Homoclinic and Heteroclinic Motions in Quantum Dynamics F. Borondo Dep. de Química; Universidad Autónoma de Madrid, Instituto Mixto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM Stability and Instability in Mechanical Systems: Applications and Numerical Barcelona, 1 December 2008 F. Borondo Homo and Heteroclinic Motions in QM 1/ 67
Outline Introduction 1 Introduction 2 3 4 F. Borondo Homo and Heteroclinic Motions in QM 2/ 67
Outline Introduction 1 Introduction 2 3 4 F. Borondo Homo and Heteroclinic Motions in QM 3/ 67
In his pioneering work on chaos Poincaré showed the importance of Periodic orbits Homoclinic solutions Heteroclinic solutions F. Borondo Homo and Heteroclinic Motions in QM 4/ 67
In this talk, we will discuss the importance of: Periodic orbits Homoclinic motions Heteroclinic motions in Quantum Mechanics F. Borondo Homo and Heteroclinic Motions in QM 5/ 67
Outline Introduction 1 Introduction 2 3 4 F. Borondo Homo and Heteroclinic Motions in QM 6/ 67
Model: Quartic oscillator H = 1 2 (P2 x + P 2 y) + 1 2 x2 y 2 + ε 4 (x4 + y 4 ), ε = 0.01 Smooth, homogeneous potential Mechanical similarity ) 1/4 (, P P 0 = E E 0 ) 1/2 ( ) 3/4 ( ) 1/4, S S 0 = E E 0, T T 0 = E E 0 ( q q 0 = E E 0 Free from hassles due to phase space evolution (bif s) Very chaotic dynamics Thought hyperbolic for ε 0 Dahlqvist and Russberg (1990) found POs for ε = 0 Also Waterland el at. for ε = 1/240 SOS: y = 0, P y > 0 F. Borondo Homo and Heteroclinic Motions in QM 7/ 67
Model: Billiards Bunimovitch stadium billiard Hyperbolic dynamics F. Borondo Homo and Heteroclinic Motions in QM 8/ 67
Billiards: in Nanotechnology Eigler F. Borondo Homo and Heteroclinic Motions in QM 9/ 67
Billiards: for Microcavity Lasers A. Douglas Stone, 1997 F. Borondo Homo and Heteroclinic Motions in QM 10/ 67
Microdisk laser, Douglas Stone, PNAS, 2004 Top and side view of a GaAs microdisk ( 5.2µm diameter) on top of an Al 0.7 Ga 0.3 pedestal. A thin InAs quantum well layer in the middle layer serves as active medium. Image obtained with a scanning electron microscope. F. Borondo Homo and Heteroclinic Motions in QM 11/ 67
Directional Laser emission Directional laser emission has direct applications in optical communications and optoelectronics F. Borondo Homo and Heteroclinic Motions in QM 12/ 67
More on microlasers... r(φ) = a(1 + η 0 cos 2φ + ɛη 0 cos 4φ) F. Borondo Homo and Heteroclinic Motions in QM 13/ 67
More on microlasers... η = 0.09 η = 0.10 η = 0.12 η = 0.16 Exp. Th. A Th. B F. Borondo Homo and Heteroclinic Motions in QM 14/ 67
More on microlasers... F. Borondo Homo and Heteroclinic Motions in QM 15/ 67
Outline Introduction 1 Introduction 2 3 4 F. Borondo Homo and Heteroclinic Motions in QM 16/ 67
Quantum Mechanics De Broglie Hypothesis: λ = h P = 2π P Wave function: ψ(q, t), q=positions, t=time Stationary Schrödinger equation: with Ĥφ n (q) = E n φ n (q) Heisenberg Uncertainty Principle: q p /2 and E τ /2 F. Borondo Homo and Heteroclinic Motions in QM 17/ 67
Example Helmholtz equation: 2 φ n = k 2 nφ n φ n (boundary) = 0 F. Borondo Homo and Heteroclinic Motions in QM 18/ 67
Simpler example (even trivial) ψ I = ψ III = 0 2 d 2 ψ II 2m dx 2 d 2 ψ II + Vψ II = Eψ II 2mE + k 2 ψ dx 2 II, k = But, don t forget the dynamics: k = P F. Borondo Homo and Heteroclinic Motions in QM 19/ 67
Solution ψ(x) = a sin kx + b cos kx First boundary condition: ψ(0) = 0 c = 0 ψ = b sin kx Normalization condition: L 0 ψ 2 2 dx = 1 a = L Second boundary condition: ψ(l) = 0 k n = nπ L Solutions: ψ n (x) = 2 L nπx sin L, n = 1, 2,... F. Borondo Homo and Heteroclinic Motions in QM 20/ 67
But, don t forget the dynamics... k = P Classical action: L Pdx = 2 0 Pdx = 2 L 0 k dx = 2k L = 2 nπ L L = nh Action is quantized in QM! F. Borondo Homo and Heteroclinic Motions in QM 21/ 67
Quantization of the action. How? Einstein Brillouin Kramers (EBK) Method N C j i P i dq i = h ( n j + α ) j 4 Classical info = Quantum condition Associated WKB (Wentzel Kramers Brillouin) wave function ψ(q) = j A j e is j(q)/ F. Borondo Homo and Heteroclinic Motions in QM 22/ 67
