Atomic physics (2005)
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Exercises for Chapter 9 215 |
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estimate ∂F/∂v and hence to determine the |
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taking into account saturation. (Use the re- |
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damping coe cient α for an atom in a pair of |
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sults of the previous exercise with the modifi- |
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counter-propagating laser beams, under these |
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cation I → 2I in both the numerator and de- |
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conditions. |
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nominator, or otherwise.) Determine the min- |
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(b) |
Estimate the damping time for a sodium atom |
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imum damping time (defined in eqn 9.19) of |
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a rubidium atom in the optical molasses tech- |
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in the optical molasses technique when each |
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nique (with two laser beams). |
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laser beam has intensity Isat and δ = −Γ/2. |
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(b) The force on an atom in an MOT is given by |
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(9.8) |
Laser cooling of a trapped ion |
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eqn 9.30. Assume the worst-case scenario in |
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A trapped Ca+ ion undergoes simple harmonic |
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motion with an oscillation frequency of Ω = 2π × |
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the calculation of the damping and the restor- |
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ing force, along a particular direction, i.e. that |
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100 kHz. |
The ion experiences a radiation force |
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the radiation force arises from two counter- |
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from laser light of wavelength 393 nm and inten- |
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propagating laser beams (each of intensity I) |
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sity I that excites a transition with Γ = 2π |
× |
23 |
× |
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but the saturation of the scattering rate de- |
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6 |
s− |
1 |
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10 |
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. The frequency detuning δ does not de- |
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pend on the ion’s position within the trap. |
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pends on the total intensity 6I of all six laser |
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beams. Show that the damping coe cient can |
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(a) |
Show that the force on the ion has the form |
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be written in the form |
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F = −κ(z − z0) − αv. Describe the ion’s mo- |
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xy |
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tion. |
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α |
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, |
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(b) |
Find the static displacement z0 of the ion from |
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(1 + y + x2)2 |
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the centre of the harmonic potential, along the |
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direction of the laser beam, for δ = −Γ/2 and |
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where x = 2δ/Γ and y = 6I/Isat. Using the |
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I = 2Isat. |
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results of the previous exercise, or otherwise, |
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(c) |
Show |
that, to a good approximation, |
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the |
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determine the nature of the motion for a ru- |
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bidium atom in an MOT with the values of |
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damping coe cient can be written in the form |
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I and δ that give maximum damping, and a |
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xy |
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α |
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field gradient 0.1 T m−1 (in the direction con- |
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, |
(9.59) |
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(1 + y + x2)2 |
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where the variables x and y are proportional |
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sidered). |
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to δ and I, respectively. Maximise this func- |
(9.10) |
Zeeman slowing in a magneto-optical trap |
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tion of two variables and hence determine the |
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intensity and frequency detuning that give the |
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(a) Instead of the optimum magnetic field profile |
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maximum value of α. |
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given in eqn 9.11, a particular apparatus to |
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(d) |
The kinetic energy of small oscillations about |
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slow sodium atoms uses a linear ramp |
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z0 decays with a damping time of τdamp = |
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z |
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M/α. |
Show that this damping time |
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in- |
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versely proportional to the recoil energy.83 |
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B(z) = B0 $1 − |
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& |
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L |
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Evaluate this minimum value of τdamp for a |
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for 0 z L, and B(z) = 0 outside this |
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calcium ion of mass M 40 a.m.u. |
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range. Explain why a suitable value for B0 |
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Comment. This treatment of Doppler cooling for |
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is the same as in eqn 9.12. Show that the |
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a single laser beam is accurate for any intensity |
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minimum value of L is 2L0, where L0 is the |
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(even above Isat), whereas the approximation that |
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stopping distance for the optimum profile. |
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two laser beams (as in the optical molasses tech- |
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(b) The capture of atoms by a magneto-optical |
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nique) give twice as much damping as a single |
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trap can be considered as Zeeman slowing in |
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beam is not accurate at high intensities. |
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a uniform magnetic field gradient, as in part |
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(9.9) |
The properties of a magneto-optical trap |
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(a). In this situation the maximum veloc- |
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(a) |
Obtain an expression for the damping |
co- |
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ity captured by an MOT with laser beams |
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e cient α for an atom in two counter- |
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of radius 0.5 cm is equivalent to the velocity |
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propagating laser beams (each of intensity I), |
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of atoms that come to rest in a distance of |
83Surprisingly, the damping time does not depend on the line width of the transition Γ, but narrow transitions lead to a small velocity capture range.