Resonance fluorescence

Resonance fluorescence is the process in which a two-level atom system interacts with the quantum electromagnetic field if the field is driven at a frequency near to the natural frequency of the atom. Typically the photon contained electromagnetic field is applied to the two-level atom through the use of a monochromatic laser. A two-level atom is a specific type of two-state system in which the atom can be found in the two possible states. The two possible states are if an electron is found in its ground state or the excited state. In many experiments an atom of lithium is used because it can be closely modeled to a two-level atom as the excited states of the singular electron are separated by large enough energy gaps to significantly reduce the possibility of the electron jumping to a higher excited state. Thus it allows for easier frequency tuning of the applied laser as frequencies further off resonance can be used while still driving the electron to jump to only the first excited state. Once the atom is excited, it will release a photon at the frequency of the absorbed photon within the range of the detuning of the laser from the natural resonance of the atom. The mechanism for this release is the spontaneous decay of the atom. The emitted photon is released in an arbitrary direction. While the transition between two specific energy levels is the dominant mechanism in resonance fluorescence, experimentally other transitions will play a very small role and thus must be taken into account when analyzing results. The other transitions will lead to emission of a photon of a different atomic transition with much lower energy which will lead to 'dark' periods of resonance fluorescence. The dynamics of the electromagnetic field of the monochromatic laser can be derived by first treating the two-level atom as a spin-1/2 system with two energy eigenstates which have energy separation of ħω0. The dynamics of the atom can then be described by the three rotation operators, R i ^ ( t ) {displaystyle {hat {R_{i}}}(t)} , R j ^ ( t ) {displaystyle {hat {R_{j}}}(t)} , R k ^ ( t ) {displaystyle {hat {R_{k}}}(t)} , acting upon the Bloch sphere. Thus the energy of the system is described entirely through an electric dipole interaction between the atom and field with the resulting hamiltonian being described by H ^ = 1 2 ∫ ( ϵ 0 E → ^ 2 ( r → , t ) + 1 μ 0 B → ^ 2 ( r → , t ) ) d 3 x + ℏ ω 0 R k ^ ( t ) + 2 ω 0 μ → ⋅ A → ^ ( 0 , t ) R j ^ ( t ) {displaystyle {hat {H}}={frac {1}{2}}int (epsilon _{0}{hat {vec {E}}}^{2}({vec {r}},t)+{frac {1}{mu _{0}}}{hat {vec {B}}}^{2}({vec {r}},t))d^{3}x+hbar omega _{0}{hat {R_{k}}}(t)+2omega _{0}{vec {mu }}cdot {hat {vec {A}}}(0,t){hat {R_{j}}}(t)} . After quantizing the electromagnetic field, the Heisenberg Equation as well as Maxwell's equations can then be used to find the resulting equations of motion for R k ^ ( t ) {displaystyle {hat {R_{k}}}(t)} as well as for b ^ ( t ) {displaystyle {hat {b}}(t)} , the annihilation operator of the field, R ^ ˙ k ( t ) = − 2 β ( R ^ k ( t ) + 1 2 ) − ( ω 0 / ℏ ) { [ b ^ ( t ) + b ^ † ( t ) ] μ → ⋅ A → ^ f r e e ( + ) ( r → , t ) + H . c . } {displaystyle {dot {hat {R}}}_{k}(t)=-2eta ({hat {R}}_{k}(t)+{frac {1}{2}})-(omega _{0}/hbar ){{vec {mu }}cdot {hat {vec {A}}}_{free}^{(+)}({vec {r}},t)+H.c.}} b ^ ˙ ( t ) = ( − i ω 0 − β + i γ ) b ^ ( t ) − ( β + i γ ) b ^ † ( t ) + 2 ( ω 0 / ℏ ) [ R ^ k ( t ) μ → ⋅ A → ^ f r e e ( + ) ( 0 , t ) + H . c . ] {displaystyle {dot {hat {b}}}(t)=(-iomega _{0}-eta +igamma ){hat {b}}(t)-(eta +igamma ){hat {b}}^{dagger }(t)+2(omega _{0}/hbar )} , where β {displaystyle eta } and γ {displaystyle gamma } are frequency parameters used to simplify equations.