Phase space representations of QM Wigner transform (1932) "On the quantum corrections to statistical thermodynamics" W(q, P) = ds e isp ψ ( ( ) q 2) s ψ q + s 2 F. Borondo Homo and Heteroclinic Motions in QM 23/ 67
But... W(q, P) can be negative Why?: Heisenberg s uncertainty principle Solution: Husimi function Gaussian average in cells of area N H(q, P) = dq dp G N q,p (q, P ) W(q, P ) Coherent state representation H(q, P) = 1 (2π ) φ N q,p ψ 2 φ minimum uncertainty coherent state φ(x, y, P x, P y ) = [ ] 2α 1/4 π e α(x x 0) 2 e α(y y0)2 e ip0 x x e ip0 y y F. Borondo Homo and Heteroclinic Motions in QM 24/ 67
But... W(q, P) can be negative Why?: Heisenberg s uncertainty principle Solution: Husimi function Gaussian average in cells of area N H(q, P) = dq dp G N q,p (q, P ) W(q, P ) Coherent state representation H(q, P) = 1 (2π ) φ N q,p ψ 2 φ minimum uncertainty coherent state φ(x, y, P x, P y ) = [ ] 2α 1/4 π e α(x x 0) 2 e α(y y0)2 e ip0 x x e ip0 y y F. Borondo Homo and Heteroclinic Motions in QM 24/ 67
But... W(q, P) can be negative Why?: Heisenberg s uncertainty principle Solution: Husimi function Gaussian average in cells of area N H(q, P) = dq dp G N q,p (q, P ) W(q, P ) Coherent state representation H(q, P) = 1 (2π ) φ N q,p ψ 2 φ minimum uncertainty coherent state φ(x, y, P x, P y ) = [ ] 2α 1/4 π e α(x x 0) 2 e α(y y0)2 e ip0 x x e ip0 y y F. Borondo Homo and Heteroclinic Motions in QM 24/ 67
Outline Introduction 1 Introduction 2 3 4 F. Borondo Homo and Heteroclinic Motions in QM 25/ 67
What are scars? Expected: Chaotic classical dynamics uniformly distributed quantum density F. Borondo Homo and Heteroclinic Motions in QM 26/ 67
Scarred functions But in numerical calculations (McDonald & Kaufman)... Heller in 1984 coined the term scar to name an enhanced localization of quantum probability density of certain eigenstates on classical unstable periodic orbits F. Borondo Homo and Heteroclinic Motions in QM 27/ 67
Scars in Optical Fibers F. Borondo Homo and Heteroclinic Motions in QM 28/ 67
F. Borondo Homo and Heteroclinic Motions in QM 29/ 67
Scars in Microcavity lasers F. Borondo Homo and Heteroclinic Motions in QM 30/ 67
Scars in Microcavity lasers F. Borondo Homo and Heteroclinic Motions in QM 31/ 67
Heller s dynamical explanation for scars Recurrences Fourier transform between: correlation function C(t) = φ(0) φ(t), and corresponding spectrum I(E) = dt e iet/ C(t) F. Borondo Homo and Heteroclinic Motions in QM 32/ 67
Peaks Where? Bohr Sommerfeld quantization condition on the action: S = P dq = 2π ( n + α ) 4 Why? Constructive interference in the WKB wavefunction ψ(q) = A e is(q)/ F. Borondo Homo and Heteroclinic Motions in QM 33/ 67
BUT... What happens to the density that does not come back in the recurrence along the scarring periodic orbit? F. Borondo Homo and Heteroclinic Motions in QM 34/ 67
Outline Introduction 1 Introduction 2 3 4 F. Borondo Homo and Heteroclinic Motions in QM 35/ 67
How to systematically construct scar function Borondo et al., PRL 73, 1613 (1994) version 2007 Wavepacket initially localized on the PO ψ tube (x, y) = N T 0 dt e αx(x xt)2 α y(y y t) 2 cos [ S t µπt 2T + P xt(x x t ) + P yt (y y t ) ] F. Borondo Homo and Heteroclinic Motions in QM 36/ 67
In phase space Quantum SOS based on Husimi function: H(x, P x ) = dx e (x x ) 2 /(2α 2 H ) ipxx ψ(x, y = 0) 2 F. Borondo Homo and Heteroclinic Motions in QM 37/ 67
This can be improved Propagate ψ tube (x, y) in time and Fourier transform at E BS ψ scar (x, y) = N ( ) T E T E dt cos πt 2T E e i(ĥ E BS )t ψ tube (x, y) F. Borondo Homo and Heteroclinic Motions in QM 38/ 67
Same in phase space ψ tube (x, y) ψ scar (x, y) F. Borondo Homo and Heteroclinic Motions in QM 39/ 67
Outline Introduction 1 Introduction 2 3 4 F. Borondo Homo and Heteroclinic Motions in QM 40/ 67
F. Borondo Homo and Heteroclinic Motions in QM 41/ 67
F. Borondo Homo and Heteroclinic Motions in QM 42/ 67
Peaks at ω = 1.67, 2.76, 2.95 F. Borondo Homo and Heteroclinic Motions in QM 43/ 67
