Lefschetz-thimble inspired analysis of the Dykhne-Davis-Pechukas - PowerPoint PPT Presentation

Lefschetz-thimble inspired analysis of the Dykhne-Davis-Pechukas method and an application for the Schwinger Mechanism iTHEMS, RIKEN Aug. 21, 2020 Takuya Shimazaki based on TS, K. Fukushima, Ann. Phys. 415 168111 (2020) The University of Tokyo

Summary 2 The Schwinger mechanism can be regarded as tunneling. Ours is inspired by the Lefschetz-thimble method. (cf. the Dykhne-Davis-Pechukas (DDP) method) • Formulation of quantum tunneling • Application to the Schwinger mechanism

Schwinger Mechanism 3 Particle Dirac sea Energies are tilted by Energy The decay width is given by We consider an electric field along the direction. the perturbative vacuum unstable. In an external field (anti)particle pair production makes states • Nonperturbative effect in quantum electrodynamics Γ ≃ exp ( − π m 2 eE ) . • Tunneling from antiparticle states to particle states m z E = (0 ,0 , E z ) z V ( z ) = − E z z . − m

Schwinger Mechanism as Tunneling Energy Hamiltonian describing Schwinger mechanism is given by This is the Landau-Zener formula. with an initial condition, where 4 Nonadiabatic energies are defined by The transition probability is analytically calculable { ⊥ = m 2 + k 2 m 2 x + k 2 y , E z = − ∂ t A ( t ) . H ( t ) ψ i ( t ) = E i ( t ) ψ i ( t ) ( i = ± ) . For A ( t ) = − Et , H is called the Landau-Zener model. E + ( t ) ψ + ( −∞ ) = 0 . δ E ≡ E + − E − ≠ 0 P ( k ) = | ψ + ( ∞ ) | 2 = exp ( − π m 2 eE ) ⊥ E − ( t ) t

Analytic Continuation 5 The DDP formula makes use of such that For t ∈ ℝ , δ E ( t ) ≡ E + ( t ) − E − ( t ) ≠ 0 . However, there can exist closing points t c ∈ ℂ such that δ E ( t c ) = 0 . 2 + m 2 ( k z + eEt ) For example, when A ( t ) = − Et , δ E ( t ) = 2 ⊥ . δ E ( t c ) = 0 ⇒ t c = − k z ± i m ⊥ eE eE t c Im t c > 0 .

DDP Approximation Formula 6 Although it is approximation scheme, it surprisingly gives the analytical result for LZ model. P DDP = exp [ − 2Im ∫ δ E ( t ) dt ] ≡ exp [ − 2Im Δ ( t c ) ] t c 0 P ( k ) = exp ( − π m 2 eE ) = P DDP ⊥ [ [ ⊥ and t c = − k z eE + i m ⊥ 2 + m 2 ( k z + eEt ) When A ( t ) = − Et , δ E ( t ) = 2 eE Then, − 2Im Δ ( t c ) = − 4Im ∫ t c ⊥ = − π m 2 2 + m 2 ⊥ ( k z + eEt ) dt eE . 0

“Derivation” of DDP Formula Contour transformation reminds us the Lefschetz-thimble method. 3. We change the original integration contour in order to only once 7 1. We expand the solution of i ∂ t ψ = H ψ as ψ = ∑ a i ψ i e − iE i t . i =± 2 . The square of a transition amplitude a + ( ∞ ) gives the probability , P = a + ( ∞ ) 2. We derive coupled equation for a ± ( t ) and solve them using first order truncation . a + ( ∞ ) ≃ ∫ E + ( t ) ∞ dt exp [ i Δ ( t ) + ln η ( t ) ] where η ( t ) = ψ * − ( t ) · ψ + ( t ) . −∞ E − ( t ) t c pick up contribution from t c . a + ( ∞ ) ≃ exp[i Δ ( t c )] { ⇒ P = exp[ − 2Im Δ ( t c )] } Re t

Lefschetz-Thimble Method thimbles Good properties of the flow 8 1. We find all saddle points, 2. We draw (dual) thimbles defined by the flow equation. = Their dual thimbles intersect the original integration contour. We employ the semiclassical approximation for Z = ∫ dz exp [ − S ( z ) ] . ∂ S z s , i . = 0 ∂ z z = z i dz d τ = ± ∂ S with z ( τ = 0) = z s , i d τ Im S [ z ( τ )] = 0 , d d ∂ z d τ Re S [ z ( τ )] ≥ 0 ( ≤ 0) 3. The thimbles contributing to Z are determined by the intersection #.

Examples of Lefschetz Thimbles 9 We complexify and focus on the blue line. Z ( ℏ ) = ∫ dz exp[ − S ( z )/ ℏ ] where S ( z ) = z 2 /2 + z 4 /4 . ⇒ z s , i = 0, ± i ℏ arg ℏ > 0 arg ℏ < 0 2 2 1 1 Im z Im z 0 0 - 1 - 1 - 2 - 2 - 2 - 1 0 1 2 - 2 - 1 0 1 2 Re z Re z The Stokes phenomenon occurs at arg ℏ = 0 .

