The Jaynes-Cummings Model F abio Danielli Bonani University of S - PowerPoint PPT Presentation

The Jaynes-Cummings Model F´ abio Danielli Bonani University of S˜ ao Paulo June 22, 2020 Bonani (USP) The Jaynes-Cummings Model June 22, 2020 1 / 24

Outline Introduction 1 Construction of the Jaynes-Cummings Hamiltonian 2 Quantization of the free electromagnetic field. Quantization of matter. Quantization of interaction. Features of the model 3 Dressed states and the Jaynes-Cummings ladder. Vacuum-field Rabi oscillations. Collapse and revival of atomic oscillations. Bonani (USP) The Jaynes-Cummings Model June 22, 2020 2 / 24

The model It’s a quantum optics model describing the interaction of a two-level atom with a single quantized mode of an optical cavity’s electromagnetic field. Initially proposed by Edwin Jaynes and Fred Cummings in 1963. First experimental demonstration in 1984 by Rempe, Walther, and Klein. Is used in cavity QED (Quantum Electrodynamics) and circuit QED, especially in relation to quantum information processing. In the context of solid state system, semiconductor quantum dots are placed inside photonic crystal, micropillar or microdisk resonators. a + 1 � a † � ˆ a † ˆ H JC = � ω k ˆ 2 � ω 0 ˆ σ z + � Ω ˆ σ + ˆ a + ˆ σ − ˆ (1) Bonani (USP) The Jaynes-Cummings Model June 22, 2020 3 / 24

Starting from a more general system with eletromagnetic field, we have: � � � 2 � � 1 p i − e e � + 1 + 1 � E | 2 + | � � ˆ � σ i · � ( | � B | 2 ) d 3 r H = � A + V ( � r ) − 2 m e c � B U ( � r i , � r j ) 2 m e c 2 2 i j � �� � � �� � � �� � � �� � spin-field coupling free field minimal coupling interaction e-e � p 2 � e 2 � e + 1 � � � E | 2 + | � � A · � � ( | � B | 2 ) d 3 r i = + V ( � r i ) − A · � p i + A 2 m e 2 m e c 2 m e c i i i � �� � � �� � interaction field-atom interaction field-field / After the approximations, we have: � p 2 � e � � � H = 1 � � E | 2 + | � ˆ ( | � B | 2 ) d 3 r � i + + V ( � r i ) − A · � (2) p i 2 2 m e m e c i i � �� � � �� � � �� � ˆ H field ˆ ˆ H atom H int Bonani (USP) The Jaynes-Cummings Model June 22, 2020 4 / 24

Quantization of the free electromagnetic field We will derive the free field Hamiltonian by quantizing the electromagnetic field in a cavity. Let’s start with Maxwell’s equations in free space: E = − ∂ � B ∇ · � ∇ × � E = 0 (3) ∂ t ∂ � B = 1 E ∇ · � ∇ × � B = 0 c 2 ∂ t We can introduce the vector � A ( � r , t ) and scalar ϕ ( � r , t ) potential that give the fields: ∂ � E = −∇ · ϕ − 1 A B = ∇ × � � � (4) A ∂ t c Bonani (USP) The Jaynes-Cummings Model June 22, 2020 5 / 24

The potentials are not unique and have gauge symmetry. They can be shifted using some gauge transformation χ without changing fields: { � A , ϕ } → { � A ′ , ϕ ′ } A ′ = � � A + ∇ χ (5) ϕ ′ = ϕ − 1 ∂χ c ∂ t In Coulomb’s gauge we have ∇ · � A = 0 and ϕ = 0. By Maxwell’s equations we can write the wave equation for the potential vector: ∂ 2 � A − 1 A ∇ 2 � ∂ t 2 = 0 (6) c 2 Bonani (USP) The Jaynes-Cummings Model June 22, 2020 6 / 24