Where do these frequencies come from? Additional quantization condition for the circuits in phase space: S i π 2 ν i = 2πn When this condition is fulfilled the recurrence along the circuit reinforce the recurrence along the scarring periodic orbit!! F. Borondo Homo and Heteroclinic Motions in QM 44/ 67
F. Borondo Homo and Heteroclinic Motions in QM 45/ 67
Circuit Type S i ω i ω i 1 Homoclinic 1.464 3.519 2.764 2 Heteroclinic 1.212 2.913 2.913 3 Heteroclinic 4.014 3.365 2.918 4 Heteroclinic 3.299 1.646 1.646 5 Heteroclinic 1.927 4.632 1.651 That agree reasonably well with the previously found values ω = 1.67, 2.76, 2.95 F. Borondo Homo and Heteroclinic Motions in QM 46/ 67
F. Borondo Homo and Heteroclinic Motions in QM 47/ 67
F. Borondo Homo and Heteroclinic Motions in QM 48/ 67
F. Borondo Homo and Heteroclinic Motions in QM 49/ 67
Homoclinic motion and wave functions Phys. Rev. Lett. 97, 094101 (2006) φ scar = T T dt cos( πt 2T) e i(e BS Ĥ)t/ φ tube F. Borondo Homo and Heteroclinic Motions in QM 50/ 67
Peaks coincide with the value of primary homoclinic areas F. Borondo Homo and Heteroclinic Motions in QM 51/ 67
Husimis for 4 scar function with quantization/antiquantization conditions on the homoclinic torus (all quantized on the PO) F. Borondo Homo and Heteroclinic Motions in QM 52/ 67
Scar function n = 224 Homoclinic quantization: n h1 = 189.01, n h2 = 168.07 Extra quantization on heteroclinic orbits: ks he = 2πn he n he1 = 19.00, n he2 = 5.98 Husimis for T = 0.9t E, 1.2t E and 3.3t E F. Borondo Homo and Heteroclinic Motions in QM 53/ 67
Classical phase space F. Borondo Homo and Heteroclinic Motions in QM 54/ 67
Relative relevance of the different circuits Why some circuits are more important than others? We propose a quantity to measure this: 1 Let us consider pieces of homo or heteroclinic trajectories, as those shown before, 2 They have initial, (x i, P xi ), and ending points, (x f, P xf ), close to the fixed point, (x F, P xf ) 3 Substract 4 Apply symplectic transformation in order to write down these differences in terms of new coordinates, (u, s), living on the unstable and stable directions 5 Define A u i s j e λt, T being the time necessary for the trajectory to go from i j 6 Remark: A is symplectic and canonical invariant F. Borondo Homo and Heteroclinic Motions in QM 55/ 67
Circuit Type S i ω i ω i A i 1 Homoclinic 1.464 3.519 2.764 2.79 2 Heteroclinic 1.212 2.913 2.913 2.64 3 Heteroclinic 4.014 3.365 2.918 2.64 4 Heteroclinic 3.299 1.646 1.646 1.34 5 Heteroclinic 1.927 4.632 1.651 1.34 The agreement between A i and the importance we found is astonishing!!!! In the sense that 1 A takes the same value for equivalent circuits 2 More important circuits have smaller values of A F. Borondo Homo and Heteroclinic Motions in QM 56/ 67
One question Is A i related to the Lazutkin s invariant? I don t know, but... ask Ernest Fontich F. Borondo Homo and Heteroclinic Motions in QM 57/ 67
Outline Introduction 1 Introduction 2 3 4 F. Borondo Homo and Heteroclinic Motions in QM 58/ 67
Scar functions Back to the quartic potential Scar functions along the vertical PO F. Borondo Homo and Heteroclinic Motions in QM 59/ 67
Quantum numbers How to compute/assign quantum numbers? Zeros in the Husimi function Leboeuf and Voros Zeros inside the homoclinic circuit F. Borondo Homo and Heteroclinic Motions in QM 60/ 67
First zero in... F. Borondo Homo and Heteroclinic Motions in QM 61/ 67
Second zero in... F. Borondo Homo and Heteroclinic Motions in QM 62/ 67
Second zero in... F. Borondo Homo and Heteroclinic Motions in QM 63/ 67
Counting the zeros in the Husimi function F. Borondo Homo and Heteroclinic Motions in QM 64/ 67
Quantizing... Let us make the argument quantitative... A i 2π (n) + µ i 4 = m i, i = 1, 2 A 1 + A 2 4π (n) + µ 1 + µ 2 = m 8 A 2 A 1 2π (n) µ 2 µ 1 = m, 4 F. Borondo Homo and Heteroclinic Motions in QM 65/ 67
Quantizing... Table: Homoclinic quantum numbers m m n m n 0 43 0 141 1 163 1 713 2 282 3 402 4 521 5 641 6 760 F. Borondo Homo and Heteroclinic Motions in QM 66/ 67
Thanks for your attention F. Borondo Homo and Heteroclinic Motions in QM 67/ 67