + K sin q

• −∞

Saddle point condition: ∂F ∂q = 0, ∂F ∂p = 0, ∂F ∂q = 0        I q        − →        q p        − →        q p        − →        I q       

R S S S S2 R S S

q Im q

Re q T1 T2 Re q q Im q real solution complex solution q p q Re q Im q turning point q q

1 β = 10

A set of saddle points

completely integrable model

• 1. turning points on the real manifold
• 2. turning points in the complex plane

Two types of turning points

• 1. Turning points on the real manifold

locally highly degenerated, reflecting tangency between I and S1(I)

• 2. Turning points in the complex plane

increase as β gets large, reflecting the increase

• f singularities, and possibly the existence of

natural boundaries

q Im q

Re q T1 T2

1 β = 10

( 0.0000000000e+00,-8.0000000000e+00)-( 6.2831853072e+00, 8.0000000000e+00) ( 0.0000000000e+00,-8.0000000000e+00)-( 6.2831853072e+00, 8.0000000000e+00) ( 0.0000000000e+00,-8.0000000000e+00)-( 6.2831853072e+00, 8.0000000000e+00)

q p q p q p q p q p q p

β = 0.1 1 β = 1 1 β = 10

I S1(I) : I : An invariant curve of the integrable map S1(I) : 1-step time evolution of I

1-step time evolution of the real manifold I

With increase in β, the initial manifold I comes closer to KAM curves, and moves very slightly within a single step.

Integrals with coalescing saddles ΨK(x) = ∞

−∞

exp(i ΦK(t; x)) dt, where ΦK(t; x) = tK+2 +

K

• m=1

xmtm Airy integral Ψ1(x) = 2π 31/3 Ai x 31/3

• Pearcey integral

Ψ2(x) = P(x2, x1) = ∞

−∞

exp

• i(t4 + x2t2 + x1t)
• dt

ΨK(x) has a convergent series expansion: ΨK(x) = 2 K + 2

∞

• n=0

in cos π(n(K + 1) − 1) 2(K + 2)

• Γ

n + 1 K + 2

• an(x)

(K : odd) where

Diffraction integrals with coalescing saddles

• 4
• 2

2 4 5 10 15 20 25 30

∂ ΦK(t, x) ∂t

x) t

2 4 6 8 10 12 14

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

β = 0.1 β = 1 β = 10

I

Im q

x = (x1, 0, · · · , 0)

Integrals with coalescing saddles ΨK(x) = ∞

−∞

exp(i ΦK(t; x)) dt, where ΦK(t; x) = tK+2 +

K

• m=1

xmtm Airy integral Ψ1(x) = 2π 31/3 Ai x 31/3

• Pearcey integral

Ψ2(x) = P(x2, x1) = ∞

−∞

exp

• i(t4 + x2t2 + x1t)
• dt

ΨK(x) has a convergent series expansion: ΨK(x) = 2 K + 2

∞

• n=0

in cos π(n(K + 1) − 1) 2(K + 2)

• Γ

n + 1 K + 2

• an(x)

(K : odd) where

Diffraction integrals with coalescing saddles

• 4
• 2

2 4 5 10 15 20 25 30

• 10
• 5

5 10 15

• 2
• 1

1 2

K = 1 (Airy) K = 3 (Pearcey) K = 5 K = 7 K = 9

ΨK(x)

∂ ΦK(t, x) ∂t

x) t

x1

x = (x1, 0, · · · , 0)

beyond leading-order saddle point approximation

• 1. Turning points on the real manifold

locally highly degenerated, reflecting tangency between I and S1(I)

• 1. turning points on the real manifold
• 2. turning points in the complex plane

1 β = 1

q Im q

Re q T1 T2

Two types of turning points

• 2. Turning points in the complex plane

increase as β gets large, reflecting the increase

• f singularities, and possibly the existence of

natural boundaries

I

Large deformation in the complex plane

( 0.0000000000e+00,-8.0000000000e+00)-( 6.2831853072e+00, 8.0000000000e+00)

1 β = 1

q p q p

I S1(I) :

1 2 3 4 5 6 7

• 6
• 4
• 2

2 4 0.5 1 1.5 2 2.5 3 q p

S1(I)

( 0.0000000000e+00,-8.0000000000e+00)-( 6.2831853072e+00, 8.0000000000e+00)

1 β = 1

q p q p

I S1(I) :

