Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model Engineering inverse power law decoherence of a qubit Filippo Giraldi and Francesco Petruccione Quantum Research Group, School of Physics and National Institute for Theoretical Physics, University of KwaZulu-Natal, South Africa Grenoble, 30 November 2010 1/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model Strategy Correct Theoretical Physics Strategy At Mathematics Conference speak about Physics! At Physics Conference speak about Mathematics! 2/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model Strategy Correct Theoretical Physics Strategy At Mathematics Conference speak about Physics! At Physics Conference speak about Mathematics! Dangerous strategy At Mathematics Conference speak about Mathematics! 2/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model Outline 1 Introduction 3/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model Outline 1 Introduction 2 Jaynes-Cummings model 3/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model Outline 1 Introduction 2 Jaynes-Cummings model 3 The Fox H -function 3/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model Outline 1 Introduction 2 Jaynes-Cummings model 3 The Fox H -function 4 Structured photonic band gap reservoirs 3/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model Outline 1 Introduction 2 Jaynes-Cummings model 3 The Fox H -function 4 Structured photonic band gap reservoirs 5 Spontaneous emission and the Dicke model 3/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model Situation and strategy Situation Analytical Exponential-like Decoherence processes for Lorentzian type distribution of field modes in Jaynes-Cummings model Oscillating decay and trapping for distribution of field modes with photonic band gap (PBG) edge near the resonant frequency of the two-level system Strategy Delay the Decoherence process by engineering the reservoirs of field modes Search for inverse power laws in the exact dynamics 4/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model The Jaynes-Cummings model The Hamiltonian of the whole system: H = H S + H E + H I , � = 1 ∞ ∞ � � ω k a † k σ − ⊗ a † � � g k σ + ⊗ a k + g ∗ H S = ω 0 σ + σ − , H E = k a k , H I = k k =1 k =1 The operators acting on the Hilbert space of the qubit: σ − = σ † σ + | 0 � = | 1 � , σ + | 1 � = 0 , + The operators acting on the Hilbert space of the field modes: a † � k | · · · , n k , · · · � E = n k + 1 | · · · , n k + 1 , · · · � E ∞ a † � N = σ + σ − + k a k , [ H , N ] = [ H I , N ] = 0 k =1 5/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model Initial condition and time evolution Initial unentangled condition between the qubit and the vacuum state of the external environment: | Ψ(0) � = ( c 0 | 0 � + c 1 (0) | 1 � ) ⊗ | 0 � E Exact time evolution ∞ � | Ψ( t ) � = c 0 | 0 � ⊗ | 0 � E + c 1 ( t ) | 1 � ⊗ | 0 � E + d k ( t ) | 0 � ⊗ | k � E k =1 where | k � E = a † k | 0 � E , k = 1 , 2 , · · · 6/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model The equations of the exact dynamics: Ansatz Interaction picture e ı ( H S + H E ) t | Ψ( t ) � | Ψ( t ) � I = ∞ � = c 0 | 0 � ⊗ | 0 � E + C 1 ( t ) | 1 � ⊗ | 0 � E + Λ k ( t ) | 0 � ⊗ | k � E k =1 where Λ k ( t ) = e ıω k t d k ( t ) , k = 1 , 2 , . . . 