Lefschetz-Thimble Inspired Method 10 We apply the Lefschetz-thimble method to cf. w/o reproduces DDP formula. We use the Gaussian approximation to get a + ( ∞ ) ≃ ∫ dt exp [ i Δ ( t ) + ln η ( t ) ] ≡ ∫ ∞ ∞ dt exp[ − S ( t )] . −∞ −∞ t s , i : i-th saddle point + ( ∞ ) = ∑ 2 π θ i a LT n i e i θ i − S ( t s , i ) ( t s , i ) | . n i : i-th intersection # | S ′ ′ t s , i i θ i : angle of i-th thimble ln η ( t ) , a + ( ∞ ) d Δ ( ∞ ) = exp [ i Δ ( t c ) ] = δ E ( t c ) = 0 ⇒ a DDP + dt t = t c

Modified Landau-Zener Model 11 We consider the following model. T → ∞ . • It reduces to the Landau-Zener model in the limit of t c = i τ but t pole = iT . • There exist not only When τ < T , DDP works. When τ ≥ T , DDP fails to work. t c = i τ t pole = iT t c = i τ t pole = iT Re t Re t

Thimble Structure 12 The red thimble Open: closing points Filled: saddle points Cross: poles Dotted: DDP contour Dashed: dual thimbles Solid: thimbles only contribute. τ ≥ T τ < T 1 � 2 × × × × � � 0 DDP 1 × × � - 1 � � Im t Im t 0 - 2 - 3 - 1 � � � � - 4 - 2 × × � - 2 - 1 0 1 2 - 1 0 1 Re t Re t Dots { Lines {

Comparison of Two Methods 13 DDP DDP log 10 [P( τ , T = 3 − τ , Λ = 10)] - 7 log 10 [ P ( � ,3 - � ;10 )] - 8 - 9 Our method DDP Full - 10 1.2 1.4 1.6 1.8 � τ = 1.5 is a boundary for DDP . • a • Fake peak occurs due to Gaussian approximation.

Schwinger Mechanism Revisited 14 There exist a closing point and a pole The Hamiltonian describing Schwinger mech. We consider a Sauter-type field and define the Keldysh parameter. it reduces the Landau-Zener model. When A ( t ) = − Et , cosh 2 ω t [ ⇒ A ( t ) = − E ω tanh ω t ] , γ ≡ m ω E E z = eE . ( − γ k z m ) t pole = i π m + i γ m ⊥ t c = 1 ω tanh − 1 2 ω .

Sauter-Type Field We define by 15 γ ≡ m ω P ( k ) ≡ exp ( − Am 2 / eE ) . A eE γ , { A → π m 2 • For small DDP works very well. ⊥ m 2 as γ → 0 2 m ( m − γ k z ) A DDP ≃ π m 2 1 1 ⊥ + m + γ k z A → ∞ as k z → ± m γ = ± eE ω γ , 2.5 Our method • For large DDP fails. DDP For simplicity, we set k = 0 . 2.0 Full A ω tan − 1 γ , t pole = i π Then, t c = i 1 2 ω . 1.5 m = 3 , eE = 3 1.0 When γ → ∞ , t c gets closer to t pole . 1 2 3 4 5 γ ≡ m ω � eE ( = ω )

Dynamically Assisted Schwinger Mech. We consider superposition of const. + Sauter-type field. 16 The worldline instanton gives the decay width as cosh 2 ω t [ ⇒ A ( t ) = − Et − ε ω tanh ω t ] ε E z = E + ( E ≫ ε ) Γ ∼ exp ( − Am 2 eE ) . 3 m ≫ ω , m 2 ≫ eE are assumed . γ = π /2 A • F S 0 Poles of tanh( ω t ) play pivotal roles . • A 2 0 1 2 3 γ ≡ m ω � 0 eE

Dynamically Assisted Schwinger Mech. 17 m = 3 , eE = 3 , E = 10 ε 3.0 2.5 A Our method DDP 2.0 Full Worldline ( E = 10 � ) 1.5 0.5 1.0 1.5 2.0 2.5 3.0 γ = m ω � eE = ω m ≫ ω , m 2 ≫ eE γ . • DDP works well for small • DDP asymptotically approaches to Worldline. • Our method always gives reasonable answer.

Summary & Prospect 18 ! Formulation of quantum tunneling ! Application to the Schwinger mechanism ? Stokes phenomenon in tunneling effects + ( ∞ ) = ∑ 2 π ( ∞ ) = exp [ i Δ ( t c ) ] a LT n i e i θ i − S ( t s , i ) cf . a DDP + | S ′ ( t s , i ) | ′ i E + ( t ) • We dealt with it as two-level systems. E − ( t ) • We apply our method comparing with DDP.