Spectral analysis of a completely positive map and thermal - PowerPoint PPT Presentation

Spectral analysis of a completely positive map and thermal relaxation of a QED cavity Joint work with Laurent Bruneau (Cergy) CIRM–January 2009 – p. 1

Overview CIRM–January 2009 – p. 2

Overview • Cavity QED and the Jaynes-Cummings Hamiltonian CIRM–January 2009 – p. 2

Overview • Cavity QED and the Jaynes-Cummings Hamiltonian • Rabi oscillations CIRM–January 2009 – p. 2

Overview • Cavity QED and the Jaynes-Cummings Hamiltonian • Rabi oscillations • The one-atom maser CIRM–January 2009 – p. 2

Overview • Cavity QED and the Jaynes-Cummings Hamiltonian • Rabi oscillations • The one-atom maser • Rabi resonances CIRM–January 2009 – p. 2

Overview • Cavity QED and the Jaynes-Cummings Hamiltonian • Rabi oscillations • The one-atom maser • Rabi resonances • Rabi sectors CIRM–January 2009 – p. 2

Overview • Cavity QED and the Jaynes-Cummings Hamiltonian • Rabi oscillations • The one-atom maser • Rabi resonances • Rabi sectors • Ergodic properties of CP maps CIRM–January 2009 – p. 2

Overview • Cavity QED and the Jaynes-Cummings Hamiltonian • Rabi oscillations • The one-atom maser • Rabi resonances • Rabi sectors • Ergodic properties of CP maps • Ergodic properties of the one-atom maser CIRM–January 2009 – p. 2

Overview • Cavity QED and the Jaynes-Cummings Hamiltonian • Rabi oscillations • The one-atom maser • Rabi resonances • Rabi sectors • Ergodic properties of CP maps • Ergodic properties of the one-atom maser • Few words about the proof CIRM–January 2009 – p. 2

Overview • Cavity QED and the Jaynes-Cummings Hamiltonian • Rabi oscillations • The one-atom maser • Rabi resonances • Rabi sectors • Ergodic properties of CP maps • Ergodic properties of the one-atom maser • Few words about the proof • Metastable states of the one-atom maser CIRM–January 2009 – p. 2

Overview • Cavity QED and the Jaynes-Cummings Hamiltonian • Rabi oscillations • The one-atom maser • Rabi resonances • Rabi sectors • Ergodic properties of CP maps • Ergodic properties of the one-atom maser • Few words about the proof • Metastable states of the one-atom maser • Open questions CIRM–January 2009 – p. 2

1. Cavity QED and the Jaynes-Cummings Hamiltonian The cavity: CIRM–January 2009 – p. 3

1. Cavity QED and the Jaynes-Cummings Hamiltonian The cavity: one mode of the quantized EM-field H cavity = ωa ∗ a CIRM–January 2009 – p. 3

1. Cavity QED and the Jaynes-Cummings Hamiltonian The cavity: one mode of the quantized EM-field H cavity = ωa ∗ a The atom: a 2 level system H atom = ω 0 b ∗ b CIRM–January 2009 – p. 3

1. Cavity QED and the Jaynes-Cummings Hamiltonian The cavity: one mode of the quantized EM-field H cavity = ωa ∗ a The atom: a 2 level system H atom = ω 0 b ∗ b The coupling: dipole approximation H dipol = λ 2 ( b + b ∗ )( a + a ∗ ) CIRM–January 2009 – p. 3

1. Cavity QED and the Jaynes-Cummings Hamiltonian The cavity: one mode of the quantized EM-field H cavity = ωa ∗ a The atom: a 2 level system H atom = ω 0 b ∗ b The coupling: dipole approximation H dipol = λ 2 ( b + b ∗ )( a + a ∗ ) The coupling: rotating wave approximation H RWA = λ 2 ( b ∗ a + ba ∗ ) CIRM–January 2009 – p. 3

1. Cavity QED and the Jaynes-Cummings Hamiltonian The cavity: one mode of the quantized EM-field H cavity = ωa ∗ a The atom: a 2 level system H atom = ω 0 b ∗ b The coupling: dipole approximation H dipol = λ 2 ( b + b ∗ )( a + a ∗ ) The Jaynes-Cummings Hamiltonian The coupling: rotating wave H JC = ωa ∗ a + ω 0 b ∗ b + λ 2 ( b ∗ a + ba ∗ ) approximation H RWA = λ 2 ( b ∗ a + ba ∗ ) CIRM–January 2009 – p. 3

2. Rabi oscillations Detuning parameter ∆ = ω − ω 0 The Rotating Wave Approximation is known (at least from numerical investigations) to be accurate as long as | ∆ | ≪ ω 0 + ω CIRM–January 2009 – p. 4

2. Rabi oscillations Detuning parameter ∆ = ω − ω 0 The Rotating Wave Approximation is known (at least from numerical investigations) to be accurate as long as | ∆ | ≪ ω 0 + ω CIRM–January 2009 – p. 4

2. Rabi oscillations Detuning parameter ∆ = ω − ω 0 2 ˛ ˛ ˛ � n − 1; ↑| e − i tH JC | n ; ↓� P ( t ) = ˛ ˛ ˛ The Rotating Wave Approximation is known (at least from numerical investigations) to be accurate as long as | ∆ | ≪ ω 0 + ω CIRM–January 2009 – p. 4

2. Rabi oscillations Detuning parameter ∆ = ω − ω 0 2 ˛ ˛ ˛ � n − 1; ↑| e − i tH JC | n ; ↓� P ( t ) = ˛ ˛ ˛ The Rotating Wave Approximation is n photons Rabi frequency known (at least from numerical investigations) to be accurate as long as p λ 2 n + ∆ 2 Ω Rabi ( n ) = | ∆ | ≪ ω 0 + ω CIRM–January 2009 – p. 4

