Week 5

Figure 2.2: Schematic setup of the experiment (from M. Brune et al., Phys. Rev. Lett. 76, 1800 (1996)). An atom flies through a cavity formed by two mirrors (C) along the arrow. The internal state of the atom is measured in D by ionizing the excited atomic state and the time Δt between the preparation of the atom in B and the detection in D is recorded to calculate the velocity of the atom. From thei sthe time t spent in the cavity can be determined. Figure 2.3: Due to the smallness of the matrix element, only resonant processes have to be taken into account, which are the emission of a cavity photon for the transition from the n =51tothen = 50 state of the Rb atom and the absorption during the reverse transition. photon mode in the cavity is given by H = ω 0 a † a + ∑ E } {{ } n c † } {{ nc n } 1 photon mode energy levels in cavity of the atom + ∑ ( g ) n,m a + a † c † nc m } {{ } dipol transsition of level m → n + h.c. }{{} the hermitian conjugate (h.c.) describes the reverse process. As we have seen, the dipole matrix element which determines the size of g n,m is extremely small. Therefore only resonant processes are relevant, i.e. transitions where |E m − E n ± ω 0 | is also small, |E m − E n ± ω 0 | |g n,m |. Therefore from all atomic levels, only two will contribute, see Fig. 2.3. These two relevant internal states, one can describe by a spin degree of freedom, where ↑ (↓) describes the state with higher (lower) energy. As for the transition from ↑ to ↓ energy will be released, a photon has to be emitted. In contrast, the process where a photon is absorbed during the transition from ↑ to ↓ can safely be neglected as it is far off-resonance, 57

|E ↑ − E ↓ + ω 0 |≫|g n,m |. After these (very precise) approximations, one obtains the famous Jaynes-Cummings model H = ω 2 σ z } {{ } 2atomic state + ω 0 a † a + g ( a } {{ } † σ − } {{ } photon energy resonant emission + aσ +) } {{ } resonant absorption (2.17) where ω = E n − E n′ is the energy difference of the two atomic levels, g = g n,n ′ the corresponding matrix element and ω 0 the energy of the photon in the cavity. The two levels ( in the ) atom and their transitions are ( described ) just by Pauli matrices, 1 0 0 0 σ z = for the energy levels, σ − = for the transition from the 0 −1 1 0 ( ) 0 1 higher- to the lower energy state and σ + = for the reverse process. 0 0 The Hamiltonian (2.17) can be simplified even more by realizing that from the infinite number of states |↑,n〉 and |↓,n〉 only pairs of two couple (here n is the number of photons in the cavity). For example, the state |↑, n− 1〉, anexcited atom with n − 1 photons can only couple to |↓, n〉, the deexcited atom with n photons but not to any other state. Using that a † σ − |↑, n− 1〉 = √ n |↓, n〉 and aσ + |↓, n〉 = √ n |↑, n− 1〉 we can therefore write in this subspace ( Δ 2 H = √ ng √ ) ng + E 0 1 (2.18) with Δ= ω ( 2 + ω 0(n − 1) − − ω ) 2 + ω 0 n = ω − ω 0 . An especially interesting point is obtained directly at resonance, i.e., for − Δ 2 ω = ω c ⇔ Δ=0 which is considered in the following. In this case the eigenvalues of H are given by E s/a = E 0 ± √ ( ) ng with eigenvectors |s/a, n〉 = √ 1 1 2 = √ 1 ±1 2 (|↑, n− 1〉±|↓,n〉). As a first example, let us consider an excited atom flying into the cavity. At t =0 the initial state is given by |↑, n− 1〉. As(1, 0) = ((1, 1)/ √ 2+(1, −1)/ √ 2)/ √ 2, the 58

- Page 1 and 2: 2.3.3 Decay rate of excited atoms A
- Page 3 and 4: [ ] Using that [⃗r i ,H 0 ]= r i
- Page 5: 2.4 Cavity QED and Rabi oscillator
- Page 9 and 10: Figure 2.4: Measurements from M. Br
- Page 11: effect is not very pronounced. This