One possible anzats is: � � � A ( � r , t ) = c k ,α ( t ) · � u k ,α ( � r ) (7) α =1 , 2 k where k represents quantum numbers which specify cavity modes (the different modes are orthogonal) and α is the polarization. Substituting 7 in 6, we can see that � u k ,α ( � r ) represents the vibrational mode of cavity (plane waves) and c k ,α ( t ) is the amplitude for a wave. For the total field in some volume V, a Fourier expansion over a collecting of these modes is used supposing boundary conditions � r + � L , t ) = � A ( � A ( � r , t ): 1 � r + c ∗ e α [ c k ,α ( t ) e i � k ,α ( t ) e − i � � k · � k · � r ] A ( � r , t ) = √ ˆ (8) V k ,α Bonani (USP) The Jaynes-Cummings Model June 22, 2020 7 / 24

The classical field energy is given by: � � � � � � 2 � ∂ � H field = 1 d 3 r = 1 1 A B | 2 + | � E | 2 � A | 2 + � � ˆ | � |∇ × � d 3 r (9) � � 2 2 � c ∂ t � Before we substitute 8 in 9: � � c 2 � � c 2 � 1 / 2 � 1 / 2 c ∗ a † c k ,α ( t ) → ˆ a k ,α ( t ) k ,α ( t ) → ˆ k ,α ( t ) 2 ω k 2 ω k � � c 2 � 1 / 2 1 � a k ,α ( t ) e i � r − ˆ k ,α ( t ) e − i � a † ∇ × � e α k × ˆ k · � k · � r ] A = √ ( i ˆ k )[ˆ 2 ω k V k ,α � � c 2 ∂ � � 1 / 2 � − i ω k ˆ � e α 1 A 1 � r − ˆ a k ,α ( t ) e i � k ,α ( t ) e − i � a † k k · � k · � r ] √ ∂ t = [ˆ 2 ω k c c V k ,α Bonani (USP) The Jaynes-Cummings Model June 22, 2020 8 / 24

After some algebra, we find: H field = 1 � ˆ a † a † 2 � ω k [ˆ a k ,α ( t )ˆ k ,α ( t ) + ˆ k ,α ( t )ˆ a k ,α ( t )] 4 k ,α a † ] = 1: We can rewrite this equation remembering [ˆ a , ˆ � � a k ,α ( t ) + 1 � a † ˆ H field = � ω k ˆ k ,α ( t )ˆ (10) 2 k ,α where ω k is the frequency of the vibrational mode. a | n � = √ n | n − 1 � √ a † | n � = ˆ ˆ n + 1 | n + 1 � Bonani (USP) The Jaynes-Cummings Model June 22, 2020 9 / 24

Quantization of matter The possible states for two-level system will be denoted | e � and | g � , which represents, respectively, excited and ground states. � � E e 0 ˆ H atom = E g | g �� g | + E e | e �� e | = 0 E g � E e + E g � � E e − E g � = 1 + 1 0 0 0 E e + E g 0 − ( E e − E g ) 2 2 = 1 I + 1 2( E e + E g )ˆ 2( E e − E g )ˆ σ z Finally, we have: H atom ≈ 1 σ z = 1 ˆ 2( E e − E g )ˆ 2 � ω 0 ˆ σ z (11) where ω 0 is the frequency of transition. Bonani (USP) The Jaynes-Cummings Model June 22, 2020 10 / 24

Quantization of interaction Let’s suppose that wavefunctions for our system are: � � r )ˆ Ψ ∗ ( � ϕ ∗ r )ˆ b † Ψ( � r ) = ϕ i ( � r ) = i ( � b i i i i � � � e � r ) | ˆ ˆ b † i ˆ � ϕ ∗ ϕ ∗ j d 3 r � Ψ i ( � H int | Ψ j ( � r ) � = b j − A · � p i i m e c i , j � � a k ,α e − i ( ω k − ω ij ) t + Ω ∗ ˆ b † i ˆ a † k ,α e i ( ω k + ω ij ) t ] = � b j [Ω ij ˆ ij ˆ i , j k ,α where Ω ij is the Rabi frequency in the dipole approximation. Utilizing the Rotating-Wave Approximation (RWA) and removing the detuning, we have: H int = � Ω[ˆ ˆ b † 2 ˆ a + ˆ b † 1 ˆ a † ] = � Ω[ˆ a † ] b 1 ˆ b 2 ˆ σ + ˆ a + ˆ σ − ˆ (12) Bonani (USP) The Jaynes-Cummings Model June 22, 2020 11 / 24