2. Rabi oscillations Detuning parameter ∆ = ω − ω 0 2 ˛ ˛ ˛ � n − 1; ↑| e − i tH JC | n ; ↓� P ( t ) = ˛ ˛ ˛ The Rotating Wave Approximation is n photons Rabi frequency known (at least from numerical investigations) to be accurate as long as p λ 2 n + ∆ 2 Ω Rabi ( n ) = | ∆ | ≪ ω 0 + ω " « 2 # „ ∆ sin 2 Ω Rabi ( n ) t/ 2 P ( t ) = 1 − Ω Rabi ( n ) CIRM–January 2009 – p. 4

3. The one-atom maser CIRM–January 2009 – p. 5

3. The one-atom maser Repeated interaction scheme O H = H cavity ⊗ H beam , H beam = H atom n n ≥ 1 CIRM–January 2009 – p. 5

3. The one-atom maser Repeated interaction scheme O H = H cavity ⊗ H beam , H beam = H atom n n ≥ 1 H n = H JC acting on H cavity ⊗ H atom n CIRM–January 2009 – p. 5

3. The one-atom maser Repeated interaction scheme O H = H cavity ⊗ H beam , H beam = H atom n n ≥ 1 H n = H JC acting on H cavity ⊗ H atom n Cavity state after n interactions " n ! # e − i τH n · · · e − i τH 1 e i τH 1 · · · e i τH n O ρ n = Tr H beam ρ 0 ⊗ ρ atom k k =1 CIRM–January 2009 – p. 5

3. The one-atom maser Thermal beam: ρ atom k = ρ β = Z − 1 e − βH atom CIRM–January 2009 – p. 6

3. The one-atom maser Thermal beam: ρ atom k = ρ β = Z − 1 e − βH atom ρ n = L β ( ρ n − 1 ) CIRM–January 2009 – p. 6

3. The one-atom maser Thermal beam: ρ atom k = ρ β = Z − 1 e − βH atom ρ n = L β ( ρ n − 1 ) Reduced dynamics h “ ρ ⊗ ρ β ” i e − i τH JC e i τH JC L β ( ρ ) = Tr H atom Completely positive, trace preserving map on the trace ideal J 1 ( H cavity ) CIRM–January 2009 – p. 6

4. Rabi resonances A resonance occurs when the interaction time τ is a multiple of the Rabi period Ω Rabi ( n ) τ ∈ 2 π Z CIRM–January 2009 – p. 7

4. Rabi resonances A resonance occurs when the interaction time τ is a multiple of the Rabi period Ω Rabi ( n ) τ ∈ 2 π Z | n � CIRM–January 2009 – p. 7

4. Rabi resonances A resonance occurs when the interaction time τ is a multiple of the Rabi period Ω Rabi ( n ) τ ∈ 2 π Z | n � CIRM–January 2009 – p. 7

4. Rabi resonances A resonance occurs when the interaction time τ is a multiple of the Rabi period Ω Rabi ( n ) τ ∈ 2 π Z Dimensionless parameters „ ∆ τ „ λτ « 2 « 2 η = , ξ = 2 π 2 π CIRM–January 2009 – p. 7

4. Rabi resonances A resonance occurs when the interaction time τ is a multiple of the Rabi period Ω Rabi ( n ) τ ∈ 2 π Z Dimensionless parameters „ ∆ τ „ λτ « 2 « 2 η = , ξ = 2 π 2 π R ( η, ξ ) = { n ∈ N ∗ | ξn + η is a perfect square } CIRM–January 2009 – p. 7

4. Rabi resonances A resonance occurs when the interaction time τ is a multiple of the Rabi period Ω Rabi ( n ) τ ∈ 2 π Z Dimensionless parameters „ ∆ τ „ λτ « 2 « 2 η = , ξ = 2 π 2 π R ( η, ξ ) = { n ∈ N ∗ | ξn + η is a perfect square } Definition. The system is • Non resonant: R ( η, ξ ) is empty. • Simply resonant: R ( η, ξ ) = { n 1 } . • Fully resonant: R ( η, ξ ) = { n 1 , n 2 , . . . } i.e. has ∞ -many resonances. • Degenerate: fully resonant and there exist n ∈ R ( η, ξ ) ∪ { 0 } and m ∈ R ( η, ξ ) such that n + 1 , m + 1 ∈ R ( η, ξ ) . CIRM–January 2009 – p. 7

4. Rabi resonances Lemma. If η and ξ are CIRM–January 2009 – p. 8

4. Rabi resonances Lemma. If η and ξ are • not both irrational or not both rational: non-resonant; CIRM–January 2009 – p. 8

4. Rabi resonances Lemma. If η and ξ are • not both irrational or not both rational: non-resonant; • both irrational: either simply resonant or non-resonant (the generic case); CIRM–January 2009 – p. 8

4. Rabi resonances Lemma. If η and ξ are • not both irrational or not both rational: non-resonant; • both irrational: either simply resonant or non-resonant (the generic case); • both rational: η = a/b , ξ = c/d (irreducible) and m = LCM( b, d ) X = { x ∈ { 0 , . . . , ξm − 1 } | x 2 m ≡ ηm (mod ξm ) } then non-resonant if X is empty or fully resonant R ( η, ξ ) = { ( k 2 − η ) /ξ | k = jmξ + x, j ∈ N ∗ , x ∈ X } ∩ N ∗ CIRM–January 2009 – p. 8