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
