Resonant Tunneling in Quantum Field Theory

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1 Resonant Tunneling in Quantum Field Theory Dan Wohns Cornell University work in progress with S.-H. Henry Tye September 18, 2009 Dan Wohns Resonant Tunneling in Quantum Field Theory 1/36

2 Resonant Tunneling in Quantum Mechanics Functional Schro dinger Method Motivation Motivation Relevant to study of potentials with many minima Generic existence of resonance effects Dan Wohns Resonant Tunneling in Quantum Field Theory 2/36

3 Results Results Show existence of resonant tunneling in QFT New effect catalytic tunneling can enhance single-barrier tunneling V Φ A Ε1 B C Φ Ε2 Dan Wohns Resonant Tunneling in Quantum Field Theory 3/36

4 General Argument A B C Tunneling rate for single-barrier tunneling is Γ A B = Ae S Tunneling probability for single-barrier tunneling is T A B = Ke S Dan Wohns Resonant Tunneling in Quantum Field Theory 4/36

5 General Argument V x E x 1 x 2 x 3 x 4 x Naive WKB analysis T A C T A B T B C is incorrect Instead t A C = t A B + t B C or equivalently T A C = T A BT B C T B C +T B C. Enhancement due to resonant tunneling effect. Dan Wohns Resonant Tunneling in Quantum Field Theory 5/36

6 Outline 1 Resonant Tunneling in Quantum Mechanics Dan Wohns Resonant Tunneling in Quantum Field Theory 6/36

7 Potential V x E x 1 x 2 x 3 x 4 x Dan Wohns Resonant Tunneling in Quantum Field Theory 7/36

8 WKB Approximation A B C Expand a general wavefunction ( Ψ(x) = e if (x)/ in powers of ψ L,R (x) 1 exp ± i ) dxk(x) in classically allowed k(x) 2m region, where k(x) = (E V (x)) ( 2 ψ ± (x) 1 exp ± ) dxκ(x) in the classically forbidden κ(x) region, where κ(x) = 2m 2 (V (x) E) Dan Wohns Resonant Tunneling in Quantum Field Theory 8/36

9 Matching Conditions Complete solution ψ(x) = α L ψ L (x) + α R ψ R (x) in vacuum A ψ(x) = α + ψ + (x) + α ψ (x) in the classically forbidden region ψ(x) = β L ψ L (x) + β R ψ R (x) in vacuum B ( ) ( αr = 1 Θ + Θ 1 i(θ Θ 1 ) ( ) ) βr α 2 L i(θ Θ 1 ) Θ + Θ 1 β L ( Θ 2 exp 1 x2 x 1 dx ) 2m(V (x) E) Tunneling probability T A B = β R α R 2 = 4 ( Θ + 1 ) 2 Θ 4 Θ 2 Dan Wohns Resonant Tunneling in Quantum Field Theory 9/36

10 Double-Barrier Tunneling Same method of analysis ( (ΘΦ ) T A C = ΘΦ cos 2 W + ( Θ Φ + Θ) Φ 2 sin 2 W ( Φ 2 exp 1 x4 x 3 dx ) 2m (V (x) E) x3 x 2 ) 1 W = 1 dx 2m(E V (x)) If B has zero width T A C 4Θ 2 Φ 2 = T A B T B C /4 4 If W = (n B + 1/2)π then T A C = (Θ/Φ+Φ/Θ) 2 Dan Wohns Resonant Tunneling in Quantum Field Theory 10/36

11 Tunneling Probability versus Energy 1 1 Copeland, Padilla, Saffin Dan Wohns Resonant Tunneling in Quantum Field Theory 11/36

12 Potential V Φ A B c Ε c Φ V (φ) = 1 4 g(φ2 c 2 ) 2 B(φ + c) Dan Wohns Resonant Tunneling in Quantum Field Theory 12/36

13 Thin-Wall Approximation Tunneling rate per unit volume 2 is Γ/V = A exp( S E / ) ( ) Euclidean EOM is φ = V (φ) τ 2 Assume O(4) symmetry Solution to Euclidean EOM ( is ) µ φ DW (τ, x, R) = c tanh 2 (r R) Inverse thickness of domain wall µ = 2gc 2 2 Coleman Dan Wohns Resonant Tunneling in Quantum Field Theory 13/36

14 Euclidean Action S E = dτd 3 x [ 1 2 ( ) ] 2 φ τ ( φ)2 + V (φ) S E = 1 2 π2 R 4 ɛ + 2π 2 R 3 S 1 Domain-wall tension is S 1 = c c dφ 2V (φ) = 2 3 µc ds E dr = 0 implies E = 4 3 πr3 ɛ + 4πR 2 S 1 = 0 R = λ c 3S 1 /ɛ Euclidean action is S E = π2 2 S 1λ 3 c = 27π2 S1 4 2 ɛ 3 Bubble grows if de/dr < 0 or R > 2λ c /3 Dan Wohns Resonant Tunneling in Quantum Field Theory 14/36