Jaynes-Cummings Hamiltonian Finally, the Jaynes-Cummings Hamiltonian can be written as: + 1 � a † � ˆ a † ˆ H JC = � ω k ˆ a 2 � ω 0 ˆ σ z + � Ω ˆ σ + ˆ a + ˆ σ − ˆ (13) � �� � � �� � � �� � ˆ H field ˆ H int ˆ H atom Bonani (USP) The Jaynes-Cummings Model June 22, 2020 12 / 24

Dressed states and the Jaynes-Cummings ladder The interaction Hamiltonian can only cause transitions of the type | e �| n � ↔ | g �| n + 1 � . The Hamiltonian can be written as: � Ω √ n + 1 � n � ω k + 1 � 2 � ω 0 ˆ � Ω √ n + 1 H JC = ( n + 1) � ω k − 1 2 � ω 0 where the eigenvalues are given by: � � n + 1 � ( ω 0 − ω k ) 2 + 4Ω 2 ( n + 1) E ± = � ω 0 ± � 2 On ressonance ( ω 0 = ω k ) and relabeling g 0 = 2Ω: � � √ n + 1 E ± = � ω 0 ± g 0 � n + 1 (14) 2 Bonani (USP) The Jaynes-Cummings Model June 22, 2020 13 / 24

Figure: Justaposition of bare states (uncloupled) and dressed states (coupled). Picture of a two level atom coupled to a single mode field. Bonani (USP) The Jaynes-Cummings Model June 22, 2020 14 / 24

Vacuum-field Rabi oscillations The Jaynes-Cummings Hamiltonian may be separated into two commuting parts: H JC = ˆ ˆ H 0 + ˆ H int All the dynamics of the system are contained in the second part. Let the initial state of the field-atom system be | i � = | e , n � and and the final state be | f � = | g , n + 1 � . Thus, the state vector may be written as: | Ψ( t ) � = C i | i � + C f | f � (15) Solving the Schrodinger equation, in the interact picture, we find: √ ˙ C i = − i Ω n + 1 C f √ ˙ C f = − i Ω n + 1 C i Bonani (USP) The Jaynes-Cummings Model June 22, 2020 15 / 24

After plugging one into the other: ¨ C i + Ω 2 ( n + 1) C i = 0 We’ll impose the initial conditions C i (0) = 1 and C f (0) = 0. Solving the pair of harmonic-oscillator-looking equations we get: √ C i ( t ) = cos(Ω n + 1 t ) √ C f ( t ) = − i sin(Ω n + 1 t ) The solution is: √ √ | Ψ( t ) � = cos(Ω n + 1 t ) | e , n � − i sin(Ω n + 1 t ) | g , n + 1 � (16) Bonani (USP) The Jaynes-Cummings Model June 22, 2020 16 / 24

The probability the system remains in the excited state is: √ P i ( t ) = | C i ( t ) | 2 = cos 2 (Ω n + 1 t ) while the probability it makes a transition to the ground state is: √ P f ( t ) = | C f ( t ) | 2 = sin 2 (Ω n + 1 t ) The atomic inversion is given by: √ W ( t ) = P i ( t ) − P f ( t ) = cos(2Ω n + 1 t ) We notice that even in the absence of light (n = 0) there is still a non-zero transition probability. Bonani (USP) The Jaynes-Cummings Model June 22, 2020 17 / 24