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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

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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

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SLIDE 2

Outline

1

Introduction

2

Construction of the Jaynes-Cummings Hamiltonian

Quantization of the free electromagnetic field. Quantization of matter. Quantization of interaction.

3

Features of the model

Dressed states and the Jaynes-Cummings ladder. Vacuum-field Rabi oscillations. Collapse and revival of atomic oscillations.

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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. ˆ HJC = ωk ˆ a†ˆ a + 1 2ω0ˆ σz + Ω

  • ˆ

σ+ˆ a + ˆ σ−ˆ a† (1)

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Starting from a more general system with eletromagnetic field, we have:

ˆ H =

  • i
  • 1

2me

  • pi − e

c

  • A

2

  • minimal coupling

+V ( r) − e 2mec σi · B

  • spin-field coupling

+ 1 2

  • (|

E|2 + | B|2)d3r

  • free field

+ 1 2

  • j

U( ri, rj)

  • interaction e-e
  • =
  • i

p2

i

2me + V ( ri)

  • −
  • i

e mec

  • A ·

pi

  • interaction field-atom

+

  • i

e2 2mec

  • A ·

A

  • interaction field-field

+ 1 2

  • (|

E|2 + | B|2)d3r

/ After the approximations, we have: ˆ H = 1 2

  • (|

E|2 + | B|2)d3r

  • ˆ

Hfield

+

  • i

p2

i

2me + V ( ri)

  • ˆ

Hatom

−

  • i

e mec

  • A ·

pi

  • ˆ

Hint

(2)

Bonani (USP) The Jaynes-Cummings Model June 22, 2020 4 / 24

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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 = 0 ∇ × E = −∂ B ∂t (3) ∇ · B = 0 ∇ × B = 1 c2 ∂ E ∂t We can introduce the vector A( r, t) and scalar ϕ( r, t) potential that give the fields:

  • B = ∇ ×

A

  • E = −∇ · ϕ − 1

c ∂ A ∂t (4)

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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 c ∂χ ∂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 c2 ∂2 A ∂t2 = 0 (6)

Bonani (USP) The Jaynes-Cummings Model June 22, 2020 6 / 24

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One possible anzats is:

  • A(

r, t) =

  • k
  • α=1,2

ck,α(t) · uk,α( r) (7) 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 uk,α( r) represents the vibrational mode

  • f cavity (plane waves) and ck,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 A( r + L, t) = A( r, t):

  • A(

r, t) = 1 √ V

  • k,α

ˆ eα[ck,α(t)ei

k· r + c∗ k,α(t)e−i k· r]

(8)

Bonani (USP) The Jaynes-Cummings Model June 22, 2020 7 / 24

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The classical field energy is given by: ˆ Hfield = 1 2 | B|2 + | E|2 d3r = 1 2 |∇ × A|2 +

  • 1

c ∂ A ∂t

  • 2

d3r (9) Before we substitute 8 in 9: ck,α(t) → c2 2ωk 1/2 ˆ ak,α(t) c∗

k,α(t) →

c2 2ωk 1/2 ˆ a†

k,α(t)

∇ × A = 1 √ V

  • k,α

c2 2ωk 1/2 (i ˆ eα

k × ˆ

k)[ˆ ak,α(t)ei

k· r − ˆ

a†

k,α(t)e−i k· r]

1 c ∂ A ∂t = 1 √ V

  • k,α

c2 2ωk 1/2−iωk ˆ eα

k

c

  • [ˆ

ak,α(t)ei

k· r − ˆ

a†

k,α(t)e−i k· r]

Bonani (USP) The Jaynes-Cummings Model June 22, 2020 8 / 24

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After some algebra, we find: ˆ Hfield = 1 4