15 Basic Idea Infinite-dimensional QFT one-dimensional QM problem In semiclassical limit, the vacuum tunneling rate is dominated by a discrete set of classical paths Equivalent to Euclidean instanton method for single-barrier tunneling Easily generalizes to multiple-barrier tunneling 3 Bender, Banks, Wu 4 Gervais, Sakita 5 Bitar, Chang Dan Wohns Resonant Tunneling in Quantum Field Theory 15/36

16 Functional Schrödinger Equation H = ( ) d 3 φ x ( φ)2 + V (φ) Quantize using [ φ(x), φ(x )] = i δ 3 (x x ) H = ( ( ) 2 ) d 3 x ( φ)2 + V (φ) δ δφ(x) Make ansatz Ψ(φ) = A exp( i S(φ)) HΨ(φ(x)) = EΨ(φ(x)) Dan Wohns Resonant Tunneling in Quantum Field Theory 16/36

17 Semiclassical Expansion S(φ) = S (0) (φ) + S (1) (φ) +... [ ( ) ] d 3 1 δs(0) (φ) 2 x 2 δφ ( φ)2 + V (φ) = E [ ] d 3 x i δ2 S (0) (φ) + 2 δs (0)(φ) δs (1) (φ) δφ 2 δφ δφ = 0 Ignore higher-order terms Goal Determine value of S (0) that gives dominant contribution to the tunneling probability Dan Wohns Resonant Tunneling in Quantum Field Theory 17/36

18 Most Probable Escape Paths φ 0 (x, λ) Trajectory in the configuration space of φ(x) parameterized by λ Give dominant contribution to tunneling probability Variation of S (0) vanishes perpendicular to the MPEP Variation of S (0) is nonvanishing along the MPEP Dan Wohns Resonant Tunneling in Quantum Field Theory 18/36

19 Most Probable Escape Paths Φ x c Λ 3Λ c 2 Λ 2Λ c 3 Λ Λ c Λ Λ c 3 x c Λ 0 Dan Wohns Resonant Tunneling in Quantum Field Theory 19/36

20 Determining S (0) Effective tunneling potential U(λ) = U(φ(x, λ)) = d 3 x ( 1 2 ( φ(x, λ))2 + V (φ(x, λ)) ) Path length (ds) 2 = d 3 x(dφ(x)) 2 = ) 2 = (dλ) 2 m(φ(x, λ)) (dλ) 2 d 3 x ( φ(x,λ) λ Zeroth-Order Solution (Classically Forbidden Region) S (0) = i s 0 ds 2[U(φ(x, s )) E] = i ( λ 2 ds 2[U(φ(x, λ 1 dλ dλ) λ)) E] Dan Wohns Resonant Tunneling in Quantum Field Theory 20/36

21 Determing the MPEP Euler-Lagrange Equation 2 φ(x,τ) τ φ(x, τ) V (φ(x,τ)) φ = 0 ds dτ = 2[U(φ(x, τ)) E] τ plays role of Euclidean time In thin-wall approximation ( solution ) is µ φ 0 (x, τ) = c tanh 2 (r λ c) Euler-Lagrange equation same as Euclidean instanton method Dan Wohns Resonant Tunneling in Quantum Field Theory 21/36

22 Parameterization of MPEP t Λ 0 Λ Λ c Λ c x Λ c Τ 0 Reparameterize solution in terms of λ = λ 2 c τ ( ) 2 µ MPEP is φ 0 (x, λ) c tanh 2 ( x λ) λ λ c Dan Wohns Resonant Tunneling in Quantum Field Theory 22/36

23 Parameterization of MPEP Φ x c Λ 3Λ c 2 Λ 2Λ c 3 Λ Λ c Λ Λ c 3 x c Λ 0 Dan Wohns Resonant Tunneling in Quantum Field Theory 23/36

24 WKB Wavefunction WKB Wavefunction Ψ(φ(x, λ)) = Ae is 0/ = A exp Position-dependent mass m(λ) ( d 3 x ( [ 1 λc 0 dλ ]) 2m(λ)[U(λ) E] ) 2 φ 0 (x,λ) λ Effective tunneling potential U(φ(x, λ)) = d 3 x ( 1 2 ( φ(x, λ))2 + V (φ(x, λ)) ) Now have one-dimensional QM problem Dan Wohns Resonant Tunneling in Quantum Field Theory 24/36

25 Equivalence of FSM with EIM U Λ m Λ Λc Integrate spatially to get U(λ) 2πS 1 λ c λ(λ 2 c λ 2 ) Position dependent mass m(λ) 4πS 1 λ 3 Amplitude is exp ( 27π 2 4 Λ λ c S 4 1 ɛ 3 ) = exp (S E /2) Λc Λ Dan Wohns Resonant Tunneling in Quantum Field Theory 25/36

