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Physics of the Jaynes-Cummings Model
Paul Eastham
February 16, 2012
Outline
1
The model
2
Solution
3
Experimental Consequences
Vacuum Rabi splitting
Rabi oscillations
4
Summary
5
Course summary
The model
= Single atom in an electromagnetic cavity
Mirrors
Single
atom
Realised experimentally
Theory:
“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 Consequences
Vacuum Rabi splitting
Rabi oscillations
4
Summary
5
Course summary
The model
Outline
1
The model
2
Solution
3
Experimental Consequences
Vacuum Rabi splitting
Rabi oscillations
4
Summary
5
Course summary
The model
Atom-field Hamiltonian
Last lecture –
Ĥ =
X
~ωn ân† ân
n
+
X
Ei |iihi|
i
+
XX
n,s
ij
En sin(kn zat )(ân + ân† )es .Dij |iihj|.
The model
→ Jaynes-Cummings Model
= One field mode, two atomic states
Energy of photon in field mode
Ĥ = (∆/2) (|eihe| − |gihg|) + ~ω ↠â +
Dipole coupling energy
Energy difference between atomic levels
~Ω
2
(â|eihg| + ↠|gihe|).
Solution
Outline
1
The model
2
Solution
3
Experimental Consequences
Vacuum Rabi splitting
Rabi oscillations
4
Summary
5
Course summary
Solution
Solving the JCM
Ĥ only connects within disjoint pairs |n, gi and |n − 1, ei
∴ eigenstates are
un,± |n, gi + vn,± |n − 1, ei.
Solution
Solving the JCM
Ĥ only connects within disjoint pairs |n, gi and |n − 1, ei
∴ eigenstates are
un,± |n, gi + vn,± |n − 1, ei.
q
1
1
⇒ En,± = ~ω(n − ) ±
(∆ − ~ω)2 + ~2 Ω2 n
2
2
and at resonance states are
1
√ (|n, gi ± |n − 1, ei).
2
Solution
Jaynes-Cummings Spectrum
Solution
Jaynes-Cummings Spectrum
Experimental Consequences
Outline
1
The model
2
Solution
3
Experimental Consequences
Vacuum Rabi splitting
Rabi oscillations
4
Summary
5
Course summary
Experimental Consequences
Vacuum Rabi splitting
Transmission experiments: idea
Laser
Detector
With no atom
Transmission
(Fabry-Perot resonator -SF Optics?)
Frequency/(Resonance frequency)
Experimental Consequences
Vacuum Rabi splitting
Transmission experiments
Transmission
0.3
4
0.2
2
0.1
T1(ωp)
4
0.2
2
Frequency/(Resonance frequency)
-2
0.1
⟨n(ω p)⟩ ×10
0.0
0.3
0.0
0.3
4
0.2
2
0.1
0.0
-40
0
-40
40
0
Probe Detuning ωp (MHz)
40
A. Boca et al., Physical Review Letters 93, 233603 (2004)
Experimental Consequences
Rabi oscillations
Rabi oscillations
Different way to observe the Jaynes-Cummings physics
Experimental Consequences
Rabi oscillations
Rabi oscillations
Different way to observe the Jaynes-Cummings physics
Suppose we start with no light, add atom in |ei
Experimental Consequences
Rabi oscillations
Rabi oscillations
Different way to observe the Jaynes-Cummings physics
Suppose we start with no light, add atom in |ei
What happens?
Experimental Consequences
Rabi oscillations
Rabi oscillations
Different way to observe the Jaynes-Cummings physics
Suppose we start with no light, add atom in |ei
What happens?
Photon number oscillates – “Rabi oscillations”
Experimental Consequences
Rabi oscillations
Rabi oscillations
Easiest for resonant case ∆ = ~ω.
1
Eigenstates with one “excitation” are |±i = √ (|0, ei ± |1, gi)
2
Energies E± and E+ − E− = ~Ω
Experimental Consequences
Rabi oscillations
Rabi oscillations
1
Eigenstates with one “excitation” are |±i = √ (|0, ei ± |1, gi)
2
Experimental Consequences
Rabi oscillations
Rabi oscillations
1
Eigenstates with one “excitation” are |±i = √ (|0, ei ± |1, gi)
2
1
∴ initial state is |0, ei = √ (|+i + |−i) .
2
Experimental Consequences
Rabi oscillations
Rabi oscillations
1
Eigenstates with one “excitation” are |±i = √ (|0, ei ± |1, gi)
2
1
∴ initial state is |0, ei = √ (|+i + |−i) .
2
⇒ state at time t is
1
√ ( |+ieiE+ t/~ + |−ieiE− t/~
2
= ei(E+ +E− )t/~ [cos (Ωt/2) |e, 0i + i sin (Ωt/2) |g, 1i] .
Experimental Consequences
Rabi oscillations
Rabi oscillations
Expected photon number is hni = 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 Consequences
Vacuum Rabi splitting
Rabi oscillations
4
Summary
5
Course summary
Summary
Summary: light-matter coupling
Interaction between light and matter is the dipole coupling
P.E.
Seen how to write this in terms of â, |iihj|
Single mode+two-level atom+Rotating-wave
approximation=Jaynes-Cummings model
Eigenstates of JCM are superpositions like
|n, gi + |n − 1, ei
Coupling splits the energy levels
Seen experimentally in optical cavities in transmission
and Rabi oscillations
Course summary
Outline
1
The model
2
Solution
3
Experimental Consequences
Vacuum Rabi splitting
Rabi oscillations
4
Summary
5
Course summary
Course summary
Course Summary: key topics
Characterisation of light by intensity fluctuations
Semiclassical (Planck) approach to
Black-body spectrum
Shot noise/photon counting
(⇒ Poisson distribution of photon number)
Canonical quantization of electromagnetism
⇒ write down useful operators for Ê, B̂
⇒ Predict distributions of measurements of Ê.
Uncertainty principles ⇒ variance in measured Ê
Course summary
Course summary: key topics
Key states:
number states (6= classical waves)
and coherent states (∼ classical waves)
. . . electric-field distributions in these states
Interaction of light and matter
Solution of the Jaynes-Cummings model
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