ECDL
3.2 Characterization and phase space density
3.2.2 Temperature
Thermometry on a trapped Maxwell-Boltzmann gas is typically performed through the analysis of the spatial extent of the gas in the time following its release from the trapping potential. Assuming the trap is turned off quickly, one can calculate what the size of an atomic cloud at temperatureT will be as a function of time. In principle, one can fit measurements of the cloud size at a series of times and extract not only the temperature but the initial sizes; for a cylindrically symmetric trap there are thus three quantities of interest, which can also be phrased in terms of temperature and the relevant trap frequencies.
We approximate the dipole trap potential (Eq. 3.1.1) as harmonic about the ori- gin; given that the trap rapidly equilibrates to η ≡ U0/T > 10 this approximation is largely valid. It is possible to calculate correction factors for peak density n0 and thus peak phase-space density ρ of a Maxwell-Boltzmann gas at temperature T in a quasi-harmonic situation such as the single-beam trap. Such calculations are detailed elsewhere [133], and while the correction factors for our trap can be as significant as 20% (on the unhelpful side!), we largely assumed this worst-case scenario when searching for BEC. Any thermal density measurements quoted in this thesis use this correction factor of 0.8, which is a constant as long as η is unchanging—a fine ap- proximation as long as free evaporation has ceased and dU/dt is slow enough for rethermalization.
The velocity distribution of a Maxwell-Boltzmann gas is well-known:
f(v) = 1
(2πσ2v)3/2e−v2/2σ2v (3.2.8) where the width of the velocity distribution is the standard thermal velocity:
σv =p
kBT /m (3.2.9)
Conveniently, the density distribution of a Maxwell-Boltzmann gas in a harmonic potential is known to be Gaussian with widths σt and σl corresponding to the two transverse directions and one longitudinal of the dipole trap:
n(r, t= 0) = 1
(2π)3/2σ2tσle−(y2+z2)/2σt2e−x2/2σl2 (3.2.10) Ballistic expansion is achieved simply by turning off the trap on a timescale much quicker than the trap frequencies. The trap turnoff condition is simply dU/dt ωmax2 ; the same condition (inverted) is encountered later as the condition for adiabatic compression or expansion of a condensate.
Obtaining the trap sizes as functions of time σl(t) andσr(t) is simple convolution:
n(r, t) = Z Z Z
∞
n(r−vt)f(v)dv (3.2.11)
This integration yields the following expression, perpetually Gaussian:
n(r, t) = 1
(2π)3/2σt(t)2σl(t)e−(y2+z2)/2σt(t)2e−x2/2σl(t)2 (3.2.12) The time-dependent widths are given by:
σi(t)2 =σi(0)2 +kBT
m t2 (3.2.13)
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What is observed, of course, is the column density along a particular line of sight:
ncol= Z ∞
−∞
n(r)dz (3.2.14)
which results in a Gaussian image with no change in the widths along the remaining di- rections. In practice, we reduce the temperature measurement to a single parameter—
the transverse width of the cloud at timet0 great enough such that the temperature is simply found through the relation σt/t =kBT /m. The σi are found through non- linear least-squares fitting of the absorption images of the expanded cloud to the ideal Gaussian form of Eq. 3.2.12, with t0 being an easily varied experimental parameter.
Conveniently, any small errors incurred through the linear expansion approximation err on the side of higher temperature, which helps avoid self-deception in the search for the BEC transition. It is also worth noting that this approximation can be more quan- titatively phrased as (ωmint0)2 1, since the radii of a Maxwell-Boltzmann gas in a harmonic trap are related to the trap frequencies via the expressionσi2 =p
kBT /mωi2. Two thermal clouds and the best-fit clouds are depicted in Fig. 3.4, and similar im- ages from after the absorption apparatus was stabilized against vibration are shown in Fig. 3.5. In addition to providing temperature information, the fit parameters provide a nice check of atom number N via the total area under the best-fit curve.
Examples of the in-situ clouds are shown in Fig. 3.6; one is shown with the absorp- tion imaging system in focus and one without. The difference between the two is the position of the objective lens with respect to the dipole trap and the emerging
‘shadow’—the imaging lens and the camera itself remain fixed.
It should go without saying that the temperature measurement process described here ceases to have meaning in the presence of a visible condensate fraction, although forcing temperature fits of this sort tend to give lower limits to phase-space density, and can be useful. In such bimodal case one would either use the ‘wings’ alone of the absorption images for the fit, or one would truly account for both the Maxwell- Boltzmann and Thomas-Fermi profiles.
0 5 10 15 20 0
100 200
ballistic delay [ms]
distance [pixels]
Figure 3.3: We finely calibrate the length scale of the absorption images using gravity.
A trapped cloud (nominally at the focus of the imaging system) is released from the trap at t= 0 and falls under the influence of gravity while ballistically expanding. The centroid of the absorption image is calculated as two of the nonlinear fitting parameters described in Fig. 3.4. The distance of this centroid at time t relative to the first measured centroid (∆ = p
δx2+δy2) is plotted above vs. time. Each point in the parabolic fit corresponds to one cloud on the image above; the image itself is the sum of many individual runs, superimposed for clarity. The points are fit to a parabolay=αt2+β; the calibration number is thenκ(pixels/µm)= 4.9×106 sinθ/α, wheresinθrepresents the viewing angle with respect to gravity. In our casesinθis known from the cube geometry to be1/√
3. Uncertainty in this length calibration is obviously not from the fit but rather pointing uncertainty, and possible distortions of the cloud positions as it falls out of focus. Unfortunately, since phase space density scales as the fourth power of κ, this renders it a major source of uncertainty.
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Figure 3.4: Two thermal clouds and the nonlinear fit results yielding values forT andN. The fit includes seven parameters: amplitude, DC offset, two widths, X and Y positions on the image, and angle of rotation in the image plane. The upper image is taken immediately after trap loading, with a ballistic expansion time of 1.5 ms. The relevant parameters are T = 200µK and N = 2.3×106. The lower image is taken after an aggressive evaporation run and 17 ms of ballistic expansion. The cloud is fit to T = 130nK and N = 2×105. The field of view is 1.2×1.4 mm; the trap is oriented along the LL-UR diagonal as in Fig. 3.6, and gravity is directed to the lower right.
Figure 3.5: The nonlinear fitting of thermal clouds with a vibration-stabilized imaging system. Observed temperature is 140µK,; observed number 2×106. The field of view is 1.2
× 1.4 mm.
Figure 3.6: In situ images of the optical trap, showing the difference that focusing makes.
The out-of-focus image at left was taken with the objective lens approximately 2 mm farther away from the optical trap than the in-focus image at right. The field of view is .9 × 1.4 mm, and peak OD (represented by red) is 2.5.
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