Splitting Reference

# Fringe spacing formula (87Rb):

(1)
\begin{align} \Delta z = \frac{ h t}{m d} = 4.602 \frac{t [ms]}{d [\mu m]} \end{align}

## Measurement of fringe spacing vs. dressing frequency

trap bottom = 773.0 kHz

## Calculations of fringe spacing vs. detuning, various models

Joseph's calculations, from email 17 March 2009

• horizontal axis is detuning (kHz)
• Quotes from email:
• These calculations are done using the parameters you gave me a few weeks ago (44Hz optical, 1340Hz undressed osc frequ, 787kHz trap boot) and 1e4 atoms. For this set of parameters, the quartic point is -6.3kHz and the cloud is split at +4kHz.
• The method is to do 3d GPE imaginary time propagation to find the in-trap distribution, and then do real-time propagation during time of flight. During TOF, the interaction energy converts to kinetic energy.
• Shown in the top graph is apparent double-well separation in microns (as would be inferred by fringe spacing) versus dressing frequency. There are four lines plotted,
1. "U" = the spacing between the minima in the potential
2. "Guess" = the fringes in the Thomas Fermi trial wavefunction
3. "GPE" = fringes in the wavefunction after GPE relaxation. This is the "real" in trap distribution.
4. "TOF" = fringes after about 1ms TOF. In all cases, less than 3% of the energy was remaining as interaction energy. # How to calculate Rabi frequency of the dressing rf (or, how to do tickle spectroscopy):

• Dress your trap at some frequency, $\nu_{dress}$
• Find the resonance, $\nu_{res}$ (ie, trap bottom) at a number of different rf amplitudes
• Fit this function, as a function of rf current amplitude, to
(2)
\begin{align} \nu_{res} = \nu_{dress} + \sqrt{\Delta^2 + (\Omega(I) * I)^2} \end{align}

where $\Delta$ is the detuning of $\nu_{dress}$ from the undressed trap bottom, and $\Omega(I)$ is the Rabi frequency as a function of current, such that $\Omega_R = \Omega(I)*I$ is the Rabi frequency.

• To extract the magnetic field amplitude,
(3)
\begin{align} B_{rf, \perp} = \frac{4 h}{\mu_B} \Omega_R = 2.9 ~ \rm{G / MHz} * \Omega_R [MHz] \end{align}

(see Book 17, p 143)

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page revision: 16, last edited: 22 Jul 2009 19:07