Quantum graphs where back-scattering is prohibited Brian Winn - PowerPoint PPT Presentation

Quantum graphs where back-scattering is prohibited Brian Winn School of Mathematics Loughborough University 26th February 2008 Brian Winn Quantum graphs without back-scattering

Credits Joint work with Jon Harrison and Uzy Smilansky Journal of Physics A 40 14181–14193. Brian Winn Quantum graphs without back-scattering

A puzzle Find an n × n unitary matrix σ : with diagonal entries 0, and off-diagonal entries with absolute value ( n − 1 ) − 1 / 2 . For example (2 × 2) � 0 � 1 . σ = 1 0 Hint: n = 3 is impossible. . . Brian Winn Quantum graphs without back-scattering

A puzzle Find an n × n unitary matrix σ : with diagonal entries 0, and off-diagonal entries with absolute value ( n − 1 ) − 1 / 2 . For example (2 × 2) � 0 � 1 . σ = 1 0 Hint: n = 3 is impossible. . . Brian Winn Quantum graphs without back-scattering

A puzzle Find an n × n unitary matrix σ : with diagonal entries 0, and off-diagonal entries with absolute value ( n − 1 ) − 1 / 2 . For example (2 × 2) � 0 � 1 . σ = 1 0 Hint: n = 3 is impossible. . . Brian Winn Quantum graphs without back-scattering

A puzzle Find an n × n unitary matrix σ : with diagonal entries 0, and off-diagonal entries with absolute value ( n − 1 ) − 1 / 2 . For example (2 × 2) � 0 � 1 . σ = 1 0 Hint: n = 3 is impossible. . . Brian Winn Quantum graphs without back-scattering

A puzzle Find an n × n unitary matrix σ : with diagonal entries 0, and off-diagonal entries with absolute value ( n − 1 ) − 1 / 2 . For example (2 × 2) � 0 � 1 . σ = 1 0 Hint: n = 3 is impossible. . . Brian Winn Quantum graphs without back-scattering

What is a quantum graph? A metric graph has bonds that have lengths L 1 , . . . , L v > 0. Standing waves satisfy − d 2 ψ j + Boundary d x 2 = k 2 ψ j j = 1, . . . , v . conditions For k = k 0 , k 1 , k 2 , . . . the spectrum of the quantum graph. Brian Winn Quantum graphs without back-scattering

What is a quantum graph? A metric graph has bonds that have lengths L 1 , . . . , L v > 0. Standing waves satisfy − d 2 ψ j + Boundary d x 2 = k 2 ψ j j = 1, . . . , v . conditions For k = k 0 , k 1 , k 2 , . . . the spectrum of the quantum graph. Brian Winn Quantum graphs without back-scattering

What is a quantum graph? A metric graph has bonds that have lengths L 1 , . . . , L v > 0. Standing waves satisfy − d 2 ψ j + Boundary d x 2 = k 2 ψ j j = 1, . . . , v . conditions For k = k 0 , k 1 , k 2 , . . . the spectrum of the quantum graph. Brian Winn Quantum graphs without back-scattering

Scattering at a vertex (Boundary conditions for the differential equation) An incoming wave is scattered at a vertex 2 3 e i kx 1 Scattering is controlled by a d × d unitary matrix σ . d is the degree of the vertex. We do not say anything about the process causing the scattering. Brian Winn Quantum graphs without back-scattering

Scattering at a vertex (Boundary conditions for the differential equation) An incoming wave is scattered at a vertex 2 σ 12 e i kx 3 e i kx σ 13 e i kx σ 11 e i kx 1 Scattering is controlled by a d × d unitary matrix σ . d is the degree of the vertex. We do not say anything about the process causing the scattering. Brian Winn Quantum graphs without back-scattering

Scattering at a vertex (Boundary conditions for the differential equation) An incoming wave is scattered at a vertex 2 σ 12 e i kx 3 e i kx σ 13 e i kx σ 11 e i kx 1 Scattering is controlled by a d × d unitary matrix σ . d is the degree of the vertex. We do not say anything about the process causing the scattering. Brian Winn Quantum graphs without back-scattering

Scattering at a vertex (Boundary conditions for the differential equation) An incoming wave is scattered at a vertex 2 σ 12 e i kx 3 e i kx σ 13 e i kx σ 11 e i kx 1 Scattering is controlled by a d × d unitary matrix σ . d is the degree of the vertex. We do not say anything about the process causing the scattering. Brian Winn Quantum graphs without back-scattering

Scattering at a vertex (Boundary conditions for the differential equation) An incoming wave is scattered at a vertex 2 σ 12 e i kx 3 e i kx σ 13 e i kx σ 11 e i kx 1 Scattering is controlled by a d × d unitary matrix σ . d is the degree of the vertex. We do not say anything about the process causing the scattering. Brian Winn Quantum graphs without back-scattering

The quantum evolution operator Collect all entries of vertex scattering matrices σ in a 2 v × 2 v matrix S . Indexing is by directed bonds. In this example  0 0 0  σ 12 σ 13 σ 11 0 0 0 0 1 0     0 0 0 0 0 1   S = .   1 0 0 0 0 0     0 0 0 σ 22 σ 23 σ 21   0 σ 32 σ 33 σ 31 0 0 Brian Winn Quantum graphs without back-scattering