1 2 3 4 5 6 7

• 6
• 4
• 2

2 4 0.5 1 1.5 2 2.5 3 q p

Large deformation in the complex plane

β = 1

• 35
• 30
• 25
• 20
• 15
• 10
• 5
• 0.2

0.2 0.4 0.6 0.8

• 35
• 30
• 25
• 20
• 15
• 10
• 5
• 0.2

0.2 0.4 0.6 0.8

1 β = 50 discontinuous line

log | I| U |I0 |2 | I

initial state I|I0

• 1. turning points on the real manifold
• 2. turning points in the complex plane

1 β = 1

q Im q

Re q

T1 T2

Range beyond the semiclassical approximation

Quantum unitary operator ˆ U = exp

• − i
• τ

V( ˆ q) 2

• exp
• − i
• τT( ˆ

p)

• exp
• − i
• τ

V( ˆ q) 2

• Integrable approximation of ˆ

U ˆ U(M) := exp

• − i
• τ ˆ

H(M)

eff ( ˆ

q, ˆ p)

• where

ˆ H(M)

eff ( ˆ

q, ˆ p) = ˆ H1( ˆ q, ˆ p) +

M

• j=3

iτ

• j−1

ˆ Hj( ˆ q, ˆ p) ˆ Hj: the j-th order term in the Baker-Campbell-Hausdorff (BCH) series. Classical Hamiltonian (“Shadow” Hamiltonian) H(M)

eff (q, p) = H1(q, p) + M

• j=3

(j∈odd int.)

iτ

• j−1

Hj(q, p). Hj(q, p): obtained by replacing commutators in the BCH series by Poisson brackets.

T(p) = p2 2 , V(q) = K sin q

q p q p

Standard map and integrable approximation

(complexified)

I = mh

β = 1 1 β = 50

• 35
• 30
• 25
• 20
• 15
• 10
• 5
• 0.2

0.2 0.4 0.6 0.8

• 35
• 30
• 25
• 20
• 15
• 10
• 5
• 0.2

0.2 0.4 0.6 0.8

10 20 30 40 50 60 70 80

I

Map with discontinuity

I(M)

log |I(M) | ˆ U | I(M) |2 log |I | ˆ U | I0|2

ˆ H(M)

eff | I(M) = E(M) eff | I(M)

Here |I(M) denotes the eigenfunction of the integrable Hamiltonian ˆ H(M)

eff :

1-step time evolution: I(M)| ˆ U |I(M) where ˆ U = e− i

T(p)e− i V(q)

Standard map

1-step time evolution in the action representation

3 M = 5

Re q q Im q real solution complex solution q p q Re q Im q turning point q q

A set of saddle points

completely integrable model

Re q q Im q

• 1. Turning points on the real manifold

locally highly degenerated, reflecting tangency between I and F(I)

• 2. Turning points in the complex plane

increases as M gets large, reflecting the increase

• f singularities, and possibly the existence of

natural boundaries

A set of saddle points

• 1. turning points on the real manifold
• 2. turning points in the complex plane

3 M = 5

Re q q Im q

Integrable approximation

I : An invariant curve of the BCH integrable Hamiltonian S2(I) : 1-step time evolution of I

M = 1 1 M = 3 3 M = 5

q p q p q p q p q p q p

I I S2(I) With increase in M, the initial manifold I comes closer to KAM curves, and moves

• nly slightly within a single step.

Around the turning point on the real manifold

• 0.2

0.2 0.4 0.6 0.8 1

• 400
• 200

200 400

M = 1 1 M = 3 3 M = 5

Im q

I(M)

0 1 2 3 4 5 6

• 2
• 1

1 2

• 2
• 1

1 2 q p

q p q p

Large deformation in the complex plane

M = 5 (complexified) I

q p q p

0 1 2 3 4 5 6

• 2
• 1

1 2

• 2
• 1

1 2 q p

Large deformation in the complex plane

M = 5 I S(I)(complexified)

Range beyond the semiclassical approximation

10 20 30 40 50 60 70 80

I(M)

log |I(M) | ˆ U | I(M) |2

• 1. turning points on the real manifold
• 2. turning points in the complex plane

3 M = 5

Re q

q Im q

Summary

• Semiclassical approximation (leading-order) in a single step propagator

breaks down in the integrable representation

• Transition from one torus to another or to chaotic regions occurs under a

purely quantum mechanism and cannot be described even by complex clas- sical dynamics.

• Purely quantum regions are sandwiched between highly degenerated turn-

ing points and turning points associated with singularities of complexified tori, and possibly with natural boundaries.

• Observed diffractive phenomena are global and beyond the treatment based
• n local diffraction integrals such as a series of diffraction catastrophes.