# Derivation of Optical-Bloch Equations

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1 Appendix C Derivation of Optical-Bloch Equations In this appendix the optical-bloch equations that give the populations and coherences for an idealized three-level Λ system, Fig. 3. on page 47, will be derived from the Liouville equation. The three-level atom has states a, b, and c. The two fields, probe and pump, are close to resonance with the transitions b a and c a, respectively. The optical-bloch equations for the three-level system can then be applied to the idealized two-level system, Fig..1 on page 1, by making small adjustments to the equations. The Hamiltonian for the three-level system with two fields is, H = H A + V pr + V pu. C.1) The interactions of the atoms with the probe and pump laser fields are given by, V pr = d E pr cos ω pr t), C.) V pu = d E pu cos ω pu t). C.3) Rotating Wave Approximation For a transition between states m and n the atomic dipole, d, is given by, d = d mn n m + m n ). C.4) 160

2 Appendix C. Optical-Bloch Equations 161 n m is the raising operator and acting on state m raises the atom to state n, m n is the lowering operator and acting on state n lowers the atom to state m. The interaction of the atom with the probe laser field can be rewritten in terms of the raising and lowering operators, d E pr cosω pr t) = Ω pr a b e iω prt + a b e iωprt C.5) + b a e iωprt + b a e iωprt), Ω pr d ba E pr, C.6) d ba = b d a. C.7) Ω pr is the Rabi frequency, this represents the coupling between the laser field and the atomic dipole. The term e iωprt, in equation C.6, is associated with absorption of a photon, whereas e iωprt is associated with the emission of a photon. For the rest of this derivation the terms where photons are absorbed and the atom falls from a to b and where photons are emitted and the atom is raised from b to a, will be neglected. This is known as the rotating-wave approximation. The neglected terms are not nearly as significant as the resonant terms, for a detailed justification see Atom Photon Interactions, [113]. Therefore making the rotating-wave approximation, V pr = Ω pr a b e iω prt + b a e iωprt), C.8) V pu = Ω pu a c e iω put + c a e iωput). C.9) The resulting Hamiltonian can be rewritten in matrix form, ω a Ω pr /) e iωprt Ω pu /) e iωput H = Ω pr /) e iωprt ω b 0. C.10) Ω pu /) e iωput 0 ω c The density matrix for the three-level system is, ρ aa ρ ab ρ ac ρ = ρ ba ρ bb ρ bc. ρ ca ρ cb ρ cc C.11)

3 Appendix C. Optical-Bloch Equations 16 From the Liouville equation equation.1), ρ = i [ρ, H ] γρ, ρ mn = i [ρ, H ] mn γρ) mn, C.1) [ρ, H ] mn = Σ k ρ mk H kn H mk ρ kn ), C.13) and γρ) mn = γ mn ρ mn, C.14) γ mn = Γ m + Γ n. C.15) Hence the equations of motion for the coherences are, ρ ab = i ω a ω b ) + γ ab ) ρ ab + iω pre iωprt ρ aa ρ bb ) C.16) e iωput ρ cb, ρ ac = i ω a ω c ) + γ ac ) ρ ac + iω pue iωput ρ aa ρ cc ) C.17) iω pre iωprt ρ bc, ρ cb = i ω c ω b ) + γ cb ) ρ cb + iω pre iωprt e +iωput ρ ab. ρ ca C.18) Similarly the equations of motion for the populations are, ρ cc = iω pu ρab e +iωprt ρ ba e iωprt) C.19) + iω pu ρac e +iωput ρ ca e iωput) Γ a ρ aa, ρba e +iωprt ρ ab e iωprt) Γ b ρ bb + Γ a ρ aa C.0) ρca e +iωput ρ ac e iωput) Γ c ρ cc + Γ a ρ aa C.1)

4 Appendix C. Optical-Bloch Equations 163 Now introduce the slow variables, ρ ab = ρ ab e iωprt, C.) ρ ac = ρ ac e iωput, C.3) ρ cb = ρ cb e iωpr ωpu)t. C.4) Substituting equations C., C.3, and C.4 into the equations of motion for the coherences, equations C.16, C.17, and C.18 leads to, ρ ab = i ω a ω b ω pr ) + γ ab ) ρ ab + iω pr ρ aa ρ bb ) C.5) ρ cb, ρ ac = i ω a ω c ω pu ) + γ ac ) ρ ac + iω pu ρ aa ρ cc ) C.6) iω pr ρ bc, ρ cb = i ω c ω b ω pr + ω pu ) + γ cb ) ρ cb + iω pr ρ ca C.7) ρ ab. Similarly for the populations, substituting equations C., C.3, and C.4 into equations C.19, C.0, and C.1 leads to, ρ ab ρ ba ) + iω pu ρ ac ρ ca ) Γ a ρ aa, C.8) ρ ba ρ ab ) Γ b ρ bb + Γ a ρ aa, C.9) ρ cc = iω pu ρ ca ρ ac ) Γ c ρ cc + Γ a ρ aa. C.30) The detuning from resonance of the probe and pump field, δ pr and δ pu, is defined as, δ pr = ω pr ω a + ω b, C.31) δ pu = ω pu ω a + ω c. C.3)

5 Appendix C. Optical-Bloch Equations 164 It follows that, ρ ab = γ ab iδ pr ) ρ ab + iω pr ρ aa ρ bb ) C.33) ρ cb, ρ ac = γ ac iδ pu ) ρ ac + iω pu ρ aa ρ cc ) C.34) iω pr ρ bc, ρ cb = γ cb i δ pr δ pu )) ρ cb + iω pr ρ ca C.35) ρ ab. This is the form of the optical-bloch equations, for the three-level Λ system, that will be used to make predictions of the absorption and dispersion of the probe beam in chapter 3. Two-level atom In the case of the two-level atom, with states b and a, equations C.8, C.9 and C.33 apply but with, Ω pu = 0. Also a only decays to b through spontaneous emission, so the term + Γ/) ρ aa is replaced by +Γ ρ aa. Hence for the two-level atom, ρ ab = γ ab iδ pr ) ρ ab + iω pr ρ aa ρ bb ), C.36) ρ ab ρ ba ) Γ a ρ aa, C.37) ρ ba ρ ab ) Γ b ρ bb + Γ a ρ aa. C.38) These equations will form the basis of the predictions of the absorption and dispersion for a two-level atom in chapter.

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