26 Advantages of FSM Same arguments lead to S (0) (φ(x, λ)) = dλ 2m((φ(x, λ))[e U((φ(x, λ))] and Lorentzian EOM in classically allowed regions If we choose λ = ( λ 2 c + t 2 as parameter, ) MPEP ( takes form ) µ φ 0 (x, λ) = c tanh 2 ( x λ) λ µ ( x λ) λ c = c tanh 2 Single real parameter describes entire system 1 λ 2 Dan Wohns Resonant Tunneling in Quantum Field Theory 26/36

27 Potential V Φ A Ε 1 B C Φ Ε 2 { 1 V (φ) = 4 g 1((φ + c 1 ) 2 c1 2)2 B 1 φ 2B 1 c 1 φ < g 2((φ c 2 ) 2 c2 2)2 B 2 φ 2B 1 c 1 φ > 0 Dan Wohns Resonant Tunneling in Quantum Field Theory 27/36

28 MPEP Τ r 2 r 1 x C B A ( µ φ 0 ( x, λ) = c 1 tanh 1 ) 1 c 2 tanh ) r 1 ( x λ) ) 2 λ ( µ 2 2 λ r 2 ( x λ ) and Lorentzian equation of motion ( λ Θ Λ + c 2 c 1 solves both Euclidean Dan Wohns Resonant Tunneling in Quantum Field Theory 28/36

29 Effective Tunneling Potential U Λ A B Λ r 1 Λ B C Λ Dan Wohns Resonant Tunneling in Quantum Field Theory 29/36

30 Position-Dependent Mass m Λ r 1 Λ Dan Wohns Resonant Tunneling in Quantum Field Theory 30/36

31 Consistency Conditions We require zero total energy at nucleation E (2) = 4π(S (1) r 1ɛ 1 )r1 2 (2) + 4π(S r 2ɛ 2 )r2 2 = 0 We do not demand that the energy of each bubble vanishes individually Equivalent to condition that action is stationary Must also ensure existence of a classically allowed region U(λ) < 0 for Λ > λ > λ B Also require the existence of a second classically forbidden region Dan Wohns Resonant Tunneling in Quantum Field Theory 31/36

32 Resonant Tunneling or Catalytic Tunneling Resonant Tunneling If the inside bubble is large enough λ 2c > r 2 > 2λ 2c /3 tunneling from A to C will complete. Catalytic Tunneling If the inside bubble is too small 0 < r 2 < 2λ 2c /3, inside bubble will collapse Effect will be tunneling from A to B Tunneling rate is exponentially enhanced by presence of C Dan Wohns Resonant Tunneling in Quantum Field Theory 32/36

33 Consistency Conditions r 2 No classically allowed region Λ B Λ 2 c No classically allowed region Resonant Tunneling 2Λ 2 c 3 Catalytic Tunneling du dλ r 0 1 Only one classically forbidden region Λ 1 c r 1 Dan Wohns Resonant Tunneling in Quantum Field Theory 33/36

34 ( (ΘΦ ) T A C = ΘΦ cos 2 W + ( Θ Φ + Θ) Φ 2 ) 1 sin 2 W W = λ 3 λ 2 W = S(1) 1 λ Λ λ B dλ 2m(λ)( U(λ)) λ 2 Λ λ2 B S (1) 1 λ B log Resonance condition is W = (n )π [ ] λ Λ + λ 2 Λ λ2 B λ B Dan Wohns Resonant Tunneling in Quantum Field Theory 34/36

35 Probability of Hitting Resonance Treat tunneling probability as function of λ Λ Expand around resonance at λ Λ = λ R of width Γ λλ ( 2 Γ λλ = Θ ΘΦ( W ) Φ + Θ) Φ λ Λ Separation between resonances λ π ( W λ Λ ) Probability of hitting resonance P(A C) = Γ Λ Λ 2 πθφ = 1 2π (T A B + T B C ) is the larger of two decay probabilities Average tunneling probability < T A C >= P(A C)T A C T A BT B C smaller of two tunneling probabilities T A B +T B C given by Dan Wohns Resonant Tunneling in Quantum Field Theory 35/36

36 Conclusions and Future Directions Used functional Schrödinger method to show how resonant tunneling takes place in QFT with a single scalar Showed existence of catalytic tunneling Expect resonance tunneling phenomena to persist outside of thin-wall approximation Interesting to study the more general case, especially including gravity Dan Wohns Resonant Tunneling in Quantum Field Theory 36/36

arxiv:hep-th/ v3 16 Apr 2007

arxiv:hep-th/ v3 16 Apr 2007 A New View of the Cosmic Landscape S.-H. Henry Tye arxiv:hep-th/0611148v3 16 Apr 2007 Laboratory for Elementary Particle Physics Cornell University, Ithaca, NY 14853 Abstract In this scenario, a generic