Physics of the Jaynes-Cummings Model

Paul Eastham

February 16, 2012

Outline

1 The model

2 Solution

3 Experimental ConsequencesVacuum Rabi splittingRabi oscillations

4 Summary

5 Course summary

The model

= Single atom in an electromagnetic cavity

MirrorsSingleatom

Realised experimentallyTheory:“Jaynes Cummings Model”⇒ Rabi oscillations

– energy levels sensitive to single atom and photon

– get inside the mechanics of “emission” and “absorption”

Outline

1 The model

2 Solution

3 Experimental ConsequencesVacuum Rabi splittingRabi oscillations

4 Summary

5 Course summary

The model

Outline

1 The model

2 Solution

3 Experimental ConsequencesVacuum Rabi splittingRabi oscillations

4 Summary

5 Course summary

The model

Atom-field Hamiltonian

Last lecture –

H =∑

n

~ωna†nan

+∑

i

Ei |i〉〈i |

+∑n,s

∑ij

En sin(knzat)(an + a†n)es.Dij |i〉〈j |.

The model

→ Jaynes-Cummings Model

= One field mode, two atomic states

Energy of photon in field mode

H = (∆/2) (|e〉〈e| − |g〉〈g|) + ~ω a†a + ~Ω2 (a|e〉〈g|+ a†|g〉〈e|).

Dipole coupling energy

Energy difference between atomic levels

Solution

Outline

1 The model

2 Solution

3 Experimental ConsequencesVacuum Rabi splittingRabi oscillations

4 Summary

5 Course summary

Solution

Solving the JCM

H only connects within disjoint pairs |n,g〉 and |n − 1,e〉∴ eigenstates are

un,±|n,g〉+ vn,±|n − 1,e〉.

⇒ En,± = ~ω(n − 12

)± 12

√(∆− ~ω)2 + ~2Ω2n

and at resonance states are

1√2

(|n,g〉 ± |n − 1,e〉).

Solution

Solving the JCM

H only connects within disjoint pairs |n,g〉 and |n − 1,e〉∴ eigenstates are

un,±|n,g〉+ vn,±|n − 1,e〉.

⇒ En,± = ~ω(n − 12

)± 12

√(∆− ~ω)2 + ~2Ω2n

and at resonance states are

1√2

(|n,g〉 ± |n − 1,e〉).

Solution

Jaynes-Cummings Spectrum

Solution

Jaynes-Cummings Spectrum

Experimental Consequences

Outline

1 The model

2 Solution

3 Experimental ConsequencesVacuum Rabi splittingRabi oscillations

4 Summary

5 Course summary

Experimental Consequences

Vacuum Rabi splitting

Transmission experiments: idea

Laser Detector

Transmission

Frequency/(Resonance frequency)

With no atom

(Fabry-Perot resonator -- SF Optics?)

Experimental Consequences

Vacuum Rabi splitting

Transmission experiments

Transmission

Frequency/(Resonance frequency)

2

4

-40 0 40Probe Detuning ωp (MHz)

-40 0 40

2

4

⟨(

nω

p )×

⟩0

12-

2

4

0.3

0.2

0.1

0.0

0.3

0.2

0.1

0.0

T1( ω

p)

0.3

0.2

0.1

0.0

A. Boca et al., Physical Review Letters 93, 233603 (2004)

Experimental Consequences

Rabi oscillations

Rabi oscillations

Different way to observe the Jaynes-Cummings physics

Suppose we start with no light, add atom in |e〉

What happens?

Photon number oscillates – “Rabi oscillations”

Experimental Consequences

Rabi oscillations

Rabi oscillations

Different way to observe the Jaynes-Cummings physics

Suppose we start with no light, add atom in |e〉

What happens?

Photon number oscillates – “Rabi oscillations”

Experimental Consequences

Rabi oscillations

Rabi oscillations

Different way to observe the Jaynes-Cummings physics

Suppose we start with no light, add atom in |e〉

What happens?

Photon number oscillates – “Rabi oscillations”

Experimental Consequences

Rabi oscillations

Rabi oscillations

Different way to observe the Jaynes-Cummings physics

Suppose we start with no light, add atom in |e〉

What happens?

Photon number oscillates – “Rabi oscillations”

Experimental Consequences

Rabi oscillations

Rabi oscillations

Easiest for resonant case ∆ = ~ω.

Eigenstates with one “excitation” are |±〉 =1√2

(|0,e〉 ± |1,g〉)

Energies E± and E+ − E− = ~Ω

Experimental Consequences

Rabi oscillations

Rabi oscillations

Eigenstates with one “excitation” are |±〉 =1√2

(|0,e〉 ± |1,g〉)

∴ initial state is |0,e〉 =1√2

(|+〉+ |−〉) .

⇒ state at time t is

1√2

( |+〉eiE+t/~ + |−〉eiE−t/~)

= ei(E++E−)t/~ [cos (Ωt/2) |e,0〉+ i sin (Ωt/2) |g,1〉] .

Experimental Consequences

Rabi oscillations

Rabi oscillations

Eigenstates with one “excitation” are |±〉 =1√2

(|0,e〉 ± |1,g〉)

∴ initial state is |0,e〉 =1√2

(|+〉+ |−〉) .

⇒ state at time t is

1√2

( |+〉eiE+t/~ + |−〉eiE−t/~)

= ei(E++E−)t/~ [cos (Ωt/2) |e,0〉+ i sin (Ωt/2) |g,1〉] .

Experimental Consequences

Rabi oscillations

Rabi oscillations

Eigenstates with one “excitation” are |±〉 =1√2

(|0,e〉 ± |1,g〉)

∴ initial state is |0,e〉 =1√2

(|+〉+ |−〉) .

⇒ state at time t is

1√2

( |+〉eiE+t/~ + |−〉eiE−t/~)

= ei(E++E−)t/~ [cos (Ωt/2) |e,0〉+ i sin (Ωt/2) |g,1〉] .

Experimental Consequences

Rabi oscillations

Rabi oscillations

Expected photon number is 〈n〉 = sin2(Ωt/2)

<n>

Time

Experimental Consequences

Rabi oscillations

Rabi oscillations

Rempe et al., Physical Review Letters 58, 393 (1987)

Summary

Outline

1 The model

2 Solution

3 Experimental ConsequencesVacuum Rabi splittingRabi oscillations

4 Summary

5 Course summary

Summary

Summary: light-matter coupling

Interaction between light and matter is the dipole couplingP.E.Seen how to write this in terms of a, |i〉〈j |Single mode+two-level atom+Rotating-waveapproximation=Jaynes-Cummings modelEigenstates of JCM are superpositions like|n,g〉+ |n − 1,e〉Coupling splits the energy levelsSeen experimentally in optical cavities in transmissionand Rabi oscillations

Course summary

Outline

1 The model

2 Solution

3 Experimental ConsequencesVacuum Rabi splittingRabi oscillations

4 Summary

5 Course summary

Course summary

Course Summary: key topics

Characterisation of light by intensity fluctuationsSemiclassical (Planck) approach to

Black-body spectrumShot noise/photon counting(⇒ Poisson distribution of photon number)

Canonical quantization of electromagnetism⇒ write down useful operators for E , B⇒ Predict distributions of measurements of E .Uncertainty principles⇒ variance in measured E

Course summary

Course summary: key topics

Key states:number states (6= classical waves)and coherent states (∼ classical waves)

. . . electric-field distributions in these statesInteraction of light and matterSolution of the Jaynes-Cummings model