  • k,α

2ωk[ˆ ak,α(t)ˆ a†

k,α(t) + ˆ

a†

k,α(t)ˆ

ak,α(t)] We can rewrite this equation remembering [ˆ a, ˆ a†] = 1: ˆ Hfield =

  • k,α

ωk

  • ˆ

a†

k,α(t)ˆ

ak,α(t) + 1 2

  • (10)

where ωk is the frequency of the vibrational mode. ˆ a|n = √n|n − 1 ˆ a†|n = √ n + 1|n + 1

Bonani (USP) The Jaynes-Cummings Model June 22, 2020 9 / 24

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Quantization of matter

The possible states for two-level system will be denoted |e and |g, which represents, respectively, excited and ground states. ˆ Hatom = Eg|gg| + Ee|ee| =

  • Ee

Eg

  • = 1

2 Ee + Eg Ee + Eg

  • + 1

2 Ee − Eg −(Ee − Eg)

  • = 1

2(Ee + Eg)ˆ I + 1 2(Ee − Eg)ˆ σz Finally, we have: ˆ Hatom ≈ 1 2(Ee − Eg)ˆ σz = 1 2ω0ˆ σz (11) where ω0 is the frequency of transition.

Bonani (USP) The Jaynes-Cummings Model June 22, 2020 10 / 24

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Quantization of interaction

Let’s suppose that wavefunctions for our system are: Ψ( r) =

  • i

ϕi( r)ˆ bi Ψ∗( r) =

  • i

ϕ∗

i (

r)ˆ b†

i

Ψi( r)| ˆ Hint|Ψj( r) =

  • i,j

ˆ b†

i ˆ

bj

  • ϕ∗

i

  • −

e mec

  • A ·

pi

  • ϕ∗

j d3r

=

  • i,j
  • k,α

ˆ b†

i ˆ

bj[Ωij ˆ ak,αe−i(ωk−ωij)t + Ω∗

ij ˆ

a†

k,αei(ωk+ωij)t]

where Ωij is the Rabi frequency in the dipole approximation. Utilizing the Rotating-Wave Approximation (RWA) and removing the detuning, we have: ˆ Hint = Ω[ˆ b†

2ˆ

b1ˆ a + ˆ b†

1ˆ

b2ˆ a†] = Ω[ˆ σ+ˆ a + ˆ σ−ˆ a†] (12)

Bonani (USP) The Jaynes-Cummings Model June 22, 2020 11 / 24

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Jaynes-Cummings Hamiltonian

Finally, the Jaynes-Cummings Hamiltonian can be written as: ˆ HJC = ωk ˆ a†ˆ a

ˆ Hfield

+ 1 2ω0ˆ σz

ˆ Hatom

+ Ω

  • ˆ

σ+ˆ a + ˆ σ−ˆ a†

  • ˆ

Hint

(13)

Bonani (USP) The Jaynes-Cummings Model June 22, 2020 12 / 24

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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: ˆ HJC = nωk + 1

2ω0

Ω√n + 1 Ω√n + 1 (n + 1)ωk − 1

2ω0

  • where the eigenvalues are given by:

E± =

  • n + 1

2

  • ω0 ±
  • (ω0 − ωk)2 + 4Ω2(n + 1)

On ressonance (ω0 = ωk) and relabeling g0 = 2Ω: E± =

  • n + 1

2

  • ω0 ± g0

√ n + 1 (14)

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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

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Vacuum-field Rabi oscillations

The Jaynes-Cummings Hamiltonian may be separated into two commuting parts: ˆ HJC = ˆ H0 + ˆ Hint 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) = Ci|i + Cf |f (15) Solving the Schrodinger equation, in the interact picture, we find: ˙ Ci = −iΩ √ n + 1Cf ˙ Cf = −iΩ √ n + 1Ci

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After plugging one into the other: ¨ Ci + Ω2(n + 1)Ci = 0 We’ll impose the initial conditions Ci(0) = 1 and Cf (0) = 0. Solving the pair of harmonic-oscillator-looking equations we get: Ci(t) = cos(Ω √ n + 1t) Cf (t) = −i sin(Ω √ n + 1t) The solution is: |Ψ(t) = cos(Ω √ n + 1t)|e, n − i sin(Ω √ n + 1t)|g, n + 1 (16)

