physics of the jaynes-cummings modelthe model!jaynes-cummings model = one field mode, two atomic...
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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