7/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model The equations of the exact dynamics: Convolution equation Equations for the coefficients: ∞ g k e ı ( ω 0 − ω k ) t Λ k ( t ) , ˙ � C 1 ( t ) = − ı k =1 k e − ı ( ω 0 − ω k ) t C 1 ( t ) ˙ − ı g ∗ Λ k ( t ) = Closed equation for C 1 ( t ) ˙ C 1 ( t ) = − ( f ∗ C 1 ) ( t ) , where two-point correlation function of the reservoir of field modes ∞ | g k | 2 e − ı ( ω k − ω 0 )( t − t ′ ) t − t ′ � � � f = k =1 8/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model The correlation function and the spectral density Two-point correlation function of the reservoir of field modes ∞ | g k | 2 e − ı ( ω k − ω 0 )( t − t ′ ) � t − t ′ � � f = k =1 For a continuous distribution of modes η ( ω ) � ∞ J ( ω ) e − ı ( ω − ω 0 ) τ d ω, f ( τ ) = 0 where spectral density function J ( ω ) = η ( ω ) | g ( ω ) | 2 frequency dependent coupling constant g ( ω ) 9/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model Reduced density matrix By tracing over the degrees of freedom of the reservoir: = 1 − ρ 0 , 0 = ρ 1 , 1 (0) | G ( t ) | 2 , ρ 1 , 1 ( t ) = ρ ∗ 0 , 1 ( t ) = ρ 1 , 0 (0) e − ıω 0 t G ( t ) ρ 1 , 0 ( t ) = The term G ( t ) fulfills ˙ G ( t ) = − ( f ∗ G ) ( t ) , with G (0) = 1 10/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model The Lorentzian spectral density and Exponential-like relaxations (1) Garraway model (Phys Rev A55 (1997) 2290): Lorentzian spectral density function γλ 2 J L ( ω ) = 1 ˜ ( ω − ω 0 ) 2 + λ 2 2 π Reservoir correlation function: � ∞ J L ( ω ) e − ı ( ω − ω 0 ) τ d ω = γλ ˜ ˜ 2 e − λ | τ | f L ( τ ) = −∞ where λ > 0: spectral width of the coupling γ > 0: relaxation rate 11/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model The Lorentzian spectral density and Exponential-like relaxations (2) The exact dynamics of the qubit: ρ 1 , 1 ( t ) = 1 − ρ 0 , 0 ( t ) = ρ 1 , 1 (0) | G L ( t ) | 2 ρ 1 , 0 ( t ) = ρ ∗ 0 , 1 ( t ) = ρ 1 , 0 (0) e − ıω 0 t G L ( t ) The weak and strong coupling regimes: � � d � + λ � d �� G L ( t ) = e − λ t / 2 cosh d sinh , λ > 2 γ 2 t 2 t � ˆ � ˆ � � �� + λ d d G L ( t ) = e − λ t / 2 cos 2 t sin 2 t , λ < 2 γ ˆ d ˆ λ 2 − 2 γλ � � 2 γλ − λ 2 , d = d = 12/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model Lorentzian type spectral densities Other known solutions for: W 1 Γ 1 W 2 Γ 2 ˜ J L + ( ω ) = + (Γ 1 / 2) 2 + � 2 � 2 � � ω − ω (1) ω − ω (2) + (Γ 2 / 2) 2 r r √ 4Γ 3 / 2 ˜ J L ′ ( ω ) = ( ω − ω r ) 4 + Γ 4 W 1 Γ 1 W 2 Γ 2 ˜ J L − ( ω ) = ( ω − ω r ) 2 + (Γ 1 / 2) 2 − ( ω − ω r ) 2 + (Γ 2 / 2) 2 , ( PBG ) (Exponential-like decay and trapping) 13/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model Literature B.M. Garraway. Phys. Rev. A 55 (1997) 2290-2303 B.M. Garraway. Phys. Rev. A 55 (1997) 4636-4639 B.M. Garraway and P.L. Knight . Phys. Rev. A 54 (1996) 3592-3602 B. Piraux, R. Bhatt and P.L. Knight. Phys. Rev. A 41 (1990) 6296-6312 S. John and T. Quang. Phys. Rev. A 50 (1994) 1764 - 1769 14/38
Introduction Jaynes-Cummings model The Fox H -function Structured photonic band gap reservoirs Spontaneous emission and the Dicke model The Fox H -function Very general function defined by: � � � ( a 1 , α 1 ) , . . . , ( a p , α p ) � H m , n z � p , q ( b 1 , β 1 ) , . . . , ( b q , β q ) � � m =1 Γ (1 − a l − α l s ) z − s Π m j =1 Γ ( b j + β j s ) Π n 1 � = j = m +1 Γ (1 − b j − β j s ) ds Π p l = n +1 Γ ( a l + α l s ) Π q 2 πı C 0 ≤ m ≤ q , 0 ≤ n ≤ p α j , β j > 0; a j , b j complex numbers such that no pole of Γ( b j + β j s ) for j = 1 , 2 , . . . , m coincides with any pole of of Γ(1 − a j + α j s ) for j = 1 , 2 , . . . , n . C is a contour in the complex s -plane from ω − i ∞ to ω + i ∞ such that ( b j + k ) /β j and ( a j − 1 − k ) /α j lie to the right and left of C , respectively. 15/38
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