The quantum evolution operator Collect all entries of vertex scattering matrices σ in a 2 v × 2 v matrix S . Indexing is by directed bonds. In this example  0 0 0  σ 12 σ 13 σ 11 0 0 0 0 1 0     0 0 0 0 0 1   S = .   1 0 0 0 0 0     0 0 0 σ 22 σ 23 σ 21   0 σ 32 σ 33 σ 31 0 0 Brian Winn Quantum graphs without back-scattering

The quantum evolution operator Collect all entries of vertex scattering matrices σ in a 2 v × 2 v matrix S . Indexing is by directed bonds. In this example  0 0 0  σ 12 σ 13 σ 11 0 0 0 0 1 0     0 0 0 0 0 1   S = .   1 0 0 0 0 0     0 0 0 σ 22 σ 23 σ 21   0 σ 32 σ 33 σ 31 0 0 Brian Winn Quantum graphs without back-scattering

The quantum evolution operator Collect all entries of vertex scattering matrices σ in a 2 v × 2 v matrix S . Indexing is by directed bonds. 2 5 4 6 1 3 In this example  0 0 0  σ 12 σ 13 σ 11 0 0 0 0 1 0     0 0 0 0 0 1   S = .   1 0 0 0 0 0     0 0 0 σ 22 σ 23 σ 21   0 σ 32 σ 33 σ 31 0 0 Brian Winn Quantum graphs without back-scattering

The quantum evolution operator Collect all entries of vertex scattering matrices σ in a 2 v × 2 v matrix S . Indexing is by directed bonds. 2 5 Scattering matrix at centre   σ 11 σ 12 σ 13 4 6 σ =  . σ 21 σ 22 σ 23  σ 31 σ 32 σ 33 1 3 In this example  0 0 0  σ 12 σ 13 σ 11 0 0 0 0 1 0     0 0 0 0 0 1   S = .   1 0 0 0 0 0     0 0 0 σ 22 σ 23 σ 21   0 σ 32 σ 33 σ 31 0 0 Brian Winn Quantum graphs without back-scattering

The quantum evolution operator Collect all entries of vertex scattering matrices σ in a 2 v × 2 v matrix S . Indexing is by directed bonds. 2 5 Scattering matrix at centre   σ 11 σ 12 σ 13 4 6 σ =  . σ 21 σ 22 σ 23  σ 31 σ 32 σ 33 1 3 In this example  0 0 0  σ 12 σ 13 σ 11 0 0 0 0 1 0     0 0 0 0 0 1   S = .   1 0 0 0 0 0     0 0 0 σ 22 σ 23 σ 21   0 σ 32 σ 33 σ 31 0 0 Brian Winn Quantum graphs without back-scattering

The quantum evolution operator Continued Waves travelling along a bond of length L acquire a phase e i kL . Put these phases into a 2 v × 2 v diagonal matrix D ( k ) . Define the quantum evolution operator U ( k ) = D ( k ) S . The spectrum There is a standing wave of energy k 2 iff det ( I − U ( k )) = 0. A sequence ( k n ) ∞ n = 1 of “eigenvalues”. Alternatively: Use the von Neumann theory to construct self-adjoint extensions of the Laplace operator. . . (Kostrykin & Schrader approach.) Brian Winn Quantum graphs without back-scattering

The quantum evolution operator Continued Waves travelling along a bond of length L acquire a phase e i kL . Put these phases into a 2 v × 2 v diagonal matrix D ( k ) . Define the quantum evolution operator U ( k ) = D ( k ) S . The spectrum There is a standing wave of energy k 2 iff det ( I − U ( k )) = 0. A sequence ( k n ) ∞ n = 1 of “eigenvalues”. Alternatively: Use the von Neumann theory to construct self-adjoint extensions of the Laplace operator. . . (Kostrykin & Schrader approach.) Brian Winn Quantum graphs without back-scattering

The quantum evolution operator Continued Waves travelling along a bond of length L acquire a phase e i kL . Put these phases into a 2 v × 2 v diagonal matrix D ( k ) . Define the quantum evolution operator U ( k ) = D ( k ) S . The spectrum There is a standing wave of energy k 2 iff det ( I − U ( k )) = 0. A sequence ( k n ) ∞ n = 1 of “eigenvalues”. Alternatively: Use the von Neumann theory to construct self-adjoint extensions of the Laplace operator. . . (Kostrykin & Schrader approach.) Brian Winn Quantum graphs without back-scattering

The quantum evolution operator Continued Waves travelling along a bond of length L acquire a phase e i kL . Put these phases into a 2 v × 2 v diagonal matrix D ( k ) . Define the quantum evolution operator U ( k ) = D ( k ) S . The spectrum There is a standing wave of energy k 2 iff det ( I − U ( k )) = 0. A sequence ( k n ) ∞ n = 1 of “eigenvalues”. Alternatively: Use the von Neumann theory to construct self-adjoint extensions of the Laplace operator. . . (Kostrykin & Schrader approach.) Brian Winn Quantum graphs without back-scattering