- #1
Poirot
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Homework Statement
A beam of $$^{39}K$$ atoms is slowed with a Zeeman slower with laser light tuned to the $$4s^2S_{1/2} --4p^2P_{3/2}$$ transition with $$F=2, m_F=2 -> F'=3,m_{F'}=3 (\lambda =767nm)$$ Nuclear spin $$I=3/2$$. The most probably velocity the atoms escape from the oven with is $$v_0 =450m/s$$ and the lifetime of the excited state is $$26nm$$.
The first few parts of the question ask for the stopping distance, maximum possible scattering force, and the number of scattering events taking place in the time it takes for the atom to slow down.
The bit I'm stuck on:
After Zeeman slowing, atoms are now placed in an optical molasses. What is the shortest possible damping time in the absence of heat due to fluctuations if the laser intensity is $$I=0.75W/m^2?$$ (Note $$I/I_{sat} << 1).$$
Homework Equations
$$F_{scatt} = \frac{\hbar k\Gamma}{2} \frac{I/I_{Sat}}{1+ I/I_{Sat} +(\frac{2\delta}{\Gamma})^2}$$
$$F_{molasses} = -2kv \frac{\partial F_{scat}}{\partial \omega}=-\alpha v$$
where
$$\alpha = 4\hbar k^2 \frac{I}{I_{sat}} \frac{-2\delta/\Gamma}{(1+(2\delta/\Gamma)^2)^2}$$
is the damping constant.
the damping time is:
$$\tau = \frac{M}{2\alpha}$$
The Attempt at a Solution
I've completed the first few parts of the question with relative ease. The issues I have are arising from calculating the damping coefficient. I also think that due to the values given in the question (which I haven't had to use most of yet) .
There's a few things I have deduced from my lecture notes such as
$$\delta = \omega -\omega_0 +kv = \frac{\mu}{\hbar} B(z)$$
where ω is laser frequency, ω0 is atomic resonance, k is wavenumber, v is the doppler shift, and the magnetic field B is there to counteract the doppler shift to stay on resonance.
$$B(z)= B_{bias} + B_0(1-\frac{z}{L_0})^{1/2}$$ where
$$ B_{bias} = \frac{\hbar}{\mu}(\omega -\omega_0)$$
which I think can be taken as 0 since it's on resonance so ω = ω0? And with the 1-z/L0, does z=L0 since the molasses comes after we've already Zeeman slowed? But with these assumptions I think this makes the B field zero and so all the preamble of the question is unnecessary so something's probably wrong.
I also need to calculate the initial magnetic field $$B_0 = v_0 \hbar k/\mu$$ where $$\mu=(g_F'm_F' - g_Fm_F)\mu_B$$ can be calculated hyperfine lande factor.
I think all of this takes care of the detuning δ in the equation of the damping coefficient but I'm not sure if I'm barking up the wrong tree entirely.
I also have that $$\tau = \frac{1}{\Gamma}$$ where tau is the lifetime of the state and is given, so this takes care of the gamma in the equation.
I'm now confused as the how I find the saturation intensity? I think this is the only value unknown now (if my assumptions are correct). I haven't calculated anything yet because my knowledge isn't there yet so I fear it would be a waste of time.
Thanks in advance for any help!