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

Completely integrable model In the real plane     q � q + ω L ( A ) � A ′ = ∅         L :      �→         p � p + K cos q    if A ′ is outside the classically allowed region. In the complex plane L ( A ) � A � � ∅ for any A and A � . L ( A � R 2 ) A � classically allowed � A q p q �� A � (complexified) q p = q � q L ( A ) (complexified)

Completely integrable model � �� � � −∞ 1-step propagator Saddle point condition � � ∞ ∂ F ( q ; p � , p ) � �� − i � � p � | ˆ F ( q ; p � , p ) U | p � = dq exp = 0 � ∂ q −∞ where F ( q ; p � , p ) : = T ( p ) + V ( q ) + q ( p � − p ) a set of saddle points Langrangian manifold complex solution p � q � q Re q Im q classically forbidden real solution p � q � q q q Re q p � q Im q classically allowed Re q Im q turning point classically forbidden q q p � q � q q q Re q q Re q

Completely integrable model Manifold around the turning point Locally, the behavior around the turning point is described by � ∞ K � t ; p �� dt , � p � = � t ; p � = t K + 2 + � x m t m exp � i Φ K where Ψ K Φ K −∞ m = 1 with K = 1, that is the Airy function. 14 12 10 Re p � 8 6 4 2 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Im q p

Completely integrable model tunneling real q p tunneling tunneling (Schematic) U | p � | 2 log | � p � | ˆ � (Schematic) p �

Completely integrable model A = { ( q , p ) ∈ C 2 | I ( q , p ) = I a ∈ R }     q � q + ω         L :     A � = { ( q , p ) ∈ C 2 | I ( q , p ) = I b ∈ R }  �→         p � p + K cos q    L ( A ) � A � = ∅ for any A and A � . A � q �� q p � A q p = q � q

Completely integrable model q p U | I � | 2 I log | � I � | ˆ I

Map with discontinuity Map:     q q + τ T � ( p )         S 1 :      �→         p p − τ V � ( q + τ T � ( p ))    where � s θ β ( p ) ≡ 1 � � � � � 2( p − d ) 2 + ω ( p − d ) T ( p ) = θ β ( p − d ) tanh( β p ) + 1 2 V ( q ) = K cos(2 π q ) � β = 0 . 1 1 β = 2 2 β = ∞ 1.4 T � ( p ) β = 5 β = 10 β = 100 1.3 q p q p q p 1.2 1.1 1.0 0.5 0.5 1.0 q p q p q p ) p

Map with discontinuity Map:     q q + τ T � ( p )         S 1 :      �→         p p − τ V � ( q + τ T � ( p ))    where � s θ β ( p ) ≡ 1 � � � � � 2( p − d ) 2 + ω ( p − d ) T ( p ) = θ β ( p − d ) tanh( β p ) + 1 2 V ( q ) = K cos(2 π q ) � β = 0 . 1 1 β = 2 2 β = ∞ 1.4 T � ( p ) β = 5 β = 10 β = 100 1.3 q p q p q p 1.2 1.1 1.0 0.5 0.5 1.0 q p q p q p ) p

Anomalous tail in the action representation 1-step time evolution: � I | U | I 0 � where U = e − i � T ( p ) e − i � V ( q ) Here | I � denotes the eigenfunction of the integrable map L : U 0 | I � = e − i where U 0 = e − i � ω p e − i � K sin q � E | I � initial state � I | I 0 � 0 0 β = 1 -5 -5 1 β = 50 log | � I | U | I 0 � | 2 -10 -10 -15 -15 -20 -20 -25 -25 discontinuous line -30 -30 -35 -35 -0.2 -0.2 0 0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 | I

