Laser Cooling of Gallium. Lauren Rutherford

Transcription

1 Laser Cooling of Gallium Lauren Rutherford

2 Laser Cooling Cooling mechanism depends on conservation of momentum during absorption and emission of radiation Incoming photons Net momentum transfer to atom Atoms are slowed in direction of laser beam. Can be extended to 3 dimensions: optical molasses Spontaneously emitted photons Incoming photons Stimulated emitted photons No net momentum transfer

3 Doppler Cooling For strong absorption the frequency of the laser, ν L, must match that of an allowed transition in the atom, ν As the atom is moving, the laser frequency will appear shifted due to the Doppler effect Slightly detuning the laser creates a velocity dependent force: if red-detuned, only atoms moving towards the laser beam will see its frequency shifted towards resonance

4 Density Matrix Approach The density operator, ρ, is used to describe the statistical state of a system w i : probability of the system being in the ith state (for a pure state w i = ) For a two-level system with ground state b and excited state a, the density matrix takes the form: ρ aa and ρ bb represent the populations of a and b ρ ab and ρ ba are coherences The density matrix evolves according to the Liouville equation Aim: to derive a master equation to describe the evolution of the system

5 Atom-Laser Interaction Total Hamiltonian is made up of two parts: atomic Hamiltonian and atom-laser interaction Substituting into the Liouville equation, and removing energy terms: where the dipole interaction Two-level atom interacts with monochromatic plane wave: Using the rotating wave approximation, the equations become: ω δ a ω b

6 Spontaneous Emission As the interaction time is much longer than the lifetime of the excited state, spontaneous emission needs to be included in the master equation a Transition rate, 2γ, is given by Density matrix equations for evolution of populations: b As state b does not decay, the evolution of the coherences is given by

7 Density-Matrix Equations To describe the complete system we use the Lindblad form of the Liouville equation: where describes spontaneous emission The complete master equations for a two-level atom are: ω δ a ω absorption/stimulated emission spontaneous emission b

8 Master equation - exact solution Exact solution known for two-level atom excited state population G γt 3 γ= As laser intensity increases, number of oscillations increases Damping rate remains constant 4 5 excited state population γ.6.8 When spontaneous emission rate, γ, is equal to G, there are no oscillations As γ decreases, oscillations increase until at γ=, pure Rabi oscillation is observed 2 4 γt G= 6 8

9 Liouville equation is ordinary differential equation for n-level system, contains three n x n matrices: ρ describes initial populations and coherences H Γ is the Hamiltonian matrix describes spontaneous emission Can be integrated using standard means... transform to rotating frame Numerical Solution ρ(t) = i[h, ρ(t)] + Γ ρ(t) t evolve by modified Euler (2 nd order Runge-Kutta) method ( Method 2 )

10 Method 2 vs Euler Method excited state population excited state population ! t (a) δ=, G= 2! t (c) δ=-2γ, G= excited state population excited state population ! t (b) δ=, G= ! t (d) δ=-2γ, G=5 exact solution Euler: h=.5 Method 2: h=.5 exact solution Euler: h=. Method 2: h=. exact solution Euler: h=.2 Method 2: h=.2 On resonance: Both methods highly accurate at low laser intensity At high laser intensity, Euler method takes longer to converge, even with very small step size Far off resonance: Euler method begins to diverge, even with very small step sizes and at low laser intensity At high laser intensity, Euler method is highly unstable, while Method 2 is much more robust

11 Method 2 - Step Size For accurate final populations, oscillations must be well resolved (a) G=, γ= (b) G=, γ= (c) G=5, γ= (d) G=, γ= Method 2 sensitivity to changes in step size exact soln h=. h=.5 h=. h=.25 h=.5

12 Error vs Step Size error linear with step size for step size up to.25 for laser intensity up to G=5 possible to predict how change in step size will affect error largest error occurs for lowest laser intensities error analysis ongoing...

