Depth, coherence time, and Q factor

We may ask what trade-offs are necessary between trap depth and coherence time. It turns out that, within the exponential approximation (8.6), this is elegantly quantifyable. We can define a `goodness factor'

$$G \equiv \frac{s_{\rm blue}({\bf r_0})- s_{\rm red}({\bf r_... ...\Gamma}{\hbar \vert\Delta\vert} \, U_{\rm max} \tau_{\rm coh},$$ (8.8)

where the trap minimum position ${\bf r_0}$ is at $(x=0, y=y_0)$, and the second equality is verified by substitution of (8.4) and (8.5), and defining $U_{\rm max} \equiv \vert U_{\rm dip}({\bf r_0})\vert$ and $\tau_{\rm coh} \equiv \Gamma_{\rm scatt}^{-1}({\bf r_0})$. We use this latter definition because we are interested in the coherence time of atoms spending time close to the trap minimum (which will certainly be true for the transverse ground state.) Using (8.6) to solve for $y_0$ and evaluate the `goodness factor', it turns out that the factor is independent of either laser power (i.e. of either $A_{\rm red}$ or $A_{\rm blue}$), giving

$$G = \frac{L_{\rm red}-L_{\rm blue}}{L_{\rm red}+L_{\rm blue}} = \frac{\alpha_L}{2 + \alpha_L} \, .$$ (8.9)

Combining (8.8) and (8.9) gives

$$U_{\rm max} \tau_{\rm coh} = \frac{\alpha_L}{2 + \alpha_L} \, \frac{\hbar \vert\Delta\vert}{\Gamma}\, ,$$ (8.10)

fixing the product of achievable depth and coherence time as a constant multiple of the detuning. This is a remarkable result since it shows that increasing $\alpha_L$ is really the only objective in the field design of two-color EW traps. We can write this in units more convenient for cesium trap design, thus,

$$\frac{U_{\rm max}}{\mu {\rm K}} \cdot \frac{\tau_{\rm coh}}{{\rm ms}} = (644.2) \frac{\alpha_L}{2 + \alpha_L} \cdot \frac{\vert\Delta\vert}{{\rm nm}}$$

$$= (122) \cdot \frac{\vert\Delta\vert}{{\rm nm}} \, ,$$ (8.11)

where the value $\alpha_L = 0.47 \pm 0.02$ (taken from best-fit exponentials to the numerically-found squared electric fields for the guide dimensions of Section 8.2.2 with $n_s = 1$) has been substituted to give the the final form. This design expression does not give the $P_{\rm tot}$ required to reach a desired balance between $U_{\rm max}$ and $\tau_{\rm coh}$, however, the total laser power is usually in the mW range, several orders of magnitude less than in most free-space trap designs. For instance, with $P_{\rm tot} = 20$mW, $p=0.4$ and $\Delta = \pm15$nm we could generate a trap of 2mK depth with the relatively short coherence time of 0.9ms. The transverse oscillation frequencies in this trap would be $\omega_x / 2\pi = 116$kHz and $\omega_y / 2\pi = 490$kHz (the field shapes fix this ratio at about 1:4), giving an atomic mode spacing due to the x motion of 5.6$\mu$K, roughly twice the cesium MOT temperature, and a characteristic ground-state size of 26nm by 12nm.

For coherent guiding, we can define a more physically meaningful figure of merit, $Q \equiv \omega_{\bot} \tau_{\rm coh}$, which tells us the typical number of coherent transverse oscillations we can expect multiplied by $2\pi$ (i.e. it is the Q-factor of the transverse oscillations). We should choose $\omega_{\bot} = \omega_x$ since this is the smaller of the transverse frequencies in our case. For $Q \gg 1$ the transverse atomic modes will be well resolved, and our guide can be a useful interferometric device. Using (8.10), in conjunction with the fact that when the trapping potential shape is fixed then $\omega_x$ is proportional to the square root of the depth, tells us that for a given trap and detuning, $Q \propto 1/\omega_x$. For a higher $Q$ we should choose smaller transverse oscillation frequencies, that is, shallower traps. For example, the 2mK trap discussed above has $Q \approx 650$, but if we reduce it to a 20$\mu$K trap of the same $\Delta$ (by changing $p$ or the laser powers), the $Q$ is 10 times larger. Increasing $\Delta$ would allow even higher $Q$ to be realized.

The dependence on detuning in (8.11) is another way of expressing the advantages already known about using far off-resonant beams[2,85]. However, our single-resonance approximation will break down if the detuning is too large: we have (somewhat arbitrarily) chosen a detuning limit of 15nm, as compared to $\Delta_{\rm fs} = 43$nm for cesium. At this limit, the additional dipole potential created due to the detunings from the D1 line is very significant. However, by removing the detuning symmetry (changing $\Delta_{+}$ from +15nm to +12.07nm and $\Delta_{-}$ from -15nm to -17.14nm), the desired D2 single-resonance approximation potential is recovered in the true physical situation of both resonances present. (These required shifts, which are of order $\Delta^2/\Delta_{\rm fs}$, can easily be found using the expression for the sum of dipole potentials from the two lines.) An additional necessity for our limit is the fact that any larger detunings start to demand separate bound-mode calculations for the two colors, a treatment we reserve for the future. This detuning limit in turn limits the depths, coherence times and Q-factors we quote here, but we anticipate similar future EW atom waveguide designs which explore the region $\vert\Delta\vert > \Delta_{\rm fs}$ (or even $\vert\Delta\vert \sim \omega_0$), and achieve much better coherence.