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The probability the system remains in the excited state is: Pi(t) = |Ci(t)|2 = cos2(Ω √ n + 1t) while the probability it makes a transition to the ground state is: Pf (t) = |Cf (t)|2 = sin2(Ω √ n + 1t) The atomic inversion is given by: W (t) = Pi(t) − Pf (t) = cos(2Ω √ n + 1t) 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

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Collapse and revival of atomic oscillations

Fock states are eigenstates of the number operator: ˆ n|n = n|n

∞

  • n=0

|nn| = 1 n|n′ = δn,n′ and can be written in terms of the vacuum state: |n = (ˆ a†)n √ n! |0 (17)

Bonani (USP) The Jaynes-Cummings Model June 22, 2020 18 / 24

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Coherent states are eigenstates of the annihilation operator: ˆ a|α = α|α (18) and we would like to express coherent states in terms of Fock states. So, we need to introduce the displacement operator: ˆ D(α) = eαˆ

a†−α∗ˆ a

It is called so because it displaces the amplitude ˆ a by the complex number α ˆ D†(α)ˆ a ˆ D(α) = ˆ a + α Finally, we have: |α = ˆ D(α)|0 = e− 1

2 |α|2

∞

  • n=0

αn √ n! |n (19)

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Let’s assume the initial atom-field state is: |Ψ(0) = |Ψatom(0) ⊗ |Ψfield(0) where |Ψatom(0) = Cg|g+Ce|e |Ψfield(0) =

∞

  • n=0

e− 1

2 |α|2 αn

√ n! |n =

∞

  • n=0

Cn|n The solution to Schrodinger’s equation is: |Ψ(t) =

∞

  • n=0
  • A(n, t)|e + B(n, t)|g
  • ⊗ |n

where: A(n, t) = CeCn cos(Ωt √ n + 1) − iCgCn+1 sin(Ωt √ n + 1) B(n, t) = iCeCn−1 sin(Ωt√n) + CgCn cos(Ωt√n)

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If we take again the case Ce(0) = 1 and Cg(0) = 0: |Ψ(t) = −i

∞

  • n=0

Cn sin(Ωt √ n + 1)|n + 1|g +

∞

  • n=0

Cn cos(Ωt √ n + 1)|n|e (20) The atomic inversion is given by: W (t) = Pi(t) − Pf (t) = e−N

∞

  • n=0

Nn √ n! cos(2Ω √ n + 1t) (21) where N = |α|2 is the average photon number. Relabeling λ = 2Ω we have:

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Figure: Collapse and revival of atomic oscillations for n=35 and n=65.

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Conclusions

To demonstrate the JC Hamiltonian, we used the dipole approximation and the rotating wave approximation (RWA), where the Jaynes-Cummings Model is valid. Simplified model allows for basic understanding about photon-atom interactions. Vacuum-field Rabi oscillations are purely quantum mechanical and are the result of the atom spontaneously emitting a photon and absorbing it. Collapse and revival of atomic oscillations is a characteristic of interaction of a two-level atom with a cavity prepared in a coherent way.

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References

1

  • E. T. Jaynes and F. W. Cummings, “Comparison of quantum and

semiclassical radiation theories with application to the beam maser”,Proceedings of the IEEE,vol. 51, no. 1, pp. 89–109, 1963.

2

  • G. Rempe, H. Walther, and N. Klein, “Observation of quantum collapse and

revival in a one-atom maser,”Physical review letters, vol. 58, no. 4, p. 353, 1987.

3

  • D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed. (Prentice Hall,

Upper Saddle River, NJ 07458, 2005).

4

  • M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University

Press,Great Britain, 1997).

5

Quantum Mechanics applied to Atoms and Light - Ph.W. Courteille (2020).

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