13 Steady State Solution Population/Force Ground state population Excited state population Radiation Pressure Force Interaction of the laser field with the induced dipole moment in an atom results in a radiation pressure force where F (z, v, t) = U(z, v, t) z U(z, v, t) = tr(ρ ˆd) E(z, t) This force can be calculated from the atomic density matrix 2! t The atomic populations and radiation pressure force both tend to a constant value - this is the steady state solution. Figure (right) shows steady state solutions for a range of atomic velocities radiation pressure force opposes motion of atom force is strongest when Doppler shift, kv, is equal to the laser detuning, δ. Normalised Populations/Force G=, δ=-5γ!!9!8!7!6!5!4!3!2! Normalised Velocity, kv/!

14 Multi-level atoms Simplest multi-level system is (+3)-level atom e- e e Transitions driven by two counter-propagating laser beams σ - g σ + E = 2 E (e + e i(kz ωt) e e i(kz ωt) ) E 2 = 2 E ( e + e i(kz+ωt) + e e i(kz+ωt) ) Normalised Force and Populations !.2!.4!.6!.8 G=, δ=-5γ!!!8!6!4! Normalised Velocity, kv/! Figure shows velocity dependence of radiation pressure force (light blue ---) for (+3)-level atom Also shown are populations g (green) e- (blue) e (red)

15 Gallium Group III elements are vital to the semi-conductor industry In recent years much research has been carried out on atom lithography and the fabrication of nanoscale structures The ability to directly control and deposit gallium atoms could lead to many exciting new developments Gallium has two stable isotopes, 69 Ga and 7 Ga; both have nuclear spin I=3/2 hyperfine structure needs to be included Unstable isotopes include 66 Ga (I=) and 57 Ga (I=/2) System to be studied is a Λ-system, using two different laser frequencies 2 P/2 2 S/2: λ = 43nm 2 P3/2 2 S/2: λ2 = 47nm

16 Gallium equations Chen, R., McCann, J.F., Lane, I.C., J. Phys. B, 4 (27) 535

17 4p 2 P 5s 2 S Oscillator Strength In laser cooling, the lifetime of the excited state is an important parameter Probability of spontaneous emission described by Einstein coefficient, A2, the inverse of the spontaneous radiative lifetime This is related to the emission oscillator strength, f2, by where γ = e 2 ω 2 /(6πε m e c 3 ) frequency ω2. f 2 = 3 A 2/γ is the classical radiative decay rate at More commonly given as absorption oscillator strength, f2 g f 2 g 2 f 2 We have calculated the absorption oscillator strength for the 4p 2 P-5s 2 S transition in gallium using the general configuration interaction code CIV3. Hibbert, A., Comp. Phys. Commun., 9, 4 (975)

18 4p 2 P 5s 2 S Oscillator Strength Table shows a range of calculations of increasing complexity α contains basic configuration set 4s 2 4p 2 P and 4s 2 5s 2 S each calculation (β to ω) adds to this set calculations γ and δ incorporate 4d orbitals ε adds 6s orbital ω adds 4f orbital to allow for core polarization Our result is in good agreement with experimental data, giving a closer result than recently published relativistic calculations. Author Type f-value Migdalek (976) 3 Rel. SE.3 Migdalek and Baylis (979) 2 Rel. HF.33 Carlsson et al (986) 4 MCHF.9 Safranova et al (26) 5 Rel. MBPT.29 Present work CIV Migdalek, J., Can. J. Phys. 54, 8 (976). Neijzen, J.H.M. & Donszelmann, A., Physica B&C 4, 24 (982) 2. Migdalek, J., & Baylis, W.E., J. Phys. B 2, 6, 2595 (979) 4. Carlsson, J., et al., Z. Phys. D 3, 345 (986) 5. Safranova, U.I., et al., J. Phys. B 39, 749 (26)

19 Present and Future Work Publish CIV3 results Complete testing of numerical method error analysis include terms beyond RWA Apply method to Doppler cooling of gallium I=, I=/2 and I=3/2 Create algorithm to automate calculations

20 Thank You