10 -15 10 -20 10 -10 10 -5 10 0 -0.5 0 0.5 1 Semiclassical analysis for discontinuous for discontinuous limit ( β = ∞ ) 1-step propagator in the action representation � ∞ � ∞ � ∞ � �� − i � � I � | ˆ dq � F ( q � , p , q ; I , I � ) U | I � = dq dp exp � −∞ −∞ −∞ 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 , �� d � + ∞ � � � � � � dq � dq � dp dq = dp + dp dq −∞ d semiclassical exact (edge contribution) � T 1 ( p ) | 0 � | 2 log | � ψ m | U | ψ 0 � | 2 Since T ( p ) has a discontinuity at p = d , 200 �� d � + ∞ � � � � � � 100 | � I | e − i dq � dq � dp dq = dp + dp dq −∞ d 0 5 10

for large but finite β Semiclassical analysis for discontinuous with an edge p d , 1-step propagator in the action representation � ∞ � ∞ � ∞ � �� − i � � I � | ˆ dq � F ( q � , p , q ; I , I � ) U | I � = dq dp exp � −∞ −∞ −∞ where F ( q � , p , q ; I , I � ) : = S ( I � , q � ) − S ( I , q ) − p ( q � − q ) + T ( p ) + V ( q ) − − − for the present map � s � 2( p − d ) 2 + ω ( p − d ) T ( p ) = θ β ( p − d ) V ( q ) = K cos(2 π q ) S ( I , q ) = Iq + K sin q

for large but finite β Semiclassical analysis for discontinuous with an edge p d , 1-step propagator in the action representation � ∞ � ∞ � ∞ � �� − i � � I � | ˆ dq � F ( q � , p , q ; I , I � ) U | I � = dq dp exp � −∞ −∞ −∞ where F ( q � , p , q ; I , I � ) : = S ( I � , q � ) − S ( I , q ) − p ( q � − q ) + T ( p ) + V ( q ) − − − for the present map � s � 2( p − d ) 2 + ω ( p − d ) T ( p ) = θ β ( p − d ) V ( q ) = K cos(2 π q ) S ( I , q ) = Iq + K sin q � � � � �� −∞   Saddle point condition:          q �   I �  I q S S 2 S S S S R R ∂ F ∂ F ∂ F                  �− →  �− →  �− →         ∂ q � = 0 , = 0 , = 0 ⇐ ⇒                 p � q � ∂ p ∂ q q p     

A set of saddle points complex solution 1 β = 10 completely integrable model p � q � q Re q Im q real solution q Im q Im q T 2 T 1 q turning point q q Re q Re q

Two types of turning points 1 β = 10 1. Turning points on the real manifold locally highly degenerated, reflecting tangency between I and S 1 ( I ) Im q T 2 T 1 2. Turning points in the complex plane q increase as β gets large, reflecting the increase of singularities, and possibly the existence of natural boundaries Re q 1. turning points on the real manifold 2. turning points in the complex plane

1-step time evolution of the real manifold I I : An invariant curve of the integrable map S 1 ( I ) : 1-step time evolution of I 1 β = 10 β = 0 . 1 1 β = 1 q p q p q p S 1 ( I ) : I ( 0.0000000000e+00,-8.0000000000e+00)-( 6.2831853072e+00, 8.0000000000e+00) q p q p ( 0.0000000000e+00,-8.0000000000e+00)-( 6.2831853072e+00, 8.0000000000e+00) q p ( 0.0000000000e+00,-8.0000000000e+00)-( 6.2831853072e+00, 8.0000000000e+00) With increase in β , the initial manifold I comes closer to KAM curves, and moves very slightly within a single step.

Di ff raction integrals with coalescing saddles Integrals with coalescing saddles K � ∞ � Φ K ( t ; x) = t K + 2 + x m t m Ψ K (x) = exp(i Φ K ( t ; x)) dt , where −∞ m = 1 x = ( x 1 , 0 , · · · , 0) Airy integral � x Ψ 1 ( x ) = 2 π � 14 3 1 / 3 Ai β = 0 . 1 3 1 / 3 β = 1 ∂ Φ K ( t , x ) 12 β = 10 ∂ t Pearcey integral 30 10 � ∞ 25 � � i( t 4 + x 2 t 2 + x 1 t ) Ψ 2 (x) = P ( x 2 , x 1 ) = exp dt 8 20 I � −∞ 15 6 Ψ K ( x ) has a convergent series expansion: 10 4 � n + 1 5 ∞ � π ( n ( K + 1) − 1) � 2 � � i n cos Ψ K (x) = Γ a n (x) ( K : odd) 2 x ) t K + 2 K + 2 - 4 - 2 0 2 4 2( K + 2) n = 